| {"pdf": "old_scans_math/1_pg72.pdf", "page": 1, "id": "1_pg72_equation_00", "type": "math", "max_diffs": 0, "checked": "verified", "url": "https://archive.org/details/in.ernet.dli.2015.222025/page/n71/mode/2up", "math": "\\therefore \\int \\frac{dx}{(a^{2}+x^{2})^{n}}=\\frac{1}{a^{2}} \\frac{x}{(2 n-2)(a^{2}+x^{2})^{n-1}}"} | |
| {"pdf": "old_scans_math/1_pg72.pdf", "page": 1, "id": "1_pg72_equation_01", "type": "math", "max_diffs": 0, "checked": "verified", "url": "https://archive.org/details/in.ernet.dli.2015.222025/page/n71/mode/2up", "math": "-\\frac{1}{a^{2}.(2 n-2)} \\int \\frac{dx}{(a^{2}+x^{2})^{n-1}}+\\frac{1}{a^{2}} \\int \\frac{dx}{(a^{2}+x^{2})^{n-1}}"} | |
| {"pdf": "old_scans_math/1_pg72.pdf", "page": 1, "id": "1_pg72_equation_02", "type": "math", "max_diffs": 0, "checked": "verified", "url": "https://archive.org/details/in.ernet.dli.2015.222025/page/n71/mode/2up", "math": "=\\frac{1}{a^{2}}\\frac{x}{(2 n-2)(a^{2}+x^{2})^{n-1}}+\\frac{2 n-3}{a^{2}(2 n-2)} \\int \\frac{dx}{(a^{2}+x^{2})^{n-1}}"} | |
| {"pdf": "old_scans_math/1_pg72.pdf", "page": 1, "id": "1_pg72_equation_03", "type": "math", "max_diffs": 0, "checked": "verified", "url": "https://archive.org/details/in.ernet.dli.2015.222025/page/n71/mode/2up", "math": "\\int \\frac{x^{m} dx}{(a^{2}+x^{2})^{n}}=\\int x^{m-1} dx \\frac{x}{(a^{2}+x^{2})^{n}}"} | |
| {"pdf": "old_scans_math/1_pg72.pdf", "page": 1, "id": "1_pg72_equation_04", "type": "math", "max_diffs": 0, "checked": "verified", "url": "https://archive.org/details/in.ernet.dli.2015.222025/page/n71/mode/2up", "math": "p=x^{m-1}, dq=x(a^{2}+x^{2})^{-n} dx"} | |
| {"pdf": "old_scans_math/1_pg72.pdf", "page": 1, "id": "1_pg72_equation_05", "type": "math", "max_diffs": 0, "checked": "verified", "url": "https://archive.org/details/in.ernet.dli.2015.222025/page/n71/mode/2up", "math": "\\int \\frac{x^{m} dx}{(a^{2}+x^{2})^{n}}"} | |
| {"pdf": "old_scans_math/1_pg72.pdf", "page": 1, "id": "1_pg72_equation_06", "type": "math", "max_diffs": 0, "checked": "verified", "url": "https://archive.org/details/in.ernet.dli.2015.222025/page/n71/mode/2up", "math": "=\\frac{1}{2-2 n} \\frac{x^{m-1}}{(a^{2}+x^{2})^{n-1}}+\\frac{m-1}{m-2} \\int \\frac{x^{m-2} dx}{(a^{2}+x^{2})^{n-1}}"} | |
| {"pdf": "old_scans_math/1_pg72.pdf", "page": 1, "id": "1_pg72_equation_07", "type": "math", "max_diffs": 0, "checked": "verified", "url": "https://archive.org/details/in.ernet.dli.2015.222025/page/n71/mode/2up", "math": "d u=(a^{2}-x^{2})^{\\frac{n}{2}} dx"} | |
| {"pdf": "old_scans_math/1_pg72.pdf", "page": 1, "id": "1_pg72_equation_08", "type": "math", "max_diffs": 0, "checked": "verified", "url": "https://archive.org/details/in.ernet.dli.2015.222025/page/n71/mode/2up", "math": "(a^{2}-x^{2})^{\\frac{n}{2}}=a^{2}(a^{2}-x^{2})^{\\frac{n-2}{2}}-x^{2}(a^{2}-x^{2})^{\\frac{n-2}{2}}"} | |
| {"pdf": "old_scans_math/1_pg72.pdf", "page": 1, "id": "1_pg72_equation_09", "type": "math", "max_diffs": 0, "checked": "verified", "url": "https://archive.org/details/in.ernet.dli.2015.222025/page/n71/mode/2up", "math": "\\therefore u=a^{2} \\int(a^{2}-x^{2})^{\\frac{n-2}{2}} dx-\\int x . x(a^{2}-x^{2})^{\\frac{n-2}{2}} dx"} | |
| {"pdf": "old_scans_math/1_pg72.pdf", "page": 1, "id": "1_pg72_equation_10", "type": "math", "max_diffs": 0, "checked": "verified", "url": "https://archive.org/details/in.ernet.dli.2015.222025/page/n71/mode/2up", "math": "=a^{2} \\int(a^{2}-x^{2})^{\\frac{n-2}{2}} dx+\\frac{x(a^{2}-x^{2})^{\\frac{n}{2}}}{n}-\\frac{u}{n}"} | |
| {"pdf": "old_scans_math/1_pg72.pdf", "page": 1, "id": "1_pg72_equation_11", "type": "math", "max_diffs": 0, "checked": "verified", "url": "https://archive.org/details/in.ernet.dli.2015.222025/page/n71/mode/2up", "math": "\\therefore u=\\frac{x(a^{2}-x^{2})^{\\frac{n}{2}}}{n+1}+\\frac{n a^{2}}{n+1} \\int(a^{2}-x^{2})^{\\frac{n-2}{2}} dx"} | |
| {"pdf": "old_scans_math/1_pg40.pdf", "page": 1, "id": "1_pg40_equation_02", "type": "math", "max_diffs": 0, "checked": "verified", "url": "https://archive.org/details/in.ernet.dli.2015.222025/page/n39/mode/2up", "math": "\\int \\frac{x^2 dx}{x^4+1}"} | |
| {"pdf": "old_scans_math/1_pg40.pdf", "page": 1, "id": "1_pg40_equation_03", "type": "math", "max_diffs": 0, "checked": "verified", "url": "https://archive.org/details/in.ernet.dli.2015.222025/page/n39/mode/2up", "math": "(x^m+1) = (x^2-2x \\cos \\frac{\\pi}{m}+1)(x^2-2x \\cos \\frac{3\\pi}{m}+1)\\dots"} | |
| {"pdf": "old_scans_math/1_pg40.pdf", "page": 1, "id": "1_pg40_equation_04", "type": "math", "max_diffs": 0, "checked": "verified", "url": "https://archive.org/details/in.ernet.dli.2015.222025/page/n39/mode/2up", "math": "(x^2-2x \\cos \\frac{m-1}{m}\\pi+1)"} | |
| {"pdf": "old_scans_math/1_pg40.pdf", "page": 1, "id": "1_pg40_equation_05", "type": "math", "max_diffs": 0, "checked": "verified", "url": "https://archive.org/details/in.ernet.dli.2015.222025/page/n39/mode/2up", "math": "(x^4+1) = (x^2-2x \\cos \\frac{\\pi}{4}+1)(x^2-2x \\cos \\frac{3\\pi}{4}+1)"} | |
| {"pdf": "old_scans_math/1_pg40.pdf", "page": 1, "id": "1_pg40_equation_06", "type": "math", "max_diffs": 0, "checked": "verified", "url": "https://archive.org/details/in.ernet.dli.2015.222025/page/n39/mode/2up", "math": "(x^4+1) = (x^2-x\\sqrt{2}+1)(x^2+x\\sqrt{2}+1)"} | |
| {"pdf": "old_scans_math/1_pg40.pdf", "page": 1, "id": "1_pg40_equation_07", "type": "math", "max_diffs": 0, "checked": "verified", "url": "https://archive.org/details/in.ernet.dli.2015.222025/page/n39/mode/2up", "math": "\\frac{\\pi}{4} = 45^{\\circ} \\cos \\frac{\\pi}{4} = \\sin \\frac{\\pi}{4} = \\frac{1}{\\sqrt{2}}, \\cos \\frac{3\\pi}{4} = -\\sin \\frac{\\pi}{4}"} | |
| {"pdf": "old_scans_math/1_pg40.pdf", "page": 1, "id": "1_pg40_equation_08", "type": "math", "max_diffs": 0, "checked": "verified", "url": "https://archive.org/details/in.ernet.dli.2015.222025/page/n39/mode/2up", "math": "\\frac{x^2}{x^4+1} = \\frac{Ax+B}{x^2-x\\sqrt{2}+1} + \\frac{Cx+D}{x^2+x\\sqrt{2}+1}"} | |
| {"pdf": "old_scans_math/1_pg40.pdf", "page": 1, "id": "1_pg40_equation_09", "type": "math", "max_diffs": 0, "checked": "verified", "url": "https://archive.org/details/in.ernet.dli.2015.222025/page/n39/mode/2up", "math": "\\therefore x^2= (Ax+B)(x^2+x\\sqrt{2}+1) + (Cx+D)(x^2-x\\sqrt{2}+1)"} | |
| {"pdf": "old_scans_math/1_pg40.pdf", "page": 1, "id": "1_pg40_equation_10", "type": "math", "max_diffs": 0, "checked": "verified", "url": "https://archive.org/details/in.ernet.dli.2015.222025/page/n39/mode/2up", "math": "x^2 + x\\sqrt{2} + 1 = 0, x = \\frac{\\sqrt{-1}-1}{\\sqrt{2}}"} | |
| {"pdf": "old_scans_math/1_pg40.pdf", "page": 1, "id": "1_pg40_equation_11", "type": "math", "max_diffs": 0, "checked": "verified", "url": "https://archive.org/details/in.ernet.dli.2015.222025/page/n39/mode/2up", "math": "x^2 - x\\sqrt{2} + 1 = 0, x = \\frac{\\sqrt{-1}+1}{\\sqrt{2}}"} | |
| {"pdf": "old_scans_math/1_pg10.pdf", "page": 1, "id": "1_pg10_equation_00", "type": "math", "max_diffs": 0, "checked": "verified", "url": "https://archive.org/details/in.ernet.dli.2015.222025/page/n9/mode/2up", "math": "\\int \\frac{dx}{x} = \\log x"} | |
| {"pdf": "old_scans_math/1_pg10.pdf", "page": 1, "id": "1_pg10_equation_01", "type": "math", "max_diffs": 0, "checked": "verified", "url": "https://archive.org/details/in.ernet.dli.2015.222025/page/n9/mode/2up", "math": "\\int \\frac{dx}{a+bx^2} = \\frac{1}{\\sqrt{ab}} \\tan^{-1} (x \\sqrt{\\frac{b}{a}})"} | |
| {"pdf": "old_scans_math/1_pg10.pdf", "page": 1, "id": "1_pg10_equation_02", "type": "math", "max_diffs": 0, "checked": "verified", "url": "https://archive.org/details/in.ernet.dli.2015.222025/page/n9/mode/2up", "math": "= \\frac{1}{a} \\tan^{-1} \\frac{x}{a}"} | |
| {"pdf": "old_scans_math/1_pg10.pdf", "page": 1, "id": "1_pg10_equation_03", "type": "math", "max_diffs": 0, "checked": "verified", "url": "https://archive.org/details/in.ernet.dli.2015.222025/page/n9/mode/2up", "math": "\\int \\frac{dx}{\\sqrt{a^2-x^2}} = \\sin^{-1} \\frac{x}{a}"} | |
| {"pdf": "old_scans_math/1_pg10.pdf", "page": 1, "id": "1_pg10_equation_04", "type": "math", "max_diffs": 0, "checked": "verified", "url": "https://archive.org/details/in.ernet.dli.2015.222025/page/n9/mode/2up", "math": "\\int \\frac{-dx}{\\sqrt{a^2-x^2}} = \\cos^{-1} \\frac{x}{a}"} | |
| {"pdf": "old_scans_math/1_pg10.pdf", "page": 1, "id": "1_pg10_equation_05", "type": "math", "max_diffs": 0, "checked": "verified", "url": "https://archive.org/details/in.ernet.dli.2015.222025/page/n9/mode/2up", "math": "\\int \\frac{dx}{\\sqrt{x^2\\pm a^2}} = \\log \\frac{(x+\\sqrt{x^2\\pm a^2})}{a}"} | |
| {"pdf": "old_scans_math/1_pg10.pdf", "page": 1, "id": "1_pg10_equation_06", "type": "math", "max_diffs": 0, "checked": "verified", "url": "https://archive.org/details/in.ernet.dli.2015.222025/page/n9/mode/2up", "math": "\\int \\frac{dx}{\\sqrt{2ax-x^2}} = \\operatorname{vers}^{-1} \\frac{x}{a}"} | |
| {"pdf": "old_scans_math/1_pg10.pdf", "page": 1, "id": "1_pg10_equation_07", "type": "math", "max_diffs": 0, "checked": "verified", "url": "https://archive.org/details/in.ernet.dli.2015.222025/page/n9/mode/2up", "math": "\\int \\frac{dx}{\\sqrt{x^2\\pm 2ax}} = \\log (x\\pm a + \\sqrt{x^2\\pm 2ax}"} | |
| {"pdf": "old_scans_math/1_pg10.pdf", "page": 1, "id": "1_pg10_equation_08", "type": "math", "max_diffs": 0, "checked": "verified", "url": "https://archive.org/details/in.ernet.dli.2015.222025/page/n9/mode/2up", "math": "\\int \\frac{dx}{x\\sqrt{x^2-a^2}} = \\frac{1}{a} \\sec^{-1} \\frac{x}{a}"} | |
| {"pdf": "old_scans_math/1_pg125.pdf", "page": 1, "id": "1_pg125_equation_00", "type": "math", "max_diffs": 0, "checked": "verified", "url": "https://archive.org/details/in.ernet.dli.2015.222025/page/n123/mode/2up", "math": "\\therefore \\text{semi-area} = \\frac{c (c^2-a^2)}{4 a} sin^{-1} \\frac{\\sqrt{c^2 - a^2}}{\\sqrt{c^2 - a^2}}"} | |
| {"pdf": "old_scans_math/1_pg125.pdf", "page": 1, "id": "1_pg125_equation_01", "type": "math", "max_diffs": 0, "checked": "verified", "url": "https://archive.org/details/in.ernet.dli.2015.222025/page/n123/mode/2up", "math": "= \\frac{c (c^2 - a^2)}{4 a} \\frac{\\pi}{2}"} | |
| {"pdf": "old_scans_math/1_pg125.pdf", "page": 1, "id": "1_pg125_equation_02", "type": "math", "max_diffs": 0, "checked": "verified", "url": "https://archive.org/details/in.ernet.dli.2015.222025/page/n123/mode/2up", "math": "\\text{area circle} = \\frac{a}{2} 2\\pi b = \\pi ab"} | |
| {"pdf": "old_scans_math/1_pg125.pdf", "page": 1, "id": "1_pg125_equation_03", "type": "math", "max_diffs": 0, "checked": "verified", "url": "https://archive.org/details/in.ernet.dli.2015.222025/page/n123/mode/2up", "math": "= \\frac{c (c^2 - a^2)}{4 a} \\pi - \\pi ab"} | |
| {"pdf": "old_scans_math/1_pg125.pdf", "page": 1, "id": "1_pg125_equation_04", "type": "math", "max_diffs": 0, "checked": "verified", "url": "https://archive.org/details/in.ernet.dli.2015.222025/page/n123/mode/2up", "math": "= \\frac{(a +2b) 4b(a + b)}{4 a} \\pi - \\pi ab"} | |
| {"pdf": "old_scans_math/1_pg125.pdf", "page": 1, "id": "1_pg125_equation_05", "type": "math", "max_diffs": 0, "checked": "verified", "url": "https://archive.org/details/in.ernet.dli.2015.222025/page/n123/mode/2up", "math": "= \\frac{\\pi}{a}(a^2b + 3ab + 2b^3 - a^2b)"} | |
| {"pdf": "old_scans_math/1_pg125.pdf", "page": 1, "id": "1_pg125_equation_06", "type": "math", "max_diffs": 0, "checked": "verified", "url": "https://archive.org/details/in.ernet.dli.2015.222025/page/n123/mode/2up", "math": "= \\pi b^2 \\left(3 + \\frac{2 b}{a}\\right)"} | |
| {"pdf": "old_scans_math/1_pg125.pdf", "page": 1, "id": "1_pg125_equation_07", "type": "math", "max_diffs": 0, "checked": "verified", "url": "https://archive.org/details/in.ernet.dli.2015.222025/page/n123/mode/2up", "math": "x^{\\frac{2}{3}} + y^{\\frac{2}{3}} = a^{\\frac{2}{3}}"} | |
| {"pdf": "old_scans_math/1_pg125.pdf", "page": 1, "id": "1_pg125_equation_08", "type": "math", "max_diffs": 0, "checked": "verified", "url": "https://archive.org/details/in.ernet.dli.2015.222025/page/n123/mode/2up", "math": "s = \\int \\sqrt{1 + \\frac{d y^2}{dx^2}} = \\int \\sqrt{1 + \\frac{y^{\\frac{2}{3}}}{x^{\\frac{2}{3}} }}"} | |
| {"pdf": "old_scans_math/1_pg125.pdf", "page": 1, "id": "1_pg125_equation_09", "type": "math", "max_diffs": 0, "checked": "verified", "url": "https://archive.org/details/in.ernet.dli.2015.222025/page/n123/mode/2up", "math": "= \\int \\sqrt{\\frac{x^{\\frac{2}{3}} + y^{\\frac{2}{3}}}{x^{\\frac{1}{3}}}} = \\int \\frac{a^{\\frac{1}{3}}}{x^{\\frac{1}{3}}} = \\frac{3}{2} a^{\\frac{1}{3}} x^{\\frac{2}{3}}"} | |
| {"pdf": "old_scans_math/1_pg125.pdf", "page": 1, "id": "1_pg125_equation_10", "type": "math", "max_diffs": 0, "checked": "verified", "url": "https://archive.org/details/in.ernet.dli.2015.222025/page/n123/mode/2up", "math": "s = \\frac{3}{2} a"} | |
| {"pdf": "old_scans_math/1_pg125.pdf", "page": 1, "id": "1_pg125_equation_11", "type": "math", "max_diffs": 0, "checked": "verified", "url": "https://archive.org/details/in.ernet.dli.2015.222025/page/n123/mode/2up", "math": "4 \\times \\frac{3}{2} a = 6 a"} | |
| {"pdf": "old_scans_math/1_pg19.pdf", "page": 1, "id": "1_pg19_equation_00", "type": "math", "max_diffs": 0, "checked": "verified", "url": "https://archive.org/details/in.ernet.dli.2015.222025/page/n17/mode/2up", "math": "\\frac{x^{2}}{(x^{2}+1)(x^{2}+4)}=\\frac{A}{(x^{2}+1)}+\\frac{B}{(x^{2}+4)}"} | |
| {"pdf": "old_scans_math/1_pg19.pdf", "page": 1, "id": "1_pg19_equation_01", "type": "math", "max_diffs": 0, "checked": "verified", "url": "https://archive.org/details/in.ernet.dli.2015.222025/page/n17/mode/2up", "math": "x^{2}=A(x^{2}+4)+B(x^{2}+1)"} | |
| {"pdf": "old_scans_math/1_pg19.pdf", "page": 1, "id": "1_pg19_equation_02", "type": "math", "max_diffs": 0, "checked": "verified", "url": "https://archive.org/details/in.ernet.dli.2015.222025/page/n17/mode/2up", "math": "x=\\sqrt{-1}"} | |
| {"pdf": "old_scans_math/1_pg19.pdf", "page": 1, "id": "1_pg19_equation_03", "type": "math", "max_diffs": 0, "checked": "verified", "url": "https://archive.org/details/in.ernet.dli.2015.222025/page/n17/mode/2up", "math": "\\therefore -1=3A, \\therefore A=-\\frac{1}{3}"} | |
| {"pdf": "old_scans_math/1_pg19.pdf", "page": 1, "id": "1_pg19_equation_04", "type": "math", "max_diffs": 0, "checked": "verified", "url": "https://archive.org/details/in.ernet.dli.2015.222025/page/n17/mode/2up", "math": "\\therefore x^{2}+\\frac{1}{3}(x^{2}+4)=B(x^{2}+1)"} | |
| {"pdf": "old_scans_math/1_pg19.pdf", "page": 1, "id": "1_pg19_equation_05", "type": "math", "max_diffs": 0, "checked": "verified", "url": "https://archive.org/details/in.ernet.dli.2015.222025/page/n17/mode/2up", "math": "\\therefore \\frac{4(x^{2}+1)}{3}=B(x^{2}+1), \\therefore B=\\frac{4}{3}"} | |
| {"pdf": "old_scans_math/1_pg19.pdf", "page": 1, "id": "1_pg19_equation_06", "type": "math", "max_diffs": 0, "checked": "verified", "url": "https://archive.org/details/in.ernet.dli.2015.222025/page/n17/mode/2up", "math": "\\int \\frac{x^{2}dx}{(x^{2}+1)(x^{2}+4)}=-\\frac{1}{3}\\int\\{\\frac{dx}{(x^{2}+1)}-\\int\\frac{4dx}{(x^{2}+4)}\\}"} | |
| {"pdf": "old_scans_math/1_pg19.pdf", "page": 1, "id": "1_pg19_equation_07", "type": "math", "max_diffs": 0, "checked": "verified", "url": "https://archive.org/details/in.ernet.dli.2015.222025/page/n17/mode/2up", "math": "=\\frac{1}{3}\\int\\{\\frac{dx}{(1+x^{2})}-\\frac{dx}{(1+\\frac{x^{2}}{4})}\\}"} | |
| {"pdf": "old_scans_math/1_pg19.pdf", "page": 1, "id": "1_pg19_equation_08", "type": "math", "max_diffs": 0, "checked": "verified", "url": "https://archive.org/details/in.ernet.dli.2015.222025/page/n17/mode/2up", "math": "=\\frac{1}{3}\\left\\{2\\tan^{-1}\\frac{x}{2}-\\tan^{-1}x\\right\\}"} | |
| {"pdf": "old_scans_math/1_pg19.pdf", "page": 1, "id": "1_pg19_equation_09", "type": "math", "max_diffs": 0, "checked": "verified", "url": "https://archive.org/details/in.ernet.dli.2015.222025/page/n17/mode/2up", "math": "\\int \\frac{x^{2}dx}{(x^{2}+1)(x^{2}+4)}=\\frac{1}{3}\\int\\frac{3x^{2}dx}{(x^{2}+1)(x^{2}+4)}"} | |
| {"pdf": "old_scans_math/1_pg19.pdf", "page": 1, "id": "1_pg19_equation_10", "type": "math", "max_diffs": 0, "checked": "verified", "url": "https://archive.org/details/in.ernet.dli.2015.222025/page/n17/mode/2up", "math": "=\\frac{1}{3}\\int\\frac{\\{4x^{2}+4-(x^{2}+4)\\}dx}{(x^{2}+1)(x^{2}+4)}"} | |
| {"pdf": "old_scans_math/1_pg19.pdf", "page": 1, "id": "1_pg19_equation_11", "type": "math", "max_diffs": 0, "checked": "verified", "url": "https://archive.org/details/in.ernet.dli.2015.222025/page/n17/mode/2up", "math": "=\\frac{1}{3}\\int\\left(\\frac{4dx}{(x^{2}+4)}-\\frac{dx}{x^{2}+1}\\right)"} | |
| {"pdf": "old_scans_math/1_pg19.pdf", "page": 1, "id": "1_pg19_equation_12", "type": "math", "max_diffs": 0, "checked": "verified", "url": "https://archive.org/details/in.ernet.dli.2015.222025/page/n17/mode/2up", "math": "=\\frac{1}{3}\\left(2\\tan^{-1}\\frac{x}{2}-\\tan^{-1}x\\right)"} | |
| {"pdf": "old_scans_math/1_pg19.pdf", "page": 1, "id": "1_pg19_equation_13", "type": "math", "max_diffs": 0, "checked": "verified", "url": "https://archive.org/details/in.ernet.dli.2015.222025/page/n17/mode/2up", "math": "du=\\frac{(3x^{2}+x-2)dx}{(x-1)^{3}(x^{2}+1)}"} | |
| {"pdf": "old_scans_math/1_pg61.pdf", "page": 1, "id": "1_pg61_equation_00", "type": "math", "max_diffs": 0, "checked": "verified", "url": "https://archive.org/details/in.ernet.dli.2015.222025/page/n59/mode/2up", "math": "\\therefore \\int \\frac{dx}{(1 + x)\\sqrt{1 - x - x^2}} = -\\int \\frac{dz}{z^2\\frac{1}{z}\\sqrt{1 + \\frac{1}{z} - \\frac{1}{z^2}}}"} | |
| {"pdf": "old_scans_math/1_pg61.pdf", "page": 1, "id": "1_pg61_equation_01", "type": "math", "max_diffs": 0, "checked": "verified", "url": "https://archive.org/details/in.ernet.dli.2015.222025/page/n59/mode/2up", "math": "=-\\int \\frac{dz}{\\sqrt{z^2 + z - 1}} = -\\int \\frac{dz}{\\sqrt{(z + \\frac{1}{2})^2 - \\frac{5}{4}}}"} | |
| {"pdf": "old_scans_math/1_pg61.pdf", "page": 1, "id": "1_pg61_equation_02", "type": "math", "max_diffs": 0, "checked": "verified", "url": "https://archive.org/details/in.ernet.dli.2015.222025/page/n59/mode/2up", "math": "=-\\log \\{\\sqrt{z^2 + z - 1} + z + \\frac{1}{2}\\}"} | |
| {"pdf": "old_scans_math/1_pg61.pdf", "page": 1, "id": "1_pg61_equation_03", "type": "math", "max_diffs": 0, "checked": "verified", "url": "https://archive.org/details/in.ernet.dli.2015.222025/page/n59/mode/2up", "math": "=-\\log \\{\\frac{\\sqrt{1 - x - x^2}}{1 + x} + \\frac{1}{1 + x} + \\frac{1}{2}\\}"} | |
| {"pdf": "old_scans_math/1_pg61.pdf", "page": 1, "id": "1_pg61_equation_04", "type": "math", "max_diffs": 0, "checked": "verified", "url": "https://archive.org/details/in.ernet.dli.2015.222025/page/n59/mode/2up", "math": "=-\\log \\{\\frac{2\\sqrt{1 - x - x^2} + x + 3}{2(1 + x)}\\}"} | |
| {"pdf": "old_scans_math/1_pg61.pdf", "page": 1, "id": "1_pg61_equation_05", "type": "math", "max_diffs": 0, "checked": "verified", "url": "https://archive.org/details/in.ernet.dli.2015.222025/page/n59/mode/2up", "math": "=\\log \\{\\frac{2(1 + x)}{2\\sqrt{1 - x - x^2} + x + 3}\\}"} | |
| {"pdf": "old_scans_math/1_pg61.pdf", "page": 1, "id": "1_pg61_equation_06", "type": "math", "max_diffs": 0, "checked": "verified", "url": "https://archive.org/details/in.ernet.dli.2015.222025/page/n59/mode/2up", "math": "=\\log \\{\\frac{2(1 + x)(x + 3) - 2\\sqrt{1 - x - x^2}}{x^2 + 6x + 9 - 4 + 4x + 4x^2}\\}"} | |
| {"pdf": "old_scans_math/1_pg61.pdf", "page": 1, "id": "1_pg61_equation_07", "type": "math", "max_diffs": 0, "checked": "verified", "url": "https://archive.org/details/in.ernet.dli.2015.222025/page/n59/mode/2up", "math": "=\\log \\frac{2(1 + x)(x + 3 - 2\\sqrt{1 - x - x^2})}{5 (1 + x)^2}"} | |
| {"pdf": "old_scans_math/1_pg61.pdf", "page": 1, "id": "1_pg61_equation_08", "type": "math", "max_diffs": 0, "checked": "verified", "url": "https://archive.org/details/in.ernet.dli.2015.222025/page/n59/mode/2up", "math": "=\\log \\frac{x + 3 - 2\\sqrt{1 - x - x^2}}{1 + x} + c"} | |
| {"pdf": "old_scans_math/1_pg61.pdf", "page": 1, "id": "1_pg61_equation_09", "type": "math", "max_diffs": 0, "checked": "verified", "url": "https://archive.org/details/in.ernet.dli.2015.222025/page/n59/mode/2up", "math": "\\int \\frac{dx}{\\sqrt{1 + 2x - x^2}} = \\int \\frac{dx}{\\sqrt{2 - (1 - x)^2}}"} | |
| {"pdf": "old_scans_math/1_pg61.pdf", "page": 1, "id": "1_pg61_equation_10", "type": "math", "max_diffs": 0, "checked": "verified", "url": "https://archive.org/details/in.ernet.dli.2015.222025/page/n59/mode/2up", "math": "=-\\int \\frac{d(1 - x)}{\\sqrt{2 - (1 - x)^2}} = \\cos^{-1}\\frac{1 - x}{\\sqrt{2}}"} | |
| {"pdf": "old_scans_math/1_pg63.pdf", "page": 1, "id": "1_pg63_equation_00", "type": "math", "max_diffs": 0, "checked": "verified", "url": "https://archive.org/details/in.ernet.dli.2015.222025/page/n62/mode/2up", "math": "\\int \\frac{\\sqrt{x} dx}{\\sqrt{a^3-x^3}}"} | |
| {"pdf": "old_scans_math/1_pg63.pdf", "page": 1, "id": "1_pg63_equation_01", "type": "math", "max_diffs": 0, "checked": "verified", "url": "https://archive.org/details/in.ernet.dli.2015.222025/page/n62/mode/2up", "math": "x^3 = z^2"} | |
| {"pdf": "old_scans_math/1_pg63.pdf", "page": 1, "id": "1_pg63_equation_02", "type": "math", "max_diffs": 0, "checked": "verified", "url": "https://archive.org/details/in.ernet.dli.2015.222025/page/n62/mode/2up", "math": "x = z^{\\frac{2}{3}}, dx = \\frac{2}{3} z^{-\\frac{1}{3}} dz, \\sqrt{x} = z^{\\frac{1}{3}}"} | |
| {"pdf": "old_scans_math/1_pg63.pdf", "page": 1, "id": "1_pg63_equation_03", "type": "math", "max_diffs": 0, "checked": "verified", "url": "https://archive.org/details/in.ernet.dli.2015.222025/page/n62/mode/2up", "math": "\\therefore \\int \\frac{\\sqrt{x} dx}{\\sqrt{a^3-x^3}} = \\frac{2}{3} \\int \\frac{dz}{\\sqrt{a^3-z^2}} = \\frac{2}{3} \\sin^{-1} \\frac{z}{a^{\\frac{3}{2}}}"} | |
| {"pdf": "old_scans_math/1_pg63.pdf", "page": 1, "id": "1_pg63_equation_04", "type": "math", "max_diffs": 0, "checked": "verified", "url": "https://archive.org/details/in.ernet.dli.2015.222025/page/n62/mode/2up", "math": "= \\frac{2}{3} \\sin^{-1} \\frac{x^{\\frac{3}{2}}}{a^{\\frac{3}{2}}}"} | |
| {"pdf": "old_scans_math/1_pg63.pdf", "page": 1, "id": "1_pg63_equation_05", "type": "math", "max_diffs": 0, "checked": "verified", "url": "https://archive.org/details/in.ernet.dli.2015.222025/page/n62/mode/2up", "math": "\\theta = \\sin^{-1} \\frac{x^{\\frac{3}{2}}}{a^{\\frac{3}{2}}}"} | |
| {"pdf": "old_scans_math/1_pg63.pdf", "page": 1, "id": "1_pg63_equation_06", "type": "math", "max_diffs": 0, "checked": "verified", "url": "https://archive.org/details/in.ernet.dli.2015.222025/page/n62/mode/2up", "math": "\\tan \\theta = \\frac{\\sin \\theta}{\\sqrt{1-\\sin^2 \\theta}} = \\frac{x^{\\frac{3}{2}}}{\\sqrt{a^3-x^3}}"} | |
| {"pdf": "old_scans_math/1_pg63.pdf", "page": 1, "id": "1_pg63_equation_07", "type": "math", "max_diffs": 0, "checked": "verified", "url": "https://archive.org/details/in.ernet.dli.2015.222025/page/n62/mode/2up", "math": "\\therefore \\int \\frac{\\sqrt{x} dx}{\\sqrt{a^3-x^3}} = \\frac{2}{3} \\tan^{-1} \\sqrt{\\frac{x^3}{a^3-x^3}}"} | |
| {"pdf": "old_scans_math/1_pg63.pdf", "page": 1, "id": "1_pg63_equation_08", "type": "math", "max_diffs": 0, "checked": "verified", "url": "https://archive.org/details/in.ernet.dli.2015.222025/page/n62/mode/2up", "math": "\\int \\frac{dx}{(2ax+x^2)^{\\frac{3}{2}}} = \\int \\frac{dx}{((x+a)^2-a^2)^{\\frac{3}{2}}}"} | |
| {"pdf": "old_scans_math/1_pg63.pdf", "page": 1, "id": "1_pg63_equation_09", "type": "math", "max_diffs": 0, "checked": "verified", "url": "https://archive.org/details/in.ernet.dli.2015.222025/page/n62/mode/2up", "math": "x + a = \\frac{1}{z}, dx = -\\frac{dz}{z^2}"} | |
| {"pdf": "old_scans_math/1_pg63.pdf", "page": 1, "id": "1_pg63_equation_10", "type": "math", "max_diffs": 0, "checked": "verified", "url": "https://archive.org/details/in.ernet.dli.2015.222025/page/n62/mode/2up", "math": "=-\\int \\frac{dz}{z^2(\\frac{1}{z^2}-a^2)^{\\frac{3}{2}}} = -\\int \\frac{z dz}{(1-a^2 z^2)^{\\frac{3}{2}}}"} | |
| {"pdf": "old_scans_math/1_pg131.pdf", "page": 1, "id": "1_pg131_equation_00", "type": "math", "max_diffs": 0, "checked": "verified", "url": "https://archive.org/details/in.ernet.dli.2015.222025/page/n129/mode/2up", "math": "\\int \\frac{dx}{\\sqrt{2 a x+x^{2}}}=\\log \\{x+a+\\sqrt{2 a x+x^{2}}\\}"} | |
| {"pdf": "old_scans_math/1_pg131.pdf", "page": 1, "id": "1_pg131_equation_01", "type": "math", "max_diffs": 0, "checked": "verified", "url": "https://archive.org/details/in.ernet.dli.2015.222025/page/n129/mode/2up", "math": "\\therefore \\int \\frac{x dx}{\\sqrt{2 a x+x^{2}}}"} | |
| {"pdf": "old_scans_math/1_pg131.pdf", "page": 1, "id": "1_pg131_equation_02", "type": "math", "max_diffs": 0, "checked": "verified", "url": "https://archive.org/details/in.ernet.dli.2015.222025/page/n129/mode/2up", "math": "=\\sqrt{2 a x+x^{2}}-a \\log \\{a+a+\\sqrt{2 a x+x^{2}}\\}"} | |
| {"pdf": "old_scans_math/1_pg131.pdf", "page": 1, "id": "1_pg131_equation_03", "type": "math", "max_diffs": 0, "checked": "verified", "url": "https://archive.org/details/in.ernet.dli.2015.222025/page/n129/mode/2up", "math": "\\therefore \\int \\frac{x^{2} dx}{\\sqrt{2 a x+x^{2}}}=\\frac{x \\sqrt{2 a x+x^{2}}}{2}-\\frac{3 a}{2} \\sqrt{2 a x+x^{2}}"} | |
| {"pdf": "old_scans_math/1_pg131.pdf", "page": 1, "id": "1_pg131_equation_04", "type": "math", "max_diffs": 0, "checked": "verified", "url": "https://archive.org/details/in.ernet.dli.2015.222025/page/n129/mode/2up", "math": "+\\frac{3 a^{2}}{2} \\log \\{x+a+\\sqrt{2 a x+x^{2}}\\}"} | |
| {"pdf": "old_scans_math/1_pg131.pdf", "page": 1, "id": "1_pg131_equation_05", "type": "math", "max_diffs": 0, "checked": "verified", "url": "https://archive.org/details/in.ernet.dli.2015.222025/page/n129/mode/2up", "math": "\\int_{a}^{0} \\frac{x^{2} dx}{(2 a x-x^{2})^{\\frac{3}{2}}}"} | |
| {"pdf": "old_scans_math/1_pg131.pdf", "page": 1, "id": "1_pg131_equation_06", "type": "math", "max_diffs": 0, "checked": "verified", "url": "https://archive.org/details/in.ernet.dli.2015.222025/page/n129/mode/2up", "math": "d q=\\frac{x dx}{(2 a x-x^{2})^{\\frac{3}{2}}}"} | |
| {"pdf": "old_scans_math/1_pg131.pdf", "page": 1, "id": "1_pg131_equation_07", "type": "math", "max_diffs": 0, "checked": "verified", "url": "https://archive.org/details/in.ernet.dli.2015.222025/page/n129/mode/2up", "math": "\\therefore q=\\int \\frac{x dx}{(2 a x-x)^{\\frac{3}{2}}}=\\int(2 a-x)^{-\\frac{3}{2}} x^{\\frac{1}{2}} dx="} | |
| {"pdf": "old_scans_math/1_pg131.pdf", "page": 1, "id": "1_pg131_equation_08", "type": "math", "max_diffs": 0, "checked": "verified", "url": "https://archive.org/details/in.ernet.dli.2015.222025/page/n129/mode/2up", "math": "\\int (2 a x^{-1}-1)^{-\\frac{3}{2}} x^{-2} dx"} | |
| {"pdf": "old_scans_math/1_pg131.pdf", "page": 1, "id": "1_pg131_equation_09", "type": "math", "max_diffs": 0, "checked": "verified", "url": "https://archive.org/details/in.ernet.dli.2015.222025/page/n129/mode/2up", "math": "=-\\frac{1}{2 a} \\int(2 a x^{-1}-1)^{-\\frac{3}{2}} \\times-2 a x^{-2} dx"} | |
| {"pdf": "old_scans_math/1_pg131.pdf", "page": 1, "id": "1_pg131_equation_10", "type": "math", "max_diffs": 0, "checked": "verified", "url": "https://archive.org/details/in.ernet.dli.2015.222025/page/n129/mode/2up", "math": "=\\frac{(2 a x^{-1}-1)^{-\\frac{1}{2}}}{a}=\\frac{1}{a} \\sqrt{\\frac{x}{2 a-x}}"} | |
| {"pdf": "old_scans_math/1_pg113.pdf", "page": 1, "id": "1_pg113_equation_00", "type": "math", "max_diffs": 0, "checked": "verified", "url": "https://archive.org/details/in.ernet.dli.2015.222025/page/n111/mode/2up", "math": "CD = h"} | |
| {"pdf": "old_scans_math/1_pg113.pdf", "page": 1, "id": "1_pg113_equation_01", "type": "math", "max_diffs": 0, "checked": "verified", "url": "https://archive.org/details/in.ernet.dli.2015.222025/page/n111/mode/2up", "math": "AB = 2c"} | |
| {"pdf": "old_scans_math/1_pg113.pdf", "page": 1, "id": "1_pg113_equation_02", "type": "math", "max_diffs": 0, "checked": "verified", "url": "https://archive.org/details/in.ernet.dli.2015.222025/page/n111/mode/2up", "math": "x = AG"} | |
| {"pdf": "old_scans_math/1_pg113.pdf", "page": 1, "id": "1_pg113_equation_03", "type": "math", "max_diffs": 0, "checked": "verified", "url": "https://archive.org/details/in.ernet.dli.2015.222025/page/n111/mode/2up", "math": "y= FG"} | |
| {"pdf": "old_scans_math/1_pg113.pdf", "page": 1, "id": "1_pg113_equation_04", "type": "math", "max_diffs": 0, "checked": "verified", "url": "https://archive.org/details/in.ernet.dli.2015.222025/page/n111/mode/2up", "math": "AD^{2}: AG.GB :: CD : EG"} | |
| {"pdf": "old_scans_math/1_pg113.pdf", "page": 1, "id": "1_pg113_equation_05", "type": "math", "max_diffs": 0, "checked": "verified", "url": "https://archive.org/details/in.ernet.dli.2015.222025/page/n111/mode/2up", "math": "c^{2}: x (2c - x) :: h: y"} | |
| {"pdf": "old_scans_math/1_pg113.pdf", "page": 1, "id": "1_pg113_equation_06", "type": "math", "max_diffs": 0, "checked": "verified", "url": "https://archive.org/details/in.ernet.dli.2015.222025/page/n111/mode/2up", "math": "y = \\frac{h(2cx - x^{2})}{c^{2}}"} | |
| {"pdf": "old_scans_math/1_pg113.pdf", "page": 1, "id": "1_pg113_equation_07", "type": "math", "max_diffs": 0, "checked": "verified", "url": "https://archive.org/details/in.ernet.dli.2015.222025/page/n111/mode/2up", "math": "y^{2} = \\frac{h^{2}}{c^{4}}(4c^{2}x^{2} - 4 cx^{3} + x^{4})"} | |
| {"pdf": "old_scans_math/1_pg113.pdf", "page": 1, "id": "1_pg113_equation_08", "type": "math", "max_diffs": 0, "checked": "verified", "url": "https://archive.org/details/in.ernet.dli.2015.222025/page/n111/mode/2up", "math": "V = \\pi \\int y^{2}dx"} | |
| {"pdf": "old_scans_math/1_pg113.pdf", "page": 1, "id": "1_pg113_equation_09", "type": "math", "max_diffs": 0, "checked": "verified", "url": "https://archive.org/details/in.ernet.dli.2015.222025/page/n111/mode/2up", "math": "\\pi\\frac{h^{2}}{c^{4}}\\int (4c^{2}x^{2} - 4 cx^{3} + x^{4})dx = \\frac{\\pi h^{2}}{c^{4}}(\\frac{4 c^{2}x^{3}}{3} - cx^{4} + \\frac{x^{5}}{5})"} | |
| {"pdf": "old_scans_math/1_pg113.pdf", "page": 1, "id": "1_pg113_equation_10", "type": "math", "max_diffs": 0, "checked": "verified", "url": "https://archive.org/details/in.ernet.dli.2015.222025/page/n111/mode/2up", "math": "\\frac{\\pi h^{2}}{c^{4}}(\\frac{4 c^{5}}{3} - c^{5} + \\frac{c^{5}}{5}) = \\frac{8}{15} \\pi h^{2} c ="} | |
| {"pdf": "old_scans_math/1_pg113.pdf", "page": 1, "id": "1_pg113_equation_11", "type": "math", "max_diffs": 0, "checked": "verified", "url": "https://archive.org/details/in.ernet.dli.2015.222025/page/n111/mode/2up", "math": "ANP = \\int y dx"} | |
| {"pdf": "old_scans_math/1_pg113.pdf", "page": 1, "id": "1_pg113_equation_12", "type": "math", "max_diffs": 0, "checked": "verified", "url": "https://archive.org/details/in.ernet.dli.2015.222025/page/n111/mode/2up", "math": "= 2 \\sqrt{a} \\int \\sqrt{x} dx = \\frac{4}{3}\\sqrt{a}x^{\\frac{3}{2}}"} | |
| {"pdf": "old_scans_math/1_pg113.pdf", "page": 1, "id": "1_pg113_equation_13", "type": "math", "max_diffs": 0, "checked": "verified", "url": "https://archive.org/details/in.ernet.dli.2015.222025/page/n111/mode/2up", "math": "A N'P' = \\int \\beta d\\alpha = \\frac{2}{3\\sqrt{3} a}\\int (\\alpha - 2a)^{\\frac{3}{2}} d\\alpha"} | |
| {"pdf": "old_scans_math/3_pg199.pdf", "page": 1, "id": "3_pg199_equation_00", "type": "math", "max_diffs": 0, "checked": "verified", "url": "https://archive.org/details/in.ernet.dli.2015.212519/page/n197/mode/2up", "math": "\\phi(x, y) = 0"} | |
| {"pdf": "old_scans_math/3_pg199.pdf", "page": 1, "id": "3_pg199_equation_01", "type": "math", "max_diffs": 0, "checked": "verified", "url": "https://archive.org/details/in.ernet.dli.2015.212519/page/n197/mode/2up", "math": "f_x + \\lambda \\phi_x = 0"} | |
| {"pdf": "old_scans_math/3_pg199.pdf", "page": 1, "id": "3_pg199_equation_02", "type": "math", "max_diffs": 0, "checked": "verified", "url": "https://archive.org/details/in.ernet.dli.2015.212519/page/n197/mode/2up", "math": "f_y + \\lambda \\phi_y = 0"} | |
| {"pdf": "old_scans_math/3_pg199.pdf", "page": 1, "id": "3_pg199_equation_03", "type": "math", "max_diffs": 0, "checked": "verified", "url": "https://archive.org/details/in.ernet.dli.2015.212519/page/n197/mode/2up", "math": "F = f + \\lambda \\phi"} | |
| {"pdf": "old_scans_math/3_pg199.pdf", "page": 1, "id": "3_pg199_equation_04", "type": "math", "max_diffs": 0, "checked": "verified", "url": "https://archive.org/details/in.ernet.dli.2015.212519/page/n197/mode/2up", "math": "\\phi = 0"} | |
| {"pdf": "old_scans_math/3_pg199.pdf", "page": 1, "id": "3_pg199_equation_05", "type": "math", "max_diffs": 0, "checked": "verified", "url": "https://archive.org/details/in.ernet.dli.2015.212519/page/n197/mode/2up", "math": "u = xy"} | |
| {"pdf": "old_scans_math/3_pg199.pdf", "page": 1, "id": "3_pg199_equation_06", "type": "math", "max_diffs": 0, "checked": "verified", "url": "https://archive.org/details/in.ernet.dli.2015.212519/page/n197/mode/2up", "math": "x^2 + y^2 - 1 = 0"} | |
| {"pdf": "old_scans_math/3_pg199.pdf", "page": 1, "id": "3_pg199_equation_07", "type": "math", "max_diffs": 0, "checked": "verified", "url": "https://archive.org/details/in.ernet.dli.2015.212519/page/n197/mode/2up", "math": "y + 2\\lambda x = 0"} | |
| {"pdf": "old_scans_math/3_pg199.pdf", "page": 1, "id": "3_pg199_equation_08", "type": "math", "max_diffs": 0, "checked": "verified", "url": "https://archive.org/details/in.ernet.dli.2015.212519/page/n197/mode/2up", "math": "x + 2\\lambda y = 0"} | |
| {"pdf": "old_scans_math/3_pg199.pdf", "page": 1, "id": "3_pg199_equation_10", "type": "math", "max_diffs": 0, "checked": "verified", "url": "https://archive.org/details/in.ernet.dli.2015.212519/page/n197/mode/2up", "math": "\\xi = \\frac{1}{2} \\sqrt{2}"} | |
| {"pdf": "old_scans_math/3_pg199.pdf", "page": 1, "id": "3_pg199_equation_11", "type": "math", "max_diffs": 0, "checked": "verified", "url": "https://archive.org/details/in.ernet.dli.2015.212519/page/n197/mode/2up", "math": "\\eta = \\frac{1}{2} \\sqrt{2}"} | |
| {"pdf": "old_scans_math/3_pg199.pdf", "page": 1, "id": "3_pg199_equation_12", "type": "math", "max_diffs": 0, "checked": "verified", "url": "https://archive.org/details/in.ernet.dli.2015.212519/page/n197/mode/2up", "math": "\\xi = -\\frac{1}{2} \\sqrt{2}"} | |
| {"pdf": "old_scans_math/3_pg199.pdf", "page": 1, "id": "3_pg199_equation_13", "type": "math", "max_diffs": 0, "checked": "verified", "url": "https://archive.org/details/in.ernet.dli.2015.212519/page/n197/mode/2up", "math": "\\eta = -\\frac{1}{2} \\sqrt{2}"} | |
| {"pdf": "old_scans_math/3_pg199.pdf", "page": 1, "id": "3_pg199_equation_18", "type": "math", "max_diffs": 0, "checked": "verified", "url": "https://archive.org/details/in.ernet.dli.2015.212519/page/n197/mode/2up", "math": "u = \\frac{1}{2}"} | |
| {"pdf": "old_scans_math/3_pg199.pdf", "page": 1, "id": "3_pg199_equation_19", "type": "math", "max_diffs": 0, "checked": "verified", "url": "https://archive.org/details/in.ernet.dli.2015.212519/page/n197/mode/2up", "math": "u = -\\frac{1}{2}"} | |
| {"pdf": "old_scans_math/3_pg634.pdf", "page": 1, "id": "3_pg634_equation_00", "type": "math", "max_diffs": 0, "checked": "verified", "url": "https://archive.org/details/in.ernet.dli.2015.212519/page/n633/mode/2up", "math": "x^{2}=1"} | |
| {"pdf": "old_scans_math/3_pg634.pdf", "page": 1, "id": "3_pg634_equation_01", "type": "math", "max_diffs": 0, "checked": "verified", "url": "https://archive.org/details/in.ernet.dli.2015.212519/page/n633/mode/2up", "math": "\\dot{x} x=0"} | |
| {"pdf": "old_scans_math/3_pg634.pdf", "page": 1, "id": "3_pg634_equation_02", "type": "math", "max_diffs": 0, "checked": "verified", "url": "https://archive.org/details/in.ernet.dli.2015.212519/page/n633/mode/2up", "math": "(y-x) x=0"} | |
| {"pdf": "old_scans_math/3_pg634.pdf", "page": 1, "id": "3_pg634_equation_03", "type": "math", "max_diffs": 0, "checked": "verified", "url": "https://archive.org/details/in.ernet.dli.2015.212519/page/n633/mode/2up", "math": "(y - x) x=1"} | |
| {"pdf": "old_scans_math/3_pg634.pdf", "page": 1, "id": "3_pg634_equation_05", "type": "math", "max_diffs": 0, "checked": "verified", "url": "https://archive.org/details/in.ernet.dli.2015.212519/page/n633/mode/2up", "math": "y-x=\\frac{[x x]}{[x x] x}"} | |
| {"pdf": "old_scans_math/3_pg634.pdf", "page": 1, "id": "3_pg634_equation_06", "type": "math", "max_diffs": 0, "checked": "verified", "url": "https://archive.org/details/in.ernet.dli.2015.212519/page/n633/mode/2up", "math": "\\xi_{1}=\\dot{x}"} | |
| {"pdf": "old_scans_math/3_pg634.pdf", "page": 1, "id": "3_pg634_equation_07", "type": "math", "max_diffs": 0, "checked": "verified", "url": "https://archive.org/details/in.ernet.dli.2015.212519/page/n633/mode/2up", "math": "\\dot{x}^{2}=1"} | |
| {"pdf": "old_scans_math/3_pg634.pdf", "page": 1, "id": "3_pg634_equation_08", "type": "math", "max_diffs": 0, "checked": "verified", "url": "https://archive.org/details/in.ernet.dli.2015.212519/page/n633/mode/2up", "math": "\\xi_{2}=x / h"} | |
| {"pdf": "old_scans_math/3_pg634.pdf", "page": 1, "id": "3_pg634_equation_09", "type": "math", "max_diffs": 0, "checked": "verified", "url": "https://archive.org/details/in.ernet.dli.2015.212519/page/n633/mode/2up", "math": "\\xi_{3}=\\left[\\xi_{1} \\xi_{2}\\right], \\pm \\sqrt{{\\xi_{3}}^{2}}=1 / \\tau"} | |
| {"pdf": "old_scans_math/3_pg634.pdf", "page": 1, "id": "3_pg634_equation_10", "type": "math", "max_diffs": 0, "checked": "verified", "url": "https://archive.org/details/in.ernet.dli.2015.212519/page/n633/mode/2up", "math": "\\xi_{1}=k \\xi_{2}"} | |
| {"pdf": "old_scans_math/3_pg634.pdf", "page": 1, "id": "3_pg634_equation_11", "type": "math", "max_diffs": 0, "checked": "verified", "url": "https://archive.org/details/in.ernet.dli.2015.212519/page/n633/mode/2up", "math": "\\xi_{2}^{2}=1"} | |
| {"pdf": "old_scans_math/3_pg634.pdf", "page": 1, "id": "3_pg634_equation_12", "type": "math", "max_diffs": 0, "checked": "verified", "url": "https://archive.org/details/in.ernet.dli.2015.212519/page/n633/mode/2up", "math": "\\xi_{3}^{2}=1"} | |
| {"pdf": "old_scans_math/3_pg634.pdf", "page": 1, "id": "3_pg634_equation_13", "type": "math", "max_diffs": 0, "checked": "verified", "url": "https://archive.org/details/in.ernet.dli.2015.212519/page/n633/mode/2up", "math": "\\xi_{1} \\xi_{2}=\\xi_{2} \\xi_{3}=\\xi_{3} \\xi_{1}=0"} | |
| {"pdf": "old_scans_math/3_pg634.pdf", "page": 1, "id": "3_pg634_equation_14", "type": "math", "max_diffs": 0, "checked": "verified", "url": "https://archive.org/details/in.ernet.dli.2015.212519/page/n633/mode/2up", "math": "\\xi_{3} \\xi_{1}=-\\xi_{1} \\xi_{3}=0"} | |
| {"pdf": "old_scans_math/3_pg634.pdf", "page": 1, "id": "3_pg634_equation_15", "type": "math", "max_diffs": 0, "checked": "verified", "url": "https://archive.org/details/in.ernet.dli.2015.212519/page/n633/mode/2up", "math": "\\xi_{2} \\xi_{3}=0"} | |
| {"pdf": "old_scans_math/3_pg634.pdf", "page": 1, "id": "3_pg634_equation_16", "type": "math", "max_diffs": 0, "checked": "verified", "url": "https://archive.org/details/in.ernet.dli.2015.212519/page/n633/mode/2up", "math": "\\xi_{3}=\\pm \\sqrt{\\left({{\\xi}_{3}}^{2}\\right)} \\xi_{2}=\\pm {\\xi_{2}}/{\\tau}"} | |
| {"pdf": "old_scans_math/3_pg634.pdf", "page": 1, "id": "3_pg634_equation_17", "type": "math", "max_diffs": 0, "checked": "verified", "url": "https://archive.org/details/in.ernet.dli.2015.212519/page/n633/mode/2up", "math": "\\xi=-{\\xi_{2}}/{\\tau}"} | |
| {"pdf": "old_scans_math/3_pg634.pdf", "page": 1, "id": "3_pg634_equation_18", "type": "math", "max_diffs": 0, "checked": "verified", "url": "https://archive.org/details/in.ernet.dli.2015.212519/page/n633/mode/2up", "math": "\\xi_{2} \\xi_{1}=-\\xi_{1} \\xi_{2}=-k"} | |
| {"pdf": "old_scans_math/3_pg634.pdf", "page": 1, "id": "3_pg634_equation_19", "type": "math", "max_diffs": 0, "checked": "verified", "url": "https://archive.org/details/in.ernet.dli.2015.212519/page/n633/mode/2up", "math": "\\xi_{2} \\xi_{2}=0"} | |
| {"pdf": "old_scans_math/3_pg634.pdf", "page": 1, "id": "3_pg634_equation_20", "type": "math", "max_diffs": 0, "checked": "verified", "url": "https://archive.org/details/in.ernet.dli.2015.212519/page/n633/mode/2up", "math": "\\xi_{2} \\xi_{3}=-\\xi_{3} \\xi_{2}=1 / \\tau"} | |
| {"pdf": "old_scans_math/3_pg634.pdf", "page": 1, "id": "3_pg634_equation_21", "type": "math", "max_diffs": 0, "checked": "verified", "url": "https://archive.org/details/in.ernet.dli.2015.212519/page/n633/mode/2up", "math": "k \\xi_{2}-l^{2} \\xi_{1}+\\frac{k}{\\tau} \\xi_{3}"} | |
| {"pdf": "old_scans_math/3_pg634.pdf", "page": 1, "id": "3_pg634_equation_22", "type": "math", "max_diffs": 0, "checked": "verified", "url": "https://archive.org/details/in.ernet.dli.2015.212519/page/n633/mode/2up", "math": "\\frac{k}{k^{3} \\tau} \\xi_{3}+\\frac{\\xi_{2}}{\\tau}"} | |
| {"pdf": "old_scans_math/3_pg634.pdf", "page": 1, "id": "3_pg634_equation_23", "type": "math", "max_diffs": 0, "checked": "verified", "url": "https://archive.org/details/in.ernet.dli.2015.212519/page/n633/mode/2up", "math": "1 /|\\tau|=\\sqrt{{{\\xi}_{3}}^{2}}=0"} | |
| {"pdf": "old_scans_math/3_pg634.pdf", "page": 1, "id": "3_pg634_equation_24", "type": "math", "max_diffs": 0, "checked": "verified", "url": "https://archive.org/details/in.ernet.dli.2015.212519/page/n633/mode/2up", "math": "x \\eta=\\xi_{1} \\eta"} | |
| {"pdf": "old_scans_math/3_pg634.pdf", "page": 1, "id": "3_pg634_equation_25", "type": "math", "max_diffs": 0, "checked": "verified", "url": "https://archive.org/details/in.ernet.dli.2015.212519/page/n633/mode/2up", "math": "=\\xi_{1} \\xi_{3}=0"} | |
| {"pdf": "old_scans_math/3_pg634.pdf", "page": 1, "id": "3_pg634_equation_26", "type": "math", "max_diffs": 0, "checked": "verified", "url": "https://archive.org/details/in.ernet.dli.2015.212519/page/n633/mode/2up", "math": "x \\eta=\\text { const. }"} | |
| {"pdf": "old_scans_math/3_pg634.pdf", "page": 1, "id": "3_pg634_equation_27", "type": "math", "max_diffs": 0, "checked": "verified", "url": "https://archive.org/details/in.ernet.dli.2015.212519/page/n633/mode/2up", "math": "x=f(t)+s f^{\\prime}(t)"} | |
| {"pdf": "old_scans_math/3_pg634.pdf", "page": 1, "id": "3_pg634_equation_28", "type": "math", "max_diffs": 0, "checked": "verified", "url": "https://archive.org/details/in.ernet.dli.2015.212519/page/n633/mode/2up", "math": "y=g(t)+s g^{\\prime}(t)"} | |
| {"pdf": "old_scans_math/3_pg634.pdf", "page": 1, "id": "3_pg634_equation_29", "type": "math", "max_diffs": 0, "checked": "verified", "url": "https://archive.org/details/in.ernet.dli.2015.212519/page/n633/mode/2up", "math": "z=h(t)+s h^{\\prime}(t)"} | |
| {"pdf": "old_scans_math/3_pg634.pdf", "page": 1, "id": "3_pg634_equation_30", "type": "math", "max_diffs": 0, "checked": "verified", "url": "https://archive.org/details/in.ernet.dli.2015.212519/page/n633/mode/2up", "math": "\\frac{\\partial^{2} z}{\\partial x^{2}}, \\frac{\\partial^{2} z}{\\partial x \\partial y}, \\frac{\\partial^{2} z}{\\partial y^{2}}"} | |
| {"pdf": "old_scans_math/3_pg39.pdf", "page": 1, "id": "3_pg39_equation_00", "type": "math", "max_diffs": 0, "checked": "verified", "url": "https://archive.org/details/in.ernet.dli.2015.212519/page/n37/mode/2up", "math": "x' = ax+by+cz"} | |
| {"pdf": "old_scans_math/3_pg39.pdf", "page": 1, "id": "3_pg39_equation_01", "type": "math", "max_diffs": 0, "checked": "verified", "url": "https://archive.org/details/in.ernet.dli.2015.212519/page/n37/mode/2up", "math": "x' = ax + by"} | |
| {"pdf": "old_scans_math/3_pg39.pdf", "page": 1, "id": "3_pg39_equation_02", "type": "math", "max_diffs": 0, "checked": "verified", "url": "https://archive.org/details/in.ernet.dli.2015.212519/page/n37/mode/2up", "math": "y'= dx + ey + fz"} | |
| {"pdf": "old_scans_math/3_pg39.pdf", "page": 1, "id": "3_pg39_equation_03", "type": "math", "max_diffs": 0, "checked": "verified", "url": "https://archive.org/details/in.ernet.dli.2015.212519/page/n37/mode/2up", "math": "y' = cx + dy"} | |
| {"pdf": "old_scans_math/3_pg39.pdf", "page": 1, "id": "3_pg39_equation_04", "type": "math", "max_diffs": 0, "checked": "verified", "url": "https://archive.org/details/in.ernet.dli.2015.212519/page/n37/mode/2up", "math": "z'=gx+hy+kz"} | |
| {"pdf": "old_scans_math/3_pg39.pdf", "page": 1, "id": "3_pg39_equation_05", "type": "math", "max_diffs": 0, "checked": "verified", "url": "https://archive.org/details/in.ernet.dli.2015.212519/page/n37/mode/2up", "math": "x = a'x' + b'y' + c'z'"} | |
| {"pdf": "old_scans_math/3_pg39.pdf", "page": 1, "id": "3_pg39_equation_06", "type": "math", "max_diffs": 0, "checked": "verified", "url": "https://archive.org/details/in.ernet.dli.2015.212519/page/n37/mode/2up", "math": "x = a'x' + b'y'"} | |
| {"pdf": "old_scans_math/3_pg39.pdf", "page": 1, "id": "3_pg39_equation_07", "type": "math", "max_diffs": 0, "checked": "verified", "url": "https://archive.org/details/in.ernet.dli.2015.212519/page/n37/mode/2up", "math": "y = d'x' + e'y'+ f'z'"} | |
| {"pdf": "old_scans_math/3_pg39.pdf", "page": 1, "id": "3_pg39_equation_08", "type": "math", "max_diffs": 0, "checked": "verified", "url": "https://archive.org/details/in.ernet.dli.2015.212519/page/n37/mode/2up", "math": "y = c'x' + d'y'"} | |
| {"pdf": "old_scans_math/3_pg39.pdf", "page": 1, "id": "3_pg39_equation_09", "type": "math", "max_diffs": 0, "checked": "verified", "url": "https://archive.org/details/in.ernet.dli.2015.212519/page/n37/mode/2up", "math": "z = g'x' + h'y' + k'z'"} | |
| {"pdf": "old_scans_math/3_pg39.pdf", "page": 1, "id": "3_pg39_equation_10", "type": "math", "max_diffs": 0, "checked": "verified", "url": "https://archive.org/details/in.ernet.dli.2015.212519/page/n37/mode/2up", "math": "x = y = z = 0"} | |
| {"pdf": "old_scans_math/3_pg39.pdf", "page": 1, "id": "3_pg39_equation_11", "type": "math", "max_diffs": 0, "checked": "verified", "url": "https://archive.org/details/in.ernet.dli.2015.212519/page/n37/mode/2up", "math": "x' = y' = z' = 0"} | |
| {"pdf": "old_scans_math/3_pg39.pdf", "page": 1, "id": "3_pg39_equation_12", "type": "math", "max_diffs": 0, "checked": "verified", "url": "https://archive.org/details/in.ernet.dli.2015.212519/page/n37/mode/2up", "math": "Ax + By + Cz + D = 0"} | |
| {"pdf": "old_scans_math/3_pg39.pdf", "page": 1, "id": "3_pg39_equation_13", "type": "math", "max_diffs": 0, "checked": "verified", "url": "https://archive.org/details/in.ernet.dli.2015.212519/page/n37/mode/2up", "math": "Ax + By + D = 0"} | |
| {"pdf": "old_scans_math/3_pg39.pdf", "page": 1, "id": "3_pg39_equation_14", "type": "math", "max_diffs": 0, "checked": "verified", "url": "https://archive.org/details/in.ernet.dli.2015.212519/page/n37/mode/2up", "math": "A(a'x' + b'y' + c'z') + B(d'x' + e'y' + f'z')"} | |
| {"pdf": "old_scans_math/3_pg39.pdf", "page": 1, "id": "3_pg39_equation_15", "type": "math", "max_diffs": 0, "checked": "verified", "url": "https://archive.org/details/in.ernet.dli.2015.212519/page/n37/mode/2up", "math": "+ C(g'x' + h'y' + k'x') + D = 0"} | |
| {"pdf": "old_scans_math/3_pg39.pdf", "page": 1, "id": "3_pg39_equation_16", "type": "math", "max_diffs": 0, "checked": "verified", "url": "https://archive.org/details/in.ernet.dli.2015.212519/page/n37/mode/2up", "math": "A(a'x' + b'y') + B(c'x' + d'y') + D = 0"} | |
| {"pdf": "old_scans_math/3_pg39.pdf", "page": 1, "id": "3_pg39_equation_17", "type": "math", "max_diffs": 0, "checked": "verified", "url": "https://archive.org/details/in.ernet.dli.2015.212519/page/n37/mode/2up", "math": "A' = a'A + d'B + g'C"} | |
| {"pdf": "old_scans_math/3_pg39.pdf", "page": 1, "id": "3_pg39_equation_18", "type": "math", "max_diffs": 0, "checked": "verified", "url": "https://archive.org/details/in.ernet.dli.2015.212519/page/n37/mode/2up", "math": "B' = b'A + e'B + h'C"} | |
| {"pdf": "old_scans_math/3_pg39.pdf", "page": 1, "id": "3_pg39_equation_19", "type": "math", "max_diffs": 0, "checked": "verified", "url": "https://archive.org/details/in.ernet.dli.2015.212519/page/n37/mode/2up", "math": "C'=c'A + f'B + k'C"} | |
| {"pdf": "old_scans_math/3_pg39.pdf", "page": 1, "id": "3_pg39_equation_20", "type": "math", "max_diffs": 0, "checked": "verified", "url": "https://archive.org/details/in.ernet.dli.2015.212519/page/n37/mode/2up", "math": "A'=a'A+c'B"} | |
| {"pdf": "old_scans_math/3_pg39.pdf", "page": 1, "id": "3_pg39_equation_21", "type": "math", "max_diffs": 0, "checked": "verified", "url": "https://archive.org/details/in.ernet.dli.2015.212519/page/n37/mode/2up", "math": "B'=b'A+d'B"} | |
| {"pdf": "old_scans_math/3_pg39.pdf", "page": 1, "id": "3_pg39_equation_22", "type": "math", "max_diffs": 0, "checked": "verified", "url": "https://archive.org/details/in.ernet.dli.2015.212519/page/n37/mode/2up", "math": "b'A + e'B+h'C=0"} | |
| {"pdf": "old_scans_math/3_pg39.pdf", "page": 1, "id": "3_pg39_equation_23", "type": "math", "max_diffs": 0, "checked": "verified", "url": "https://archive.org/details/in.ernet.dli.2015.212519/page/n37/mode/2up", "math": "a'A + c'B = 0"} | |
| {"pdf": "old_scans_math/3_pg39.pdf", "page": 1, "id": "3_pg39_equation_24", "type": "math", "max_diffs": 0, "checked": "verified", "url": "https://archive.org/details/in.ernet.dli.2015.212519/page/n37/mode/2up", "math": "b'A + d'B = 0"} | |
| {"pdf": "old_scans_math/3_pg39.pdf", "page": 1, "id": "3_pg39_equation_25", "type": "math", "max_diffs": 0, "checked": "verified", "url": "https://archive.org/details/in.ernet.dli.2015.212519/page/n37/mode/2up", "math": "c'A + f'B + h'C = 0"} | |
| {"pdf": "old_scans_math/3_pg204.pdf", "page": 1, "id": "3_pg204_equation_00", "type": "math", "max_diffs": 0, "checked": "verified", "url": "https://archive.org/details/in.ernet.dli.2015.212519/page/n203/mode/2up", "math": "f_{x} + \\lambda \\phi_{x} + \\mu \\psi_{x} = 0"} | |
| {"pdf": "old_scans_math/3_pg204.pdf", "page": 1, "id": "3_pg204_equation_01", "type": "math", "max_diffs": 0, "checked": "verified", "url": "https://archive.org/details/in.ernet.dli.2015.212519/page/n203/mode/2up", "math": "\\phi_{y} + \\phi_{z} \\frac{\\partial z}{\\partial y} + \\phi_{t} \\frac{\\partial t}{\\partial y} = 0"} | |
| {"pdf": "old_scans_math/3_pg204.pdf", "page": 1, "id": "3_pg204_equation_02", "type": "math", "max_diffs": 0, "checked": "verified", "url": "https://archive.org/details/in.ernet.dli.2015.212519/page/n203/mode/2up", "math": "\\psi_{y} + \\psi_{z} \\frac{\\partial z}{\\partial y} + \\psi_{t} \\frac{\\partial t}{\\partial y} = 0"} | |
| {"pdf": "old_scans_math/3_pg204.pdf", "page": 1, "id": "3_pg204_equation_03", "type": "math", "max_diffs": 0, "checked": "verified", "url": "https://archive.org/details/in.ernet.dli.2015.212519/page/n203/mode/2up", "math": "f_{y} + f{z} \\frac{\\partial z}{\\partial y} + f_{t} \\frac{\\partial t}{\\partial y} = 0"} | |
| {"pdf": "old_scans_math/3_pg204.pdf", "page": 1, "id": "3_pg204_equation_04", "type": "math", "max_diffs": 0, "checked": "verified", "url": "https://archive.org/details/in.ernet.dli.2015.212519/page/n203/mode/2up", "math": "f_{y} + \\lambda \\phi_{y} + \\mu \\psi_{y} = 0"} | |
| {"pdf": "old_scans_math/3_pg204.pdf", "page": 1, "id": "3_pg204_equation_05", "type": "math", "max_diffs": 0, "checked": "verified", "url": "https://archive.org/details/in.ernet.dli.2015.212519/page/n203/mode/2up", "math": "\\phi(x, y, z, t) = 0"} | |
| {"pdf": "old_scans_math/3_pg204.pdf", "page": 1, "id": "3_pg204_equation_06", "type": "math", "max_diffs": 0, "checked": "verified", "url": "https://archive.org/details/in.ernet.dli.2015.212519/page/n203/mode/2up", "math": "\\psi(x, y, z, t) = 0"} | |
| {"pdf": "old_scans_math/3_pg204.pdf", "page": 1, "id": "3_pg204_equation_07", "type": "math", "max_diffs": 0, "checked": "verified", "url": "https://archive.org/details/in.ernet.dli.2015.212519/page/n203/mode/2up", "math": "\\frac{\\partial(\\phi, \\psi)}{\\partial(z, t)}"} | |
| {"pdf": "old_scans_math/3_pg204.pdf", "page": 1, "id": "3_pg204_equation_10", "type": "math", "max_diffs": 0, "checked": "verified", "url": "https://archive.org/details/in.ernet.dli.2015.212519/page/n203/mode/2up", "math": "f_{z} + \\lambda \\phi_{z} + \\mu \\psi_{z} = 0"} | |
| {"pdf": "old_scans_math/3_pg204.pdf", "page": 1, "id": "3_pg204_equation_11", "type": "math", "max_diffs": 0, "checked": "verified", "url": "https://archive.org/details/in.ernet.dli.2015.212519/page/n203/mode/2up", "math": "f_{t} + \\lambda \\phi_{t} + \\mu \\psi_{t} = 0"} | |
| {"pdf": "old_scans_math/3_pg204.pdf", "page": 1, "id": "3_pg204_equation_12", "type": "math", "max_diffs": 0, "checked": "verified", "url": "https://archive.org/details/in.ernet.dli.2015.212519/page/n203/mode/2up", "math": "\\frac{\\partial(\\phi, \\psi)}{\\partial(z, t)} \\neq 0"} | |
| {"pdf": "old_scans_math/3_pg204.pdf", "page": 1, "id": "3_pg204_equation_13", "type": "math", "max_diffs": 0, "checked": "verified", "url": "https://archive.org/details/in.ernet.dli.2015.212519/page/n203/mode/2up", "math": "\\frac{\\partial(\\phi, \\psi)}{\\partial(x, y)}, \\frac{\\partial(\\phi, \\psi)}{\\partial(x, z)}, \\dots, \\frac{\\partial(\\phi, \\psi)}{\\partial(z, t)}"} | |
| {"pdf": "old_scans_math/3_pg101.pdf", "page": 1, "id": "3_pg101_equation_02", "type": "math", "max_diffs": 0, "checked": "verified", "url": "https://archive.org/details/in.ernet.dli.2015.212519/page/n99/mode/2up", "math": "\\text{div} \\mathbf{u} = \\frac{\\partial u_{1}}{\\partial x_{1}} + \\frac{\\partial u_{2}}{\\partial x_{2}} + \\frac{\\partial u_{3}}{\\partial x_{3}}"} | |
| {"pdf": "old_scans_math/3_pg101.pdf", "page": 1, "id": "3_pg101_equation_03", "type": "math", "max_diffs": 0, "checked": "verified", "url": "https://archive.org/details/in.ernet.dli.2015.212519/page/n99/mode/2up", "math": "\\text{div} \\mathbf{u} = \\frac{\\partial \\omega_{1}}{\\partial \\xi_{1}} + \\frac{\\partial \\omega_{2}}{\\partial \\xi_{2}} + \\frac{\\partial \\omega_{3}}{\\partial \\xi_{3}}"} | |
| {"pdf": "old_scans_math/3_pg101.pdf", "page": 1, "id": "3_pg101_equation_04", "type": "math", "max_diffs": 0, "checked": "verified", "url": "https://archive.org/details/in.ernet.dli.2015.212519/page/n99/mode/2up", "math": "\\frac{\\partial u_{1}}{\\partial x_{1}} + \\frac{\\partial u_{2}}{\\partial x_{2}} + \\frac{\\partial u_{3}}{\\partial x_{3}} = \\frac{\\partial \\omega_{1}}{\\partial \\xi_{1}} + \\frac{\\partial \\omega{2}}{\\partial \\xi_{2}} + \\frac{\\partial \\omega_{3}}{\\partial \\xi_{3}}"} | |
| {"pdf": "old_scans_math/3_pg270.pdf", "page": 1, "id": "3_pg270_equation_00", "type": "math", "max_diffs": 0, "checked": "verified", "url": "https://archive.org/details/in.ernet.dli.2015.212519/page/n269/mode/2up", "math": "\\iint_{K_{m}} e^{-x^{2}-y^{2}}dxdy \\leqq \\iint_{R_{v}} e^{-x^{2}-y^{2}}dxdy \\leqq \\iint_{K_{M}} e^{-x^{2}-y^{2}}dxdy"} | |
| {"pdf": "old_scans_math/3_pg270.pdf", "page": 1, "id": "3_pg270_equation_01", "type": "math", "max_diffs": 0, "checked": "verified", "url": "https://archive.org/details/in.ernet.dli.2015.212519/page/n269/mode/2up", "math": "\\iint_{R_{v}} e^{-x^{2}-y^{2}}dxdy=\\int_{-v}^{v}e^{-x^{2}}dx\\int_{-v}^{1}e^{-y^{2}}dy=\\left(\\int_{-v}^{v}e^{-x^{2}}dx\\right)^{2}=\\left(2\\int_{0}^{v}e^{-x^{2}}dx\\right)^{2}"} | |
| {"pdf": "old_scans_math/3_pg270.pdf", "page": 1, "id": "3_pg270_equation_02", "type": "math", "max_diffs": 0, "checked": "verified", "url": "https://archive.org/details/in.ernet.dli.2015.212519/page/n269/mode/2up", "math": "\\left(2\\int_{0}^{\\infty}e^{-x^{2}}dx\\right)^{2}=\\pi"} | |
| {"pdf": "old_scans_math/3_pg270.pdf", "page": 1, "id": "3_pg270_equation_03", "type": "math", "max_diffs": 0, "checked": "verified", "url": "https://archive.org/details/in.ernet.dli.2015.212519/page/n269/mode/2up", "math": "\\int_{0}^{\\infty}e^{-x^{2}}dx = \\frac{1}{2}\\sqrt{\\pi}"} | |
| {"pdf": "old_scans_math/3_pg94.pdf", "page": 1, "id": "3_pg94_equation_00", "type": "math", "max_diffs": 0, "checked": "verified", "url": "https://archive.org/details/in.ernet.dli.2015.212519/page/n93/mode/2up", "math": "\\omega_1 = \\alpha_1u_1 + \\beta_1u_2 + \\gamma_1u_3"} | |
| {"pdf": "old_scans_math/3_pg94.pdf", "page": 1, "id": "3_pg94_equation_01", "type": "math", "max_diffs": 0, "checked": "verified", "url": "https://archive.org/details/in.ernet.dli.2015.212519/page/n93/mode/2up", "math": "\\omega_2 = \\alpha_2u_1 + \\beta_2u_2 + \\gamma_2u_3"} | |
| {"pdf": "old_scans_math/3_pg94.pdf", "page": 1, "id": "3_pg94_equation_02", "type": "math", "max_diffs": 0, "checked": "verified", "url": "https://archive.org/details/in.ernet.dli.2015.212519/page/n93/mode/2up", "math": "\\omega_3 = \\alpha_3u_1 + \\beta_3u_2 + \\gamma_3u_3"} | |
| {"pdf": "old_scans_math/3_pg94.pdf", "page": 1, "id": "3_pg94_equation_03", "type": "math", "max_diffs": 0, "checked": "verified", "url": "https://archive.org/details/in.ernet.dli.2015.212519/page/n93/mode/2up", "math": "u_1 = \\alpha_1\\omega_1 + \\alpha_2\\omega_2 + \\alpha_3\\omega_3"} | |
| {"pdf": "old_scans_math/3_pg94.pdf", "page": 1, "id": "3_pg94_equation_04", "type": "math", "max_diffs": 0, "checked": "verified", "url": "https://archive.org/details/in.ernet.dli.2015.212519/page/n93/mode/2up", "math": "u_2 = \\beta_1\\omega_1 + \\beta_2\\omega_2 + \\beta_3\\omega_3"} | |
| {"pdf": "old_scans_math/3_pg94.pdf", "page": 1, "id": "3_pg94_equation_05", "type": "math", "max_diffs": 0, "checked": "verified", "url": "https://archive.org/details/in.ernet.dli.2015.212519/page/n93/mode/2up", "math": "u_3 = \\gamma_1\\omega_1 + \\gamma_2\\omega_2 + \\gamma_3\\omega_3"} | |
| {"pdf": "old_scans_math/3_pg94.pdf", "page": 1, "id": "3_pg94_equation_06", "type": "math", "max_diffs": 0, "checked": "verified", "url": "https://archive.org/details/in.ernet.dli.2015.212519/page/n93/mode/2up", "math": "u = f(x_1, x_2, x_3)"} | |
| {"pdf": "old_scans_math/3_pg94.pdf", "page": 1, "id": "3_pg94_equation_07", "type": "math", "max_diffs": 0, "checked": "verified", "url": "https://archive.org/details/in.ernet.dli.2015.212519/page/n93/mode/2up", "math": "|u|^2 = u_1^2 + u_2^2 + u_3^2"} | |
| {"pdf": "old_scans_math/3_pg148.pdf", "page": 1, "id": "3_pg148_equation_00", "type": "math", "max_diffs": 0, "checked": "verified", "url": "https://archive.org/details/in.ernet.dli.2015.212519/page/n147/mode/2up", "math": "\\xi = r = \\sqrt{(x^2 + y^2)}"} | |
| {"pdf": "old_scans_math/3_pg148.pdf", "page": 1, "id": "3_pg148_equation_01", "type": "math", "max_diffs": 0, "checked": "verified", "url": "https://archive.org/details/in.ernet.dli.2015.212519/page/n147/mode/2up", "math": "\\eta = \\theta = \\arctan (y/x)"} | |
| {"pdf": "old_scans_math/3_pg148.pdf", "page": 1, "id": "3_pg148_equation_02", "type": "math", "max_diffs": 0, "checked": "verified", "url": "https://archive.org/details/in.ernet.dli.2015.212519/page/n147/mode/2up", "math": "(0\\leq\\theta <2\\pi)"} | |
| {"pdf": "old_scans_math/3_pg148.pdf", "page": 1, "id": "3_pg148_equation_03", "type": "math", "max_diffs": 0, "checked": "verified", "url": "https://archive.org/details/in.ernet.dli.2015.212519/page/n147/mode/2up", "math": "\\xi = \\phi(x, y)"} | |
| {"pdf": "old_scans_math/3_pg148.pdf", "page": 1, "id": "3_pg148_equation_04", "type": "math", "max_diffs": 0, "checked": "verified", "url": "https://archive.org/details/in.ernet.dli.2015.212519/page/n147/mode/2up", "math": "\\eta = \\psi(x, y)"} | |
| {"pdf": "old_scans_math/3_pg148.pdf", "page": 1, "id": "3_pg148_equation_05", "type": "math", "max_diffs": 0, "checked": "verified", "url": "https://archive.org/details/in.ernet.dli.2015.212519/page/n147/mode/2up", "math": "\\phi(x, y) = const"} | |
| {"pdf": "old_scans_math/3_pg148.pdf", "page": 1, "id": "3_pg148_equation_06", "type": "math", "max_diffs": 0, "checked": "verified", "url": "https://archive.org/details/in.ernet.dli.2015.212519/page/n147/mode/2up", "math": "\\psi(x, y) = const"} | |
| {"pdf": "old_scans_math/3_pg148.pdf", "page": 1, "id": "3_pg148_equation_07", "type": "math", "max_diffs": 0, "checked": "verified", "url": "https://archive.org/details/in.ernet.dli.2015.212519/page/n147/mode/2up", "math": "x = g(\\xi, \\eta)"} | |
| {"pdf": "old_scans_math/3_pg148.pdf", "page": 1, "id": "3_pg148_equation_08", "type": "math", "max_diffs": 0, "checked": "verified", "url": "https://archive.org/details/in.ernet.dli.2015.212519/page/n147/mode/2up", "math": "y = h(\\xi, \\eta)"} | |
| {"pdf": "old_scans_math/3_pg251.pdf", "page": 1, "id": "3_pg251_equation_00", "type": "math", "max_diffs": 0, "checked": "verified", "url": "https://archive.org/details/in.ernet.dli.2015.212519/page/n249/mode/2up", "math": "\\lim_{\\delta \\to 0} \\int_{\\psi_{1}(x)}^{\\overline{\\psi_{2}}(x)} f(x, y) dy = \\int_{\\psi_{1}(x)}^{\\psi_{2}(x)} f(x, y) dy"} | |
| {"pdf": "old_scans_math/3_pg251.pdf", "page": 1, "id": "3_pg251_equation_01", "type": "math", "max_diffs": 0, "checked": "verified", "url": "https://archive.org/details/in.ernet.dli.2015.212519/page/n249/mode/2up", "math": "\\lim_{\\delta \\to 0} \\int_{x_{0}'}^{x_{1}'} dx \\int_{\\overline{\\psi_{1}}(x)}^{\\overline{\\psi_{2}}(x)} f(x, y) dx = \\int_{x_{0}}^{x_{1}} dx \\int_{\\psi_{1}(x)}^{\\psi_{2}(x)} f(x, y) dx"} | |
| {"pdf": "old_scans_math/3_pg251.pdf", "page": 1, "id": "3_pg251_equation_02", "type": "math", "max_diffs": 0, "checked": "verified", "url": "https://archive.org/details/in.ernet.dli.2015.212519/page/n249/mode/2up", "math": "\\lim_{\\delta \\to 0} \\iint_{R} f(x, y) dS = \\iint_{R} f(x, y) dS"} | |
| {"pdf": "old_scans_math/3_pg251.pdf", "page": 1, "id": "3_pg251_equation_03", "type": "math", "max_diffs": 0, "checked": "verified", "url": "https://archive.org/details/in.ernet.dli.2015.212519/page/n249/mode/2up", "math": "\\iint_{R} f(x, y) dS = \\int_{x_{0}}^{x_{1}} dx \\int_{\\psi{1}(x)}^{\\psi_{2}(x)} f(x, y) dy"} | |
| {"pdf": "old_scans_math/3_pg251.pdf", "page": 1, "id": "3_pg251_equation_04", "type": "math", "max_diffs": 0, "checked": "verified", "url": "https://archive.org/details/in.ernet.dli.2015.212519/page/n249/mode/2up", "math": "\\iint f(x, y) dy"} | |
| {"pdf": "old_scans_math/2_pg39.pdf", "page": 1, "id": "2_pg39_equation_00", "type": "math", "max_diffs": 0, "checked": "verified", "url": "https://archive.org/details/integralcalculus00philrich/page/n37/mode/2up", "math": "dy = \\frac{(x^2 - 1) d (x^2 + 1) - (x^2 + 1) d (x^2 - 1)}{(x^2 - 1)^2}"} | |
| {"pdf": "old_scans_math/2_pg39.pdf", "page": 1, "id": "2_pg39_equation_01", "type": "math", "max_diffs": 0, "checked": "verified", "url": "https://archive.org/details/integralcalculus00philrich/page/n37/mode/2up", "math": " = \\frac{(x^2 - 1) 2 x dx - (x^2 + 1) 2 x dx}{(x^2 - 1)^2}"} | |
| {"pdf": "old_scans_math/2_pg39.pdf", "page": 1, "id": "2_pg39_equation_02", "type": "math", "max_diffs": 0, "checked": "verified", "url": "https://archive.org/details/integralcalculus00philrich/page/n37/mode/2up", "math": "= -\\frac{4 x dx}{(x^2 - 1)^2}"} | |
| {"pdf": "old_scans_math/2_pg39.pdf", "page": 1, "id": "2_pg39_equation_03", "type": "math", "max_diffs": 0, "checked": "verified", "url": "https://archive.org/details/integralcalculus00philrich/page/n37/mode/2up", "math": "y = \\sqrt{x^2 - 1}"} | |
| {"pdf": "old_scans_math/2_pg39.pdf", "page": 1, "id": "2_pg39_equation_04", "type": "math", "max_diffs": 0, "checked": "verified", "url": "https://archive.org/details/integralcalculus00philrich/page/n37/mode/2up", "math": "u = x^2 - 1"} | |
| {"pdf": "old_scans_math/2_pg39.pdf", "page": 1, "id": "2_pg39_equation_05", "type": "math", "max_diffs": 0, "checked": "verified", "url": "https://archive.org/details/integralcalculus00philrich/page/n37/mode/2up", "math": "\\frac{dy}{dx} = \\frac{d}{dx}(x^2 - 1)^{\\frac{1}{2}} = \\frac{1}{2} (x^2 - 1)^{-\\frac{1}{2}}\\frac{d}{dx} (x^2 - 1)"} | |
| {"pdf": "old_scans_math/2_pg39.pdf", "page": 1, "id": "2_pg39_equation_06", "type": "math", "max_diffs": 0, "checked": "verified", "url": "https://archive.org/details/integralcalculus00philrich/page/n37/mode/2up", "math": "= \\frac{1}{2} (x^2 - 1)^{-\\frac{1}{2}} (2 x) = \\frac{x}{\\sqrt{x^2 - 1}}"} | |
| {"pdf": "old_scans_math/2_pg39.pdf", "page": 1, "id": "2_pg39_equation_07", "type": "math", "max_diffs": 0, "checked": "verified", "url": "https://archive.org/details/integralcalculus00philrich/page/n37/mode/2up", "math": "x^2 + xy - y^2 = 1"} | |
| {"pdf": "old_scans_math/2_pg39.pdf", "page": 1, "id": "2_pg39_equation_08", "type": "math", "max_diffs": 0, "checked": "verified", "url": "https://archive.org/details/integralcalculus00philrich/page/n37/mode/2up", "math": "d (x^2) + d (xy) - d (y^2) = d (1) = 0"} | |
| {"pdf": "old_scans_math/2_pg39.pdf", "page": 1, "id": "2_pg39_equation_09", "type": "math", "max_diffs": 0, "checked": "verified", "url": "https://archive.org/details/integralcalculus00philrich/page/n37/mode/2up", "math": "2 x dx + x dy + y dx - 2 y dy = 0"} | |
| {"pdf": "old_scans_math/2_pg39.pdf", "page": 1, "id": "2_pg39_equation_10", "type": "math", "max_diffs": 0, "checked": "verified", "url": "https://archive.org/details/integralcalculus00philrich/page/n37/mode/2up", "math": "(2 x + y) dx + (x - 2 y) dy = 0"} | |
| {"pdf": "old_scans_math/2_pg39.pdf", "page": 1, "id": "2_pg39_equation_11", "type": "math", "max_diffs": 0, "checked": "verified", "url": "https://archive.org/details/integralcalculus00philrich/page/n37/mode/2up", "math": "\\frac{dy}{dx} = \\frac{2 x + y}{2 y - x}"} | |
| {"pdf": "old_scans_math/2_pg39.pdf", "page": 1, "id": "2_pg39_equation_12", "type": "math", "max_diffs": 0, "checked": "verified", "url": "https://archive.org/details/integralcalculus00philrich/page/n37/mode/2up", "math": "x = t + \\frac{1}{t}"} | |
| {"pdf": "old_scans_math/2_pg39.pdf", "page": 1, "id": "2_pg39_equation_13", "type": "math", "max_diffs": 0, "checked": "verified", "url": "https://archive.org/details/integralcalculus00philrich/page/n37/mode/2up", "math": "dx = dt - \\frac{dt}{t^2}"} | |
| {"pdf": "old_scans_math/2_pg39.pdf", "page": 1, "id": "2_pg39_equation_14", "type": "math", "max_diffs": 0, "checked": "verified", "url": "https://archive.org/details/integralcalculus00philrich/page/n37/mode/2up", "math": "\\frac{dy}{dx} = \\frac{1 + \\frac{1}{t^2}}{1 - \\frac{1}{t^2}} = \\frac{t^2 + 1}{t^2 - 1}"} | |
| {"pdf": "old_scans_math/2_pg39.pdf", "page": 1, "id": "2_pg39_equation_15", "type": "math", "max_diffs": 0, "checked": "verified", "url": "https://archive.org/details/integralcalculus00philrich/page/n37/mode/2up", "math": "y = (\\frac{1 - x}{1 + x})^{\\frac{1}{3}}"} | |
| {"pdf": "old_scans_math/2_pg39.pdf", "page": 1, "id": "2_pg39_equation_16", "type": "math", "max_diffs": 0, "checked": "verified", "url": "https://archive.org/details/integralcalculus00philrich/page/n37/mode/2up", "math": "x = 0.2"} | |
| {"pdf": "old_scans_math/2_pg65.pdf", "page": 1, "id": "2_pg65_equation_00", "type": "math", "max_diffs": 0, "checked": "verified", "url": "https://archive.org/details/integralcalculus00philrich/page/n63/mode/2up", "math": "d cos u = d sin \\left(\\frac{\\pi}{2}-u\\right) = cos \\left(\\frac{\\pi}{2}-u\\right) d\\left(\\frac{\\pi}{2}-u\\right) = - sin u du"} | |
| {"pdf": "old_scans_math/2_pg65.pdf", "page": 1, "id": "2_pg65_equation_01", "type": "math", "max_diffs": 0, "checked": "verified", "url": "https://archive.org/details/integralcalculus00philrich/page/n63/mode/2up", "math": "tan u = \\frac{sin u}{cos u}"} | |
| {"pdf": "old_scans_math/2_pg65.pdf", "page": 1, "id": "2_pg65_equation_02", "type": "math", "max_diffs": 0, "checked": "verified", "url": "https://archive.org/details/integralcalculus00philrich/page/n63/mode/2up", "math": "d tan u = \\frac{cos u d sin u - sin u d cos u}{cos^2 u} = \\frac{cos^2 u du + sin^2 u du}{cos^2 u}"} | |
| {"pdf": "old_scans_math/2_pg65.pdf", "page": 1, "id": "2_pg65_equation_03", "type": "math", "max_diffs": 0, "checked": "verified", "url": "https://archive.org/details/integralcalculus00philrich/page/n63/mode/2up", "math": "= sec^2 u du"} | |
| {"pdf": "old_scans_math/2_pg65.pdf", "page": 1, "id": "2_pg65_equation_04", "type": "math", "max_diffs": 0, "checked": "verified", "url": "https://archive.org/details/integralcalculus00philrich/page/n63/mode/2up", "math": "cot u = \\frac{cos u}{sin u}"} | |
| {"pdf": "old_scans_math/2_pg65.pdf", "page": 1, "id": "2_pg65_equation_05", "type": "math", "max_diffs": 0, "checked": "verified", "url": "https://archive.org/details/integralcalculus00philrich/page/n63/mode/2up", "math": "y = sin^2 (x^2 + 3)"} | |
| {"pdf": "old_scans_math/2_pg65.pdf", "page": 1, "id": "2_pg65_equation_06", "type": "math", "max_diffs": 0, "checked": "verified", "url": "https://archive.org/details/integralcalculus00philrich/page/n63/mode/2up", "math": "sin^2 (x^2 + 3) = [sin (x^2 +3)]^2"} | |
| {"pdf": "old_scans_math/2_pg65.pdf", "page": 1, "id": "2_pg65_equation_07", "type": "math", "max_diffs": 0, "checked": "verified", "url": "https://archive.org/details/integralcalculus00philrich/page/n63/mode/2up", "math": "dy = 2 sin (x^2 + 3) d sin (x^2 + 3)"} | |
| {"pdf": "old_scans_math/2_pg65.pdf", "page": 1, "id": "2_pg65_equation_08", "type": "math", "max_diffs": 0, "checked": "verified", "url": "https://archive.org/details/integralcalculus00philrich/page/n63/mode/2up", "math": "= 2 sin (x^2 + 3) cos (x^2 + 3) d (x^2 + 3)"} | |
| {"pdf": "old_scans_math/2_pg65.pdf", "page": 1, "id": "2_pg65_equation_09", "type": "math", "max_diffs": 0, "checked": "verified", "url": "https://archive.org/details/integralcalculus00philrich/page/n63/mode/2up", "math": "= 4 x sin (x^2 + 3) cos (x^2 + 3) dx"} | |
| {"pdf": "old_scans_math/2_pg65.pdf", "page": 1, "id": "2_pg65_equation_10", "type": "math", "max_diffs": 0, "checked": "verified", "url": "https://archive.org/details/integralcalculus00philrich/page/n63/mode/2up", "math": "y = sec 2 x tan 2 x"} | |
| {"pdf": "old_scans_math/2_pg65.pdf", "page": 1, "id": "2_pg65_equation_11", "type": "math", "max_diffs": 0, "checked": "verified", "url": "https://archive.org/details/integralcalculus00philrich/page/n63/mode/2up", "math": "\\frac{dy}{dx} = sec 2 x \\frac{d}{dx} tan 2 x + tan 2 x \\frac{d}{dx} sec 2 x"} | |
| {"pdf": "old_scans_math/2_pg65.pdf", "page": 1, "id": "2_pg65_equation_12", "type": "math", "max_diffs": 0, "checked": "verified", "url": "https://archive.org/details/integralcalculus00philrich/page/n63/mode/2up", "math": "= sec 2 x sec^2 2 x (2) + tan-2 x sec 2 x tan 2 x (2)"} | |
| {"pdf": "old_scans_math/2_pg65.pdf", "page": 1, "id": "2_pg65_equation_13", "type": "math", "max_diffs": 0, "checked": "verified", "url": "https://archive.org/details/integralcalculus00philrich/page/n63/mode/2up", "math": "= 2 sec 2 x (sec^2 2 x + tan^2 2 x)"} | |
| {"pdf": "old_scans_math/2_pg65.pdf", "page": 1, "id": "2_pg65_equation_14", "type": "math", "max_diffs": 0, "checked": "verified", "url": "https://archive.org/details/integralcalculus00philrich/page/n63/mode/2up", "math": " \\quad \\frac{dy}{dx} = 6 (cos 3 x - sin 2 x)"} | |
| {"pdf": "old_scans_math/2_pg189.pdf", "page": 1, "id": "2_pg189_equation_00", "type": "math", "max_diffs": 0, "checked": "verified", "url": "https://archive.org/details/integralcalculus00philrich/page/n187/mode/2up", "math": "\\frac{dx}{dt} = C_1"} | |
| {"pdf": "old_scans_math/2_pg189.pdf", "page": 1, "id": "2_pg189_equation_01", "type": "math", "max_diffs": 0, "checked": "verified", "url": "https://archive.org/details/integralcalculus00philrich/page/n187/mode/2up", "math": "\\frac{dy}{dt} = -gt + C_2"} | |
| {"pdf": "old_scans_math/2_pg189.pdf", "page": 1, "id": "2_pg189_equation_02", "type": "math", "max_diffs": 0, "checked": "verified", "url": "https://archive.org/details/integralcalculus00philrich/page/n187/mode/2up", "math": "\\frac{dx}{dt}"} | |
| {"pdf": "old_scans_math/2_pg189.pdf", "page": 1, "id": "2_pg189_equation_03", "type": "math", "max_diffs": 0, "checked": "verified", "url": "https://archive.org/details/integralcalculus00philrich/page/n187/mode/2up", "math": "C_2 = v_0 \\sin{\\alpha}"} | |
| {"pdf": "old_scans_math/2_pg189.pdf", "page": 1, "id": "2_pg189_equation_04", "type": "math", "max_diffs": 0, "checked": "verified", "url": "https://archive.org/details/integralcalculus00philrich/page/n187/mode/2up", "math": "C_1=v_0 \\cos{\\alpha}"} | |
| {"pdf": "old_scans_math/2_pg189.pdf", "page": 1, "id": "2_pg189_equation_05", "type": "math", "max_diffs": 0, "checked": "verified", "url": "https://archive.org/details/integralcalculus00philrich/page/n187/mode/2up", "math": "\\frac{dx}{dt} = v_0 \\cos{\\alpha}"} | |
| {"pdf": "old_scans_math/2_pg189.pdf", "page": 1, "id": "2_pg189_equation_06", "type": "math", "max_diffs": 0, "checked": "verified", "url": "https://archive.org/details/integralcalculus00philrich/page/n187/mode/2up", "math": "\\frac{dy}{dt} = v_0 \\sin{\\alpha} - gt"} | |
| {"pdf": "old_scans_math/2_pg189.pdf", "page": 1, "id": "2_pg189_equation_07", "type": "math", "max_diffs": 0, "checked": "verified", "url": "https://archive.org/details/integralcalculus00philrich/page/n187/mode/2up", "math": "x = v_0 t \\cos{\\alpha}"} | |
| {"pdf": "old_scans_math/2_pg189.pdf", "page": 1, "id": "2_pg189_equation_08", "type": "math", "max_diffs": 0, "checked": "verified", "url": "https://archive.org/details/integralcalculus00philrich/page/n187/mode/2up", "math": "y = v_0 t \\sin{\\alpha} - \\frac{1}{2} gt^2"} | |
| {"pdf": "old_scans_math/2_pg189.pdf", "page": 1, "id": "2_pg189_equation_09", "type": "math", "max_diffs": 0, "checked": "verified", "url": "https://archive.org/details/integralcalculus00philrich/page/n187/mode/2up", "math": "\\frac{dy}{dx} = f(x)"} | |
| {"pdf": "old_scans_math/2_pg189.pdf", "page": 1, "id": "2_pg189_equation_10", "type": "math", "max_diffs": 0, "checked": "verified", "url": "https://archive.org/details/integralcalculus00philrich/page/n187/mode/2up", "math": "dy = f(x) dx"} | |
| {"pdf": "old_scans_math/2_pg189.pdf", "page": 1, "id": "2_pg189_equation_11", "type": "math", "max_diffs": 0, "checked": "verified", "url": "https://archive.org/details/integralcalculus00philrich/page/n187/mode/2up", "math": "y = \\int f(x) dx + C"} | |
| {"pdf": "old_scans_math/2_pg349.pdf", "page": 1, "id": "2_pg349_equation_00", "type": "math", "max_diffs": 0, "checked": "verified", "url": "https://archive.org/details/integralcalculus00philrich/page/n347/mode/2up", "math": "x^{2} + y^{2} = 2 ax"} | |
| {"pdf": "old_scans_math/2_pg349.pdf", "page": 1, "id": "2_pg349_equation_01", "type": "math", "max_diffs": 0, "checked": "verified", "url": "https://archive.org/details/integralcalculus00philrich/page/n347/mode/2up", "math": "z^{2} = x^{2} + y^{2}"} | |
| {"pdf": "old_scans_math/2_pg349.pdf", "page": 1, "id": "2_pg349_equation_02", "type": "math", "max_diffs": 0, "checked": "verified", "url": "https://archive.org/details/integralcalculus00philrich/page/n347/mode/2up", "math": "y dx + (x - xy) dy = 0"} | |
| {"pdf": "old_scans_math/2_pg349.pdf", "page": 1, "id": "2_pg349_equation_03", "type": "math", "max_diffs": 0, "checked": "verified", "url": "https://archive.org/details/integralcalculus00philrich/page/n347/mode/2up", "math": "sin x sin y dx + cos x cos y dy = 0"} | |
| {"pdf": "old_scans_math/2_pg349.pdf", "page": 1, "id": "2_pg349_equation_04", "type": "math", "max_diffs": 0, "checked": "verified", "url": "https://archive.org/details/integralcalculus00philrich/page/n347/mode/2up", "math": "(2 xy - y^{2} + 6 x^{2}) dx + (3y^{2} + x^{2} - 2 xy) dy = 0"} | |
| {"pdf": "old_scans_math/2_pg349.pdf", "page": 1, "id": "2_pg349_equation_05", "type": "math", "max_diffs": 0, "checked": "verified", "url": "https://archive.org/details/integralcalculus00philrich/page/n347/mode/2up", "math": "x \\frac{dy}{dx} + y = x^{3}y"} | |
| {"pdf": "old_scans_math/2_pg349.pdf", "page": 1, "id": "2_pg349_equation_06", "type": "math", "max_diffs": 0, "checked": "verified", "url": "https://archive.org/details/integralcalculus00philrich/page/n347/mode/2up", "math": "x \\frac{dy}{dx} + y = cot x"} | |
| {"pdf": "old_scans_math/2_pg349.pdf", "page": 1, "id": "2_pg349_equation_07", "type": "math", "max_diffs": 0, "checked": "verified", "url": "https://archive.org/details/integralcalculus00philrich/page/n347/mode/2up", "math": "x dy - (y + e^{\\frac{1}{x}}) dx = 0"} | |
| {"pdf": "old_scans_math/2_pg349.pdf", "page": 1, "id": "2_pg349_equation_08", "type": "math", "max_diffs": 0, "checked": "verified", "url": "https://archive.org/details/integralcalculus00philrich/page/n347/mode/2up", "math": "(1 + x^{2}) dy + (xy + x) dx = 0"} | |
| {"pdf": "old_scans_math/2_pg349.pdf", "page": 1, "id": "2_pg349_equation_09", "type": "math", "max_diffs": 0, "checked": "verified", "url": "https://archive.org/details/integralcalculus00philrich/page/n347/mode/2up", "math": "x dx + y dy = x dy - y dx"} | |
| {"pdf": "old_scans_math/2_pg349.pdf", "page": 1, "id": "2_pg349_equation_10", "type": "math", "max_diffs": 0, "checked": "verified", "url": "https://archive.org/details/integralcalculus00philrich/page/n347/mode/2up", "math": "(sin x + y) dy + (y cos x - x^{2}) dx = 0"} | |
| {"pdf": "old_scans_math/2_pg349.pdf", "page": 1, "id": "2_pg349_equation_11", "type": "math", "max_diffs": 0, "checked": "verified", "url": "https://archive.org/details/integralcalculus00philrich/page/n347/mode/2up", "math": "y (e^{x} + 2) dx + (e^{x} + 2x) dy = 0"} | |
| {"pdf": "old_scans_math/2_pg349.pdf", "page": 1, "id": "2_pg349_equation_12", "type": "math", "max_diffs": 0, "checked": "verified", "url": "https://archive.org/details/integralcalculus00philrich/page/n347/mode/2up", "math": "(xy^{2} - x) dx + (y + xy) dy = 0"} | |
| {"pdf": "old_scans_math/2_pg349.pdf", "page": 1, "id": "2_pg349_equation_13", "type": "math", "max_diffs": 0, "checked": "verified", "url": "https://archive.org/details/integralcalculus00philrich/page/n347/mode/2up", "math": "(1 + x^{2}) \\frac{dy}{dx} + xy = 2 y"} | |
| {"pdf": "old_scans_math/2_pg349.pdf", "page": 1, "id": "2_pg349_equation_14", "type": "math", "max_diffs": 0, "checked": "verified", "url": "https://archive.org/details/integralcalculus00philrich/page/n347/mode/2up", "math": "x dy - y dx = \\sqrt{x^{2} + y^{2}} dx"} | |
| {"pdf": "old_scans_math/2_pg349.pdf", "page": 1, "id": "2_pg349_equation_15", "type": "math", "max_diffs": 0, "checked": "verified", "url": "https://archive.org/details/integralcalculus00philrich/page/n347/mode/2up", "math": "(x - y) dx + x dy = 0"} | |
| {"pdf": "old_scans_math/2_pg349.pdf", "page": 1, "id": "2_pg349_equation_16", "type": "math", "max_diffs": 0, "checked": "verified", "url": "https://archive.org/details/integralcalculus00philrich/page/n347/mode/2up", "math": "x dy - y dx = x \\sqrt{x^{2} + y^{2}} dx"} | |
| {"pdf": "old_scans_math/2_pg349.pdf", "page": 1, "id": "2_pg349_equation_17", "type": "math", "max_diffs": 0, "checked": "verified", "url": "https://archive.org/details/integralcalculus00philrich/page/n347/mode/2up", "math": "e^{x+y} dy + (1+e^{y}) dx = 0"} | |
| {"pdf": "old_scans_math/2_pg349.pdf", "page": 1, "id": "2_pg349_equation_18", "type": "math", "max_diffs": 0, "checked": "verified", "url": "https://archive.org/details/integralcalculus00philrich/page/n347/mode/2up", "math": "(2x+3y - 1) dx + (4x+6y-5) dy = 0"} | |
| {"pdf": "old_scans_math/2_pg349.pdf", "page": 1, "id": "2_pg349_equation_19", "type": "math", "max_diffs": 0, "checked": "verified", "url": "https://archive.org/details/integralcalculus00philrich/page/n347/mode/2up", "math": "(3y^{2} + 3 xy + x^{2}) dx = (x^{2} + 2 xy) dy"} | |
| {"pdf": "old_scans_math/2_pg349.pdf", "page": 1, "id": "2_pg349_equation_20", "type": "math", "max_diffs": 0, "checked": "verified", "url": "https://archive.org/details/integralcalculus00philrich/page/n347/mode/2up", "math": "(1 + x^{2}) dy + (xy - x^{2}) dx = 0"} | |
| {"pdf": "old_scans_math/2_pg238.pdf", "page": 1, "id": "2_pg238_equation_00", "type": "math", "max_diffs": 0, "checked": "verified", "url": "https://archive.org/details/integralcalculus00philrich/page/n237/mode/2up", "math": "x^{\\frac{1}{2}} + y^{\\frac{1}{2}} = a^{\\frac{1}{2}}"} | |
| {"pdf": "old_scans_math/2_pg238.pdf", "page": 1, "id": "2_pg238_equation_01", "type": "math", "max_diffs": 0, "checked": "verified", "url": "https://archive.org/details/integralcalculus00philrich/page/n237/mode/2up", "math": "y = \\frac{a}{2} (e^{\\frac{x}{a}} + e^{-\\frac{x}{a}})"} | |
| {"pdf": "old_scans_math/2_pg238.pdf", "page": 1, "id": "2_pg238_equation_02", "type": "math", "max_diffs": 0, "checked": "verified", "url": "https://archive.org/details/integralcalculus00philrich/page/n237/mode/2up", "math": "y = sin x"} | |
| {"pdf": "old_scans_math/2_pg238.pdf", "page": 1, "id": "2_pg238_equation_03", "type": "math", "max_diffs": 0, "checked": "verified", "url": "https://archive.org/details/integralcalculus00philrich/page/n237/mode/2up", "math": "y^2 = 4 ax"} | |
| {"pdf": "old_scans_math/2_pg238.pdf", "page": 1, "id": "2_pg238_equation_05", "type": "math", "max_diffs": 0, "checked": "verified", "url": "https://archive.org/details/integralcalculus00philrich/page/n237/mode/2up", "math": "y = -2 a"} | |
| {"pdf": "old_scans_math/2_pg238.pdf", "page": 1, "id": "2_pg238_equation_07", "type": "math", "max_diffs": 0, "checked": "verified", "url": "https://archive.org/details/integralcalculus00philrich/page/n237/mode/2up", "math": "x = a (\\phi - sin \\phi)"} | |
| {"pdf": "old_scans_math/2_pg238.pdf", "page": 1, "id": "2_pg238_equation_08", "type": "math", "max_diffs": 0, "checked": "verified", "url": "https://archive.org/details/integralcalculus00philrich/page/n237/mode/2up", "math": "y = a (1 - cos \\phi)"} | |
| {"pdf": "old_scans_math/2_pg238.pdf", "page": 1, "id": "2_pg238_equation_09", "type": "math", "max_diffs": 0, "checked": "verified", "url": "https://archive.org/details/integralcalculus00philrich/page/n237/mode/2up", "math": "x = a cos^3 \\phi"} | |
| {"pdf": "old_scans_math/2_pg238.pdf", "page": 1, "id": "2_pg238_equation_10", "type": "math", "max_diffs": 0, "checked": "verified", "url": "https://archive.org/details/integralcalculus00philrich/page/n237/mode/2up", "math": "y = a sin^3 \\phi"} | |
| {"pdf": "old_scans_math/2_pg238.pdf", "page": 1, "id": "2_pg238_equation_11", "type": "math", "max_diffs": 0, "checked": "verified", "url": "https://archive.org/details/integralcalculus00philrich/page/n237/mode/2up", "math": "r = a (1 + cos \\theta)"} | |
| {"pdf": "old_scans_math/2_pg238.pdf", "page": 1, "id": "2_pg238_equation_12", "type": "math", "max_diffs": 0, "checked": "verified", "url": "https://archive.org/details/integralcalculus00philrich/page/n237/mode/2up", "math": "a (1 + cos \\theta)"} | |
| {"pdf": "old_scans_math/2_pg238.pdf", "page": 1, "id": "2_pg238_equation_13", "type": "math", "max_diffs": 0, "checked": "verified", "url": "https://archive.org/details/integralcalculus00philrich/page/n237/mode/2up", "math": "x = -\\frac{a}{4}"} | |
| {"pdf": "old_scans_math/2_pg238.pdf", "page": 1, "id": "2_pg238_equation_14", "type": "math", "max_diffs": 0, "checked": "verified", "url": "https://archive.org/details/integralcalculus00philrich/page/n237/mode/2up", "math": "x^2 + xy + y^2 = 3"} | |
| {"pdf": "old_scans_math/2_pg238.pdf", "page": 1, "id": "2_pg238_equation_15", "type": "math", "max_diffs": 0, "checked": "verified", "url": "https://archive.org/details/integralcalculus00philrich/page/n237/mode/2up", "math": "y = x"} | |
| {"pdf": "old_scans_math/2_pg238.pdf", "page": 1, "id": "2_pg238_equation_17", "type": "math", "max_diffs": 0, "checked": "verified", "url": "https://archive.org/details/integralcalculus00philrich/page/n237/mode/2up", "math": "x + y = a"} | |
| {"pdf": "old_scans_math/2_pg238.pdf", "page": 1, "id": "2_pg238_equation_18", "type": "math", "max_diffs": 0, "checked": "verified", "url": "https://archive.org/details/integralcalculus00philrich/page/n237/mode/2up", "math": "PQR \\cdot \\Delta x = X \\Delta x"} | |
| {"pdf": "old_scans_math/4_pg433.pdf", "page": 1, "id": "4_pg433_equation_00", "type": "math", "max_diffs": 0, "checked": "verified", "url": "https://archive.org/details/advancedalgebra00schu/page/416/mode/2up", "math": "\\begin{vmatrix} \na_1 \\pm mb_1 & b_1 & c_1 \\\\ \na_2 \\pm mb_2 & b_2 & c_2 \\\\ \na_3 \\pm mb_3 & b_3 & c_3 \n\\end{vmatrix} = \n\\begin{vmatrix} \na_1 & b_1 & c_1 \\\\ \na_2 & b_2 & c_2 \\\\ \na_3 & b_3 & c_3 \n\\end{vmatrix} \\pm \n\\begin{vmatrix} \nmb_1 & b_1 & c_1 \\\\ \nmb_2 & b_2 & c_2 \\\\ \nmb_3 & b_3 & c_3 \n\\end{vmatrix}\n"} | |
| {"pdf": "old_scans_math/4_pg433.pdf", "page": 1, "id": "4_pg433_equation_01", "type": "math", "max_diffs": 0, "checked": "verified", "url": "https://archive.org/details/advancedalgebra00schu/page/416/mode/2up", "math": "m\\begin{vmatrix} b_1 & b_1 & c_1 \\\\ b_2 & b_2 & c_2 \\\\ b_3 & b_3 & c_3 \\end{vmatrix}"} | |
| {"pdf": "old_scans_math/4_pg433.pdf", "page": 1, "id": "4_pg433_equation_02", "type": "math", "max_diffs": 0, "checked": "verified", "url": "https://archive.org/details/advancedalgebra00schu/page/416/mode/2up", "math": "\\begin{vmatrix} 4244 & 4245 \\\\ 4246 & 4247 \\end{vmatrix}"} | |
| {"pdf": "old_scans_math/4_pg433.pdf", "page": 1, "id": "4_pg433_equation_03", "type": "math", "max_diffs": 0, "checked": "verified", "url": "https://archive.org/details/advancedalgebra00schu/page/416/mode/2up", "math": "\\begin{vmatrix} 4244 & 4245 \\\\ 2 & 2 \\end{vmatrix} = \\begin{vmatrix} 4244 & 1 \\\\ 2 & 0 \\end{vmatrix} = -2"} | |
| {"pdf": "old_scans_math/4_pg98.pdf", "page": 1, "id": "4_pg98_equation_00", "type": "math", "max_diffs": 0, "checked": "verified", "url": "https://archive.org/details/advancedalgebra00schu/page/82/mode/2up", "math": "50 (11-x) + 10 x = 310"} | |
| {"pdf": "old_scans_math/4_pg98.pdf", "page": 1, "id": "4_pg98_equation_01", "type": "math", "max_diffs": 0, "checked": "verified", "url": "https://archive.org/details/advancedalgebra00schu/page/82/mode/2up", "math": "550-50 x + 10 x = 310"} | |
| {"pdf": "old_scans_math/4_pg98.pdf", "page": 1, "id": "4_pg98_equation_02", "type": "math", "max_diffs": 0, "checked": "verified", "url": "https://archive.org/details/advancedalgebra00schu/page/82/mode/2up", "math": "- 50 x + 10 x = - 550 + 310"} | |
| {"pdf": "old_scans_math/4_pg98.pdf", "page": 1, "id": "4_pg98_equation_03", "type": "math", "max_diffs": 0, "checked": "verified", "url": "https://archive.org/details/advancedalgebra00schu/page/82/mode/2up", "math": "- 40 x = - 240"} | |
| {"pdf": "old_scans_math/4_pg98.pdf", "page": 1, "id": "4_pg98_equation_04", "type": "math", "max_diffs": 0, "checked": "verified", "url": "https://archive.org/details/advancedalgebra00schu/page/82/mode/2up", "math": "x = 6"} | |
| {"pdf": "old_scans_math/4_pg98.pdf", "page": 1, "id": "4_pg98_equation_05", "type": "math", "max_diffs": 0, "checked": "verified", "url": "https://archive.org/details/advancedalgebra00schu/page/82/mode/2up", "math": "11-x=5"} | |
| {"pdf": "old_scans_math/4_pg451.pdf", "page": 1, "id": "4_pg451_equation_00", "type": "math", "max_diffs": 0, "checked": "verified", "url": "https://archive.org/details/advancedalgebra00schu/page/435/mode/2up", "math": "x^3-3x^2 - 18 x + 40 = 0"} | |
| {"pdf": "old_scans_math/4_pg451.pdf", "page": 1, "id": "4_pg451_equation_01", "type": "math", "max_diffs": 0, "checked": "verified", "url": "https://archive.org/details/advancedalgebra00schu/page/435/mode/2up", "math": "(x-a) Q(x) + R=f(x)"} | |
| {"pdf": "old_scans_math/4_pg451.pdf", "page": 1, "id": "4_pg451_equation_02", "type": "math", "max_diffs": 0, "checked": "verified", "url": "https://archive.org/details/advancedalgebra00schu/page/435/mode/2up", "math": "R=f(a)"} | |
| {"pdf": "old_scans_math/4_pg451.pdf", "page": 1, "id": "4_pg451_equation_03", "type": "math", "max_diffs": 0, "checked": "verified", "url": "https://archive.org/details/advancedalgebra00schu/page/435/mode/2up", "math": "f(x) = x^4 - 2 x^3 - 9 x^2 + 2"} | |
| {"pdf": "old_scans_math/4_pg451.pdf", "page": 1, "id": "4_pg451_equation_04", "type": "math", "max_diffs": 0, "checked": "verified", "url": "https://archive.org/details/advancedalgebra00schu/page/435/mode/2up", "math": "f(4) = - 14"} | |
| {"pdf": "old_scans_math/4_pg451.pdf", "page": 1, "id": "4_pg451_equation_05", "type": "math", "max_diffs": 0, "checked": "verified", "url": "https://archive.org/details/advancedalgebra00schu/page/435/mode/2up", "math": "x^4 - 2 x^3 - 9 x^2 + 0x + 2"} | |
| {"pdf": "old_scans_math/4_pg451.pdf", "page": 1, "id": "4_pg451_equation_06", "type": "math", "max_diffs": 0, "checked": "verified", "url": "https://archive.org/details/advancedalgebra00schu/page/435/mode/2up", "math": "x^4 =4x^3"} | |
| {"pdf": "old_scans_math/4_pg451.pdf", "page": 1, "id": "4_pg451_equation_07", "type": "math", "max_diffs": 0, "checked": "verified", "url": "https://archive.org/details/advancedalgebra00schu/page/435/mode/2up", "math": "2 x^3 =8x^2"} | |
| {"pdf": "old_scans_math/4_pg451.pdf", "page": 1, "id": "4_pg451_equation_08", "type": "math", "max_diffs": 0, "checked": "verified", "url": "https://archive.org/details/advancedalgebra00schu/page/435/mode/2up", "math": "-x^2=-4x"} | |
| {"pdf": "old_scans_math/4_pg451.pdf", "page": 1, "id": "4_pg451_equation_09", "type": "math", "max_diffs": 0, "checked": "verified", "url": "https://archive.org/details/advancedalgebra00schu/page/435/mode/2up", "math": "-4x=-16"} | |
| {"pdf": "old_scans_math/4_pg48.pdf", "page": 1, "id": "4_pg48_equation_00", "type": "math", "max_diffs": 0, "checked": "verified", "url": "https://archive.org/details/advancedalgebra00schu/page/32/mode/2up", "math": "a+[b-(a - b)]"} | |
| {"pdf": "old_scans_math/4_pg48.pdf", "page": 1, "id": "4_pg48_equation_01", "type": "math", "max_diffs": 0, "checked": "verified", "url": "https://archive.org/details/advancedalgebra00schu/page/32/mode/2up", "math": "a+b -[(b+d) - (a - b)]"} | |
| {"pdf": "old_scans_math/4_pg48.pdf", "page": 1, "id": "4_pg48_equation_02", "type": "math", "max_diffs": 0, "checked": "verified", "url": "https://archive.org/details/advancedalgebra00schu/page/32/mode/2up", "math": "m-(n-p) +[3m-\\overline{3n-6 m}]"} | |
| {"pdf": "old_scans_math/4_pg48.pdf", "page": 1, "id": "4_pg48_equation_03", "type": "math", "max_diffs": 0, "checked": "verified", "url": "https://archive.org/details/advancedalgebra00schu/page/32/mode/2up", "math": "a-[a-\\{a-(-a)\\}]"} | |
| {"pdf": "old_scans_math/4_pg48.pdf", "page": 1, "id": "4_pg48_equation_04", "type": "math", "max_diffs": 0, "checked": "verified", "url": "https://archive.org/details/advancedalgebra00schu/page/32/mode/2up", "math": "x-[3y+\\{3 z - (z - x) + y\\} - 2x]"} | |
| {"pdf": "old_scans_math/4_pg48.pdf", "page": 1, "id": "4_pg48_equation_05", "type": "math", "max_diffs": 0, "checked": "verified", "url": "https://archive.org/details/advancedalgebra00schu/page/32/mode/2up", "math": "-[m-(m+n) - (m-n) - (- m + n)]"} | |
| {"pdf": "old_scans_math/4_pg48.pdf", "page": 1, "id": "4_pg48_equation_06", "type": "math", "max_diffs": 0, "checked": "verified", "url": "https://archive.org/details/advancedalgebra00schu/page/32/mode/2up", "math": "12a -\\{(a+b) - [b - (a - b)] - a\\}"} | |
| {"pdf": "old_scans_math/4_pg48.pdf", "page": 1, "id": "4_pg48_equation_07", "type": "math", "max_diffs": 0, "checked": "verified", "url": "https://archive.org/details/advancedalgebra00schu/page/32/mode/2up", "math": "2 x -\\{x - (x - y) - [x -\\overline{x - y}] - y\\}"} | |
| {"pdf": "old_scans_math/4_pg48.pdf", "page": 1, "id": "4_pg48_equation_08", "type": "math", "max_diffs": 0, "checked": "verified", "url": "https://archive.org/details/advancedalgebra00schu/page/32/mode/2up", "math": "12-2a-\\{- a - [2a - (a-\\overline{7-a})]\\}"} | |
| {"pdf": "old_scans_math/4_pg48.pdf", "page": 1, "id": "4_pg48_equation_09", "type": "math", "max_diffs": 0, "checked": "verified", "url": "https://archive.org/details/advancedalgebra00schu/page/32/mode/2up", "math": "14-3 a -\\{9 a - [10 a - (11 a -\\overline{6 - 6 a})]\\}"} | |
| {"pdf": "old_scans_math/4_pg48.pdf", "page": 1, "id": "4_pg48_equation_10", "type": "math", "max_diffs": 0, "checked": "verified", "url": "https://archive.org/details/advancedalgebra00schu/page/32/mode/2up", "math": "a-[-\\{-(-a)\\}]"} | |
| {"pdf": "old_scans_math/4_pg48.pdf", "page": 1, "id": "4_pg48_equation_11", "type": "math", "max_diffs": 0, "checked": "verified", "url": "https://archive.org/details/advancedalgebra00schu/page/32/mode/2up", "math": "a-3-[-\\{-(-a + \\overline{a + b})\\}]"} | |
| {"pdf": "old_scans_math/4_pg48.pdf", "page": 1, "id": "4_pg48_equation_12", "type": "math", "max_diffs": 0, "checked": "verified", "url": "https://archive.org/details/advancedalgebra00schu/page/32/mode/2up", "math": "x + y -[-(x - y) + \\{ - x + (x -\\overline{x - y}) \\} ]"} | |
| {"pdf": "old_scans_math/4_pg48.pdf", "page": 1, "id": "4_pg48_equation_13", "type": "math", "max_diffs": 0, "checked": "verified", "url": "https://archive.org/details/advancedalgebra00schu/page/32/mode/2up", "math": "1 - \\{ - a -(a+1)-[-a-(a -\\overline{a - 1})]\\}"} | |
| {"pdf": "old_scans_math/4_pg48.pdf", "page": 1, "id": "4_pg48_equation_14", "type": "math", "max_diffs": 0, "checked": "verified", "url": "https://archive.org/details/advancedalgebra00schu/page/32/mode/2up", "math": "a-(-{\\{-[-(-a)]\\}})"} | |
| {"pdf": "old_scans_math/4_pg48.pdf", "page": 1, "id": "4_pg48_equation_15", "type": "math", "max_diffs": 0, "checked": "verified", "url": "https://archive.org/details/advancedalgebra00schu/page/32/mode/2up", "math": "1-(-{\\{a+(-a +1)\\} }) - \\{a-\\overline{a-1}\\}"} | |
| {"pdf": "old_scans_math/4_pg48.pdf", "page": 1, "id": "4_pg48_equation_16", "type": "math", "max_diffs": 0, "checked": "verified", "url": "https://archive.org/details/advancedalgebra00schu/page/32/mode/2up", "math": "6m+\\{4m-[8n- (2m +4 n) - 22 n]-7 n\\}+[9m-(3n+ 4 m) +14 n]"} | |
| {"pdf": "old_scans_math/4_pg48.pdf", "page": 1, "id": "4_pg48_equation_17", "type": "math", "max_diffs": 0, "checked": "verified", "url": "https://archive.org/details/advancedalgebra00schu/page/32/mode/2up", "math": "1247 - [1722-\\{1722+(933 - 1247)\\}]"} | |
| {"pdf": "old_scans_math/4_pg48.pdf", "page": 1, "id": "4_pg48_equation_18", "type": "math", "max_diffs": 0, "checked": "verified", "url": "https://archive.org/details/advancedalgebra00schu/page/32/mode/2up", "math": "a +\\{(4 - b) + (a - 4) -\\overline{a-7}\\}"} | |
| {"pdf": "old_scans_math/4_pg48.pdf", "page": 1, "id": "4_pg48_equation_19", "type": "math", "max_diffs": 0, "checked": "verified", "url": "https://archive.org/details/advancedalgebra00schu/page/32/mode/2up", "math": "a -\\{(6-b)+(6a - 6) - (5a - 7)\\}"} | |
| {"pdf": "old_scans_math/4_pg48.pdf", "page": 1, "id": "4_pg48_equation_20", "type": "math", "max_diffs": 0, "checked": "verified", "url": "https://archive.org/details/advancedalgebra00schu/page/32/mode/2up", "math": "a + \\{a-(b - c)\\}"} | |
| {"pdf": "old_scans_math/4_pg48.pdf", "page": 1, "id": "4_pg48_equation_21", "type": "math", "max_diffs": 0, "checked": "verified", "url": "https://archive.org/details/advancedalgebra00schu/page/32/mode/2up", "math": "- a+[4a-(5 b + c)]"} | |
| {"pdf": "old_scans_math/4_pg48.pdf", "page": 1, "id": "4_pg48_equation_22", "type": "math", "max_diffs": 0, "checked": "verified", "url": "https://archive.org/details/advancedalgebra00schu/page/32/mode/2up", "math": "4a-[6b+(3a-c) - \\{5b-\\overline{c- a}\\}]"} | |
| {"pdf": "old_scans_math/4_pg48.pdf", "page": 1, "id": "4_pg48_equation_23", "type": "math", "max_diffs": 0, "checked": "verified", "url": "https://archive.org/details/advancedalgebra00schu/page/32/mode/2up", "math": "9a-[-7a+\\{56 - (a - b) + \\overline{a -b}\\}]"} | |
| {"pdf": "old_scans_math/4_pg396.pdf", "page": 1, "id": "4_pg396_equation_00", "type": "math", "max_diffs": 0, "checked": "verified", "url": "https://archive.org/details/advancedalgebra00schu/page/380/mode/2up", "math": "OA=a"} | |
| {"pdf": "old_scans_math/4_pg396.pdf", "page": 1, "id": "4_pg396_equation_01", "type": "math", "max_diffs": 0, "checked": "verified", "url": "https://archive.org/details/advancedalgebra00schu/page/380/mode/2up", "math": "a + bi"} | |
| {"pdf": "old_scans_math/4_pg396.pdf", "page": 1, "id": "4_pg396_equation_02", "type": "math", "max_diffs": 0, "checked": "verified", "url": "https://archive.org/details/advancedalgebra00schu/page/380/mode/2up", "math": "-4-2 i"} | |
| {"pdf": "old_scans_math/4_pg396.pdf", "page": 1, "id": "4_pg396_equation_03", "type": "math", "max_diffs": 0, "checked": "verified", "url": "https://archive.org/details/advancedalgebra00schu/page/380/mode/2up", "math": "a + bi = OB"} | |
| {"pdf": "old_scans_math/4_pg396.pdf", "page": 1, "id": "4_pg396_equation_04", "type": "math", "max_diffs": 0, "checked": "verified", "url": "https://archive.org/details/advancedalgebra00schu/page/380/mode/2up", "math": "4 - 2 i = \\sqrt{4^2 + 2^2} = 2\\sqrt{5}"} | |
| {"pdf": "old_scans_math/4_pg396.pdf", "page": 1, "id": "4_pg396_equation_05", "type": "math", "max_diffs": 0, "checked": "verified", "url": "https://archive.org/details/advancedalgebra00schu/page/380/mode/2up", "math": "OA = OB = OC = OD = 1"} | |
| {"pdf": "old_scans_math/4_pg396.pdf", "page": 1, "id": "4_pg396_equation_06", "type": "math", "max_diffs": 0, "checked": "verified", "url": "https://archive.org/details/advancedalgebra00schu/page/380/mode/2up", "math": "\\angle AOB = \\angle BOC = \\angle COD = 60\\degree"} | |
| {"pdf": "old_scans_math/4_pg396.pdf", "page": 1, "id": "4_pg396_equation_07", "type": "math", "max_diffs": 0, "checked": "verified", "url": "https://archive.org/details/advancedalgebra00schu/page/380/mode/2up", "math": "OB = x"} | |
| {"pdf": "old_scans_math/4_pg396.pdf", "page": 1, "id": "4_pg396_equation_08", "type": "math", "max_diffs": 0, "checked": "verified", "url": "https://archive.org/details/advancedalgebra00schu/page/380/mode/2up", "math": "x^3 = - 1"} | |
| {"pdf": "old_scans_math/4_pg396.pdf", "page": 1, "id": "4_pg396_equation_09", "type": "math", "max_diffs": 0, "checked": "verified", "url": "https://archive.org/details/advancedalgebra00schu/page/380/mode/2up", "math": "x=\\sqrt[3]{-1}"} | |
| {"pdf": "old_scans_math/4_pg396.pdf", "page": 1, "id": "4_pg396_equation_10", "type": "math", "max_diffs": 0, "checked": "verified", "url": "https://archive.org/details/advancedalgebra00schu/page/380/mode/2up", "math": "\\sqrt[3]{- 1} = \\frac{1}{2} + \\frac{1}{2}\\sqrt{3} \\cdot i"} | |
| {"pdf": "old_scans_math/4_pg186.pdf", "page": 1, "id": "4_pg186_equation_00", "type": "math", "max_diffs": 0, "checked": "verified", "url": "https://archive.org/details/advancedalgebra00schu/page/170/mode/2up", "math": "a:b=c:d"} | |
| {"pdf": "old_scans_math/4_pg186.pdf", "page": 1, "id": "4_pg186_equation_01", "type": "math", "max_diffs": 0, "checked": "verified", "url": "https://archive.org/details/advancedalgebra00schu/page/170/mode/2up", "math": "\\frac{a}{b} = \\frac{c}{d}"} | |
| {"pdf": "old_scans_math/4_pg186.pdf", "page": 1, "id": "4_pg186_equation_02", "type": "math", "max_diffs": 0, "checked": "verified", "url": "https://archive.org/details/advancedalgebra00schu/page/170/mode/2up", "math": "ad = bc"} | |
| {"pdf": "old_scans_math/4_pg186.pdf", "page": 1, "id": "4_pg186_equation_03", "type": "math", "max_diffs": 0, "checked": "verified", "url": "https://archive.org/details/advancedalgebra00schu/page/170/mode/2up", "math": "a : b = b : c"} | |
| {"pdf": "old_scans_math/4_pg186.pdf", "page": 1, "id": "4_pg186_equation_04", "type": "math", "max_diffs": 0, "checked": "verified", "url": "https://archive.org/details/advancedalgebra00schu/page/170/mode/2up", "math": "b^{2}=ac"} | |
| {"pdf": "old_scans_math/4_pg186.pdf", "page": 1, "id": "4_pg186_equation_05", "type": "math", "max_diffs": 0, "checked": "verified", "url": "https://archive.org/details/advancedalgebra00schu/page/170/mode/2up", "math": "b=\\sqrt{ac}"} | |
| {"pdf": "old_scans_math/4_pg186.pdf", "page": 1, "id": "4_pg186_equation_06", "type": "math", "max_diffs": 0, "checked": "verified", "url": "https://archive.org/details/advancedalgebra00schu/page/170/mode/2up", "math": "mn=pq"} | |
| {"pdf": "old_scans_math/4_pg186.pdf", "page": 1, "id": "4_pg186_equation_07", "type": "math", "max_diffs": 0, "checked": "verified", "url": "https://archive.org/details/advancedalgebra00schu/page/170/mode/2up", "math": "\\frac{m}{q} = \\frac{p}{n}"} | |
| {"pdf": "old_scans_math/4_pg186.pdf", "page": 1, "id": "4_pg186_equation_08", "type": "math", "max_diffs": 0, "checked": "verified", "url": "https://archive.org/details/advancedalgebra00schu/page/170/mode/2up", "math": "x=12: 7"} | |
| {"pdf": "old_scans_math/4_pg186.pdf", "page": 1, "id": "4_pg186_equation_09", "type": "math", "max_diffs": 0, "checked": "verified", "url": "https://archive.org/details/advancedalgebra00schu/page/170/mode/2up", "math": "12 x = 42"} | |
| {"pdf": "old_scans_math/4_pg186.pdf", "page": 1, "id": "4_pg186_equation_10", "type": "math", "max_diffs": 0, "checked": "verified", "url": "https://archive.org/details/advancedalgebra00schu/page/170/mode/2up", "math": "x = \\frac{4}{1}\\frac{2}{2} = 3\\frac{1}{2}"} | |
| {"pdf": "old_scans_math/4_pg186.pdf", "page": 1, "id": "4_pg186_equation_11", "type": "math", "max_diffs": 0, "checked": "verified", "url": "https://archive.org/details/advancedalgebra00schu/page/170/mode/2up", "math": "8:5=7:4\\frac{3}{8}"} | |
| {"pdf": "old_scans_math/4_pg186.pdf", "page": 1, "id": "4_pg186_equation_12", "type": "math", "max_diffs": 0, "checked": "verified", "url": "https://archive.org/details/advancedalgebra00schu/page/170/mode/2up", "math": "8 \\times 4\\frac{3}{8} = 35"} | |
| {"pdf": "old_scans_math/4_pg186.pdf", "page": 1, "id": "4_pg186_equation_13", "type": "math", "max_diffs": 0, "checked": "verified", "url": "https://archive.org/details/advancedalgebra00schu/page/170/mode/2up", "math": "5 \\times 7 = 35"} | |
| {"pdf": "old_scans_math/4_pg281.pdf", "page": 1, "id": "4_pg281_equation_00", "type": "math", "max_diffs": 0, "checked": "verified", "url": "https://archive.org/details/advancedalgebra00schu/page/264/mode/2up", "math": "5+\\sqrt{x}=2"} | |
| {"pdf": "old_scans_math/4_pg281.pdf", "page": 1, "id": "4_pg281_equation_01", "type": "math", "max_diffs": 0, "checked": "verified", "url": "https://archive.org/details/advancedalgebra00schu/page/264/mode/2up", "math": "(-3)^{2}=(+3)^{2}"} | |
| {"pdf": "old_scans_math/4_pg281.pdf", "page": 1, "id": "4_pg281_equation_02", "type": "math", "max_diffs": 0, "checked": "verified", "url": "https://archive.org/details/advancedalgebra00schu/page/264/mode/2up", "math": "x = a"} | |
| {"pdf": "old_scans_math/4_pg281.pdf", "page": 1, "id": "4_pg281_equation_03", "type": "math", "max_diffs": 0, "checked": "verified", "url": "https://archive.org/details/advancedalgebra00schu/page/264/mode/2up", "math": "x^{2} = a^{2}"} | |
| {"pdf": "old_scans_math/4_pg281.pdf", "page": 1, "id": "4_pg281_equation_04", "type": "math", "max_diffs": 0, "checked": "verified", "url": "https://archive.org/details/advancedalgebra00schu/page/264/mode/2up", "math": "x+4=\\sqrt{x}"} | |
| {"pdf": "old_scans_math/4_pg281.pdf", "page": 1, "id": "4_pg281_equation_05", "type": "math", "max_diffs": 0, "checked": "verified", "url": "https://archive.org/details/advancedalgebra00schu/page/264/mode/2up", "math": "x=\\sqrt{x}-4"} | |
| {"pdf": "old_scans_math/4_pg355.pdf", "page": 1, "id": "4_pg355_equation_00", "type": "math", "max_diffs": 0, "checked": "verified", "url": "https://archive.org/details/advancedalgebra00schu/page/338/mode/2up", "math": "a \\text{ and } 5 a"} | |
| {"pdf": "old_scans_math/4_pg355.pdf", "page": 1, "id": "4_pg355_equation_01", "type": "math", "max_diffs": 0, "checked": "verified", "url": "https://archive.org/details/advancedalgebra00schu/page/338/mode/2up", "math": "\\frac{1}{a} \\text{ and } \\frac{1}{b}"} | |
| {"pdf": "old_scans_math/4_pg355.pdf", "page": 1, "id": "4_pg355_equation_02", "type": "math", "max_diffs": 0, "checked": "verified", "url": "https://archive.org/details/advancedalgebra00schu/page/338/mode/2up", "math": "\\frac{x^2-y^2}{x} \\text{ and } \\frac{x^2 + y^2}{x}"} | |
| {"pdf": "old_scans_math/4_pg355.pdf", "page": 1, "id": "4_pg355_equation_03", "type": "math", "max_diffs": 0, "checked": "verified", "url": "https://archive.org/details/advancedalgebra00schu/page/338/mode/2up", "math": "\\frac{a-b}{2} \\text{ and } \\frac{3a+b}{2}"} | |
| {"pdf": "old_scans_math/4_pg380.pdf", "page": 1, "id": "4_pg380_equation_00", "type": "math", "max_diffs": 0, "checked": "verified", "url": "https://archive.org/details/advancedalgebra00schu/page/364/mode/2up", "math": "3x^{2}+2x+1"} | |
| {"pdf": "old_scans_math/4_pg380.pdf", "page": 1, "id": "4_pg380_equation_01", "type": "math", "max_diffs": 0, "checked": "verified", "url": "https://archive.org/details/advancedalgebra00schu/page/364/mode/2up", "math": "\\sqrt{x}"} | |
| {"pdf": "old_scans_math/4_pg380.pdf", "page": 1, "id": "4_pg380_equation_02", "type": "math", "max_diffs": 0, "checked": "verified", "url": "https://archive.org/details/advancedalgebra00schu/page/364/mode/2up", "math": "4^{x}+2x^{3}"} | |
| {"pdf": "old_scans_math/4_pg380.pdf", "page": 1, "id": "4_pg380_equation_03", "type": "math", "max_diffs": 0, "checked": "verified", "url": "https://archive.org/details/advancedalgebra00schu/page/364/mode/2up", "math": "f(x) = 2x^{2} + 2x + 3"} | |
| {"pdf": "old_scans_math/4_pg380.pdf", "page": 1, "id": "4_pg380_equation_04", "type": "math", "max_diffs": 0, "checked": "verified", "url": "https://archive.org/details/advancedalgebra00schu/page/364/mode/2up", "math": "f(3)=2\\cdot3^{2}+2\\cdot3+3=27"} | |
| {"pdf": "old_scans_math/4_pg380.pdf", "page": 1, "id": "4_pg380_equation_05", "type": "math", "max_diffs": 0, "checked": "verified", "url": "https://archive.org/details/advancedalgebra00schu/page/364/mode/2up", "math": "f(a) = 2a^{2}+2a+3"} | |
| {"pdf": "old_scans_math/4_pg380.pdf", "page": 1, "id": "4_pg380_equation_06", "type": "math", "max_diffs": 0, "checked": "verified", "url": "https://archive.org/details/advancedalgebra00schu/page/364/mode/2up", "math": "f(0) = 0 + 0 + 3 = 3"} | |
| {"pdf": "old_scans_math/4_pg380.pdf", "page": 1, "id": "4_pg380_equation_07", "type": "math", "max_diffs": 0, "checked": "verified", "url": "https://archive.org/details/advancedalgebra00schu/page/364/mode/2up", "math": "y=f(x)"} | |
| {"pdf": "old_scans_math/4_pg380.pdf", "page": 1, "id": "4_pg380_equation_08", "type": "math", "max_diffs": 0, "checked": "verified", "url": "https://archive.org/details/advancedalgebra00schu/page/364/mode/2up", "math": "x^{2} + 3 xy + y^{2}"} | |
| {"pdf": "old_scans_math/4_pg380.pdf", "page": 1, "id": "4_pg380_equation_09", "type": "math", "max_diffs": 0, "checked": "verified", "url": "https://archive.org/details/advancedalgebra00schu/page/364/mode/2up", "math": "\\sqrt{x}+\\sqrt{y}"} | |
| {"pdf": "old_scans_math/4_pg380.pdf", "page": 1, "id": "4_pg380_equation_10", "type": "math", "max_diffs": 0, "checked": "verified", "url": "https://archive.org/details/advancedalgebra00schu/page/364/mode/2up", "math": "\\frac{x}{y}"} | |
| {"pdf": "old_scans_math/4_pg380.pdf", "page": 1, "id": "4_pg380_equation_11", "type": "math", "max_diffs": 0, "checked": "verified", "url": "https://archive.org/details/advancedalgebra00schu/page/364/mode/2up", "math": "f(x, y) = xy + x"} | |
| {"pdf": "old_scans_math/4_pg380.pdf", "page": 1, "id": "4_pg380_equation_12", "type": "math", "max_diffs": 0, "checked": "verified", "url": "https://archive.org/details/advancedalgebra00schu/page/364/mode/2up", "math": "f(2, 3) = 2^{3} + 2 = 10"} | |
| {"pdf": "old_scans_math/4_pg380.pdf", "page": 1, "id": "4_pg380_equation_13", "type": "math", "max_diffs": 0, "checked": "verified", "url": "https://archive.org/details/advancedalgebra00schu/page/364/mode/2up", "math": "f(x) = x^{2} +3x+2"} | |
| {"pdf": "old_scans_math/4_pg380.pdf", "page": 1, "id": "4_pg380_equation_14", "type": "math", "max_diffs": 0, "checked": "verified", "url": "https://archive.org/details/advancedalgebra00schu/page/364/mode/2up", "math": "f(1)"} | |
| {"pdf": "old_scans_math/4_pg380.pdf", "page": 1, "id": "4_pg380_equation_15", "type": "math", "max_diffs": 0, "checked": "verified", "url": "https://archive.org/details/advancedalgebra00schu/page/364/mode/2up", "math": "f(0)"} | |
| {"pdf": "old_scans_math/4_pg380.pdf", "page": 1, "id": "4_pg380_equation_16", "type": "math", "max_diffs": 0, "checked": "verified", "url": "https://archive.org/details/advancedalgebra00schu/page/364/mode/2up", "math": "f(-1)"} | |
| {"pdf": "old_scans_math/4_pg380.pdf", "page": 1, "id": "4_pg380_equation_17", "type": "math", "max_diffs": 0, "checked": "verified", "url": "https://archive.org/details/advancedalgebra00schu/page/364/mode/2up", "math": "f(x) = 5 x^{3}-2x-3"} | |
| {"pdf": "old_scans_math/4_pg380.pdf", "page": 1, "id": "4_pg380_equation_18", "type": "math", "max_diffs": 0, "checked": "verified", "url": "https://archive.org/details/advancedalgebra00schu/page/364/mode/2up", "math": "f(2)"} | |
| {"pdf": "old_scans_math/4_pg380.pdf", "page": 1, "id": "4_pg380_equation_19", "type": "math", "max_diffs": 0, "checked": "verified", "url": "https://archive.org/details/advancedalgebra00schu/page/364/mode/2up", "math": "f(a)"} | |
| {"pdf": "old_scans_math/4_pg380.pdf", "page": 1, "id": "4_pg380_equation_21", "type": "math", "max_diffs": 0, "checked": "verified", "url": "https://archive.org/details/advancedalgebra00schu/page/364/mode/2up", "math": "f(x) = 4^{x}"} | |
| {"pdf": "old_scans_math/4_pg380.pdf", "page": 1, "id": "4_pg380_equation_24", "type": "math", "max_diffs": 0, "checked": "verified", "url": "https://archive.org/details/advancedalgebra00schu/page/364/mode/2up", "math": "f(\\frac{1}{2})"} | |
| {"pdf": "old_scans_math/4_pg512.pdf", "page": 1, "id": "4_pg512_equation_00", "type": "math", "max_diffs": 0, "checked": "verified", "url": "https://archive.org/details/advancedalgebra00schu/page/496/mode/2up", "math": "x^3-3x-2=0"} | |
| {"pdf": "old_scans_math/4_pg512.pdf", "page": 1, "id": "4_pg512_equation_01", "type": "math", "max_diffs": 0, "checked": "verified", "url": "https://archive.org/details/advancedalgebra00schu/page/496/mode/2up", "math": "x^3-9x + 28 = 0"} | |
| {"pdf": "old_scans_math/4_pg512.pdf", "page": 1, "id": "4_pg512_equation_02", "type": "math", "max_diffs": 0, "checked": "verified", "url": "https://archive.org/details/advancedalgebra00schu/page/496/mode/2up", "math": "x^3 - 18 x - 35 = 0"} | |
| {"pdf": "old_scans_math/4_pg512.pdf", "page": 1, "id": "4_pg512_equation_03", "type": "math", "max_diffs": 0, "checked": "verified", "url": "https://archive.org/details/advancedalgebra00schu/page/496/mode/2up", "math": "x^3 - 36 x - 91 = 0"} | |
| {"pdf": "old_scans_math/4_pg512.pdf", "page": 1, "id": "4_pg512_equation_04", "type": "math", "max_diffs": 0, "checked": "verified", "url": "https://archive.org/details/advancedalgebra00schu/page/496/mode/2up", "math": "x^3-3x+2=0"} | |
| {"pdf": "old_scans_math/4_pg512.pdf", "page": 1, "id": "4_pg512_equation_05", "type": "math", "max_diffs": 0, "checked": "verified", "url": "https://archive.org/details/advancedalgebra00schu/page/496/mode/2up", "math": "x^3- 27 x - 54 = 0"} | |
| {"pdf": "old_scans_math/4_pg512.pdf", "page": 1, "id": "4_pg512_equation_06", "type": "math", "max_diffs": 0, "checked": "verified", "url": "https://archive.org/details/advancedalgebra00schu/page/496/mode/2up", "math": "x^3+ 9x + 26 = 0"} | |
| {"pdf": "old_scans_math/4_pg512.pdf", "page": 1, "id": "4_pg512_equation_07", "type": "math", "max_diffs": 0, "checked": "verified", "url": "https://archive.org/details/advancedalgebra00schu/page/496/mode/2up", "math": "x^3- 72 x - 280 = 0"} | |
| {"pdf": "old_scans_math/4_pg512.pdf", "page": 1, "id": "4_pg512_equation_08", "type": "math", "max_diffs": 0, "checked": "verified", "url": "https://archive.org/details/advancedalgebra00schu/page/496/mode/2up", "math": "x^3-6x+9 = 0"} | |
| {"pdf": "old_scans_math/4_pg512.pdf", "page": 1, "id": "4_pg512_equation_09", "type": "math", "max_diffs": 0, "checked": "verified", "url": "https://archive.org/details/advancedalgebra00schu/page/496/mode/2up", "math": "x^3+9x - 26 = 0"} | |
| {"pdf": "old_scans_math/4_pg512.pdf", "page": 1, "id": "4_pg512_equation_10", "type": "math", "max_diffs": 0, "checked": "verified", "url": "https://archive.org/details/advancedalgebra00schu/page/496/mode/2up", "math": "x^3-9x - 28 = 0"} | |
| {"pdf": "old_scans_math/4_pg512.pdf", "page": 1, "id": "4_pg512_equation_11", "type": "math", "max_diffs": 0, "checked": "verified", "url": "https://archive.org/details/advancedalgebra00schu/page/496/mode/2up", "math": "x^3- 72 x + 280 = 0"} | |
| {"pdf": "old_scans_math/4_pg512.pdf", "page": 1, "id": "4_pg512_equation_12", "type": "math", "max_diffs": 0, "checked": "verified", "url": "https://archive.org/details/advancedalgebra00schu/page/496/mode/2up", "math": "x^3-6 x^2 - 12 x + 112 = 0"} | |
| {"pdf": "old_scans_math/4_pg512.pdf", "page": 1, "id": "4_pg512_equation_13", "type": "math", "max_diffs": 0, "checked": "verified", "url": "https://archive.org/details/advancedalgebra00schu/page/496/mode/2up", "math": "x^3+ 5 x^2 + 8x + 6 = 0"} | |
| {"pdf": "old_scans_math/4_pg512.pdf", "page": 1, "id": "4_pg512_equation_14", "type": "math", "max_diffs": 0, "checked": "verified", "url": "https://archive.org/details/advancedalgebra00schu/page/496/mode/2up", "math": "x^4 + px^2 + qx + r = 0"} | |
| {"pdf": "old_scans_math/4_pg512.pdf", "page": 1, "id": "4_pg512_equation_15", "type": "math", "max_diffs": 0, "checked": "verified", "url": "https://archive.org/details/advancedalgebra00schu/page/496/mode/2up", "math": "x^4+px^2+qx+r=(x^2+lx+m) (x^2-lx+n)"} | |
| {"pdf": "old_scans_math/4_pg512.pdf", "page": 1, "id": "4_pg512_equation_16", "type": "math", "max_diffs": 0, "checked": "verified", "url": "https://archive.org/details/advancedalgebra00schu/page/496/mode/2up", "math": "=x^4+(-l^2+m+n)x^2-l(m-n)x+mn"} | |
| {"pdf": "old_scans_math/4_pg512.pdf", "page": 1, "id": "4_pg512_equation_17", "type": "math", "max_diffs": 0, "checked": "verified", "url": "https://archive.org/details/advancedalgebra00schu/page/496/mode/2up", "math": "-l^2+m+n=p, -l(m-n)=q, mn = r"} | |
| {"pdf": "old_scans_math/4_pg512.pdf", "page": 1, "id": "4_pg512_equation_18", "type": "math", "max_diffs": 0, "checked": "verified", "url": "https://archive.org/details/advancedalgebra00schu/page/496/mode/2up", "math": "(m+n)^2 - (m-n)^2 - 4 mn = 0"} | |
| {"pdf": "old_scans_math/4_pg512.pdf", "page": 1, "id": "4_pg512_equation_19", "type": "math", "max_diffs": 0, "checked": "verified", "url": "https://archive.org/details/advancedalgebra00schu/page/496/mode/2up", "math": "l^6 + 2 pl^4 + (p^2 - 4 r)l^2 - q^2 = 0"} | |
| {"pdf": "old_scans_math/4_pg512.pdf", "page": 1, "id": "4_pg512_equation_20", "type": "math", "max_diffs": 0, "checked": "verified", "url": "https://archive.org/details/advancedalgebra00schu/page/496/mode/2up", "math": "x^2+lx + m = 0"} | |
| {"pdf": "old_scans_math/4_pg512.pdf", "page": 1, "id": "4_pg512_equation_21", "type": "math", "max_diffs": 0, "checked": "verified", "url": "https://archive.org/details/advancedalgebra00schu/page/496/mode/2up", "math": "x^2 - lx + n = 0"} | |
| {"pdf": "old_scans_math/5_pg558.pdf", "page": 1, "id": "5_pg558_equation_00", "type": "math", "max_diffs": 0, "checked": "verified", "url": "https://archive.org/details/doctrinepermuta00newtgoog/page/527/mode/2up", "math": "\\frac{N-a^m}{ma^{m-1}}"} | |
| {"pdf": "old_scans_math/5_pg558.pdf", "page": 1, "id": "5_pg558_equation_01", "type": "math", "max_diffs": 0, "checked": "verified", "url": "https://archive.org/details/doctrinepermuta00newtgoog/page/527/mode/2up", "math": "a^m + ma^{m-1} z+m \\times \\frac{m-1}{2} \\times a^{m-2} z \\times \\sqrt{\\frac{N-a^m}{ma^{m-1}}} = N"} | |
| {"pdf": "old_scans_math/5_pg558.pdf", "page": 1, "id": "5_pg558_equation_02", "type": "math", "max_diffs": 0, "checked": "verified", "url": "https://archive.org/details/doctrinepermuta00newtgoog/page/527/mode/2up", "math": "a^m + ma^{m-1} z +m \\times \\frac{m-1}{2} \\times a^{m-2} z \\times \\frac{N-a^m}{ma^{m-1}} = N"} | |
| {"pdf": "old_scans_math/5_pg558.pdf", "page": 1, "id": "5_pg558_equation_04", "type": "math", "max_diffs": 0, "checked": "verified", "url": "https://archive.org/details/doctrinepermuta00newtgoog/page/527/mode/2up", "math": "ma^{m-1} z+m \\times \\frac{m-1}{2} \\times a^{m-2} \\times \\sqrt{\\frac{N-a^m}{ma^{m-1}}} \\times z = N-a^m"} | |
| {"pdf": "old_scans_math/5_pg558.pdf", "page": 1, "id": "5_pg558_equation_05", "type": "math", "max_diffs": 0, "checked": "verified", "url": "https://archive.org/details/doctrinepermuta00newtgoog/page/527/mode/2up", "math": "z \\times \\sqrt{ma^{m-1} +m \\times \\frac{m-1}{2} \\times a^{m-2} \\times \\sqrt{\\frac{N-a^m}{ma^{m-1}}}} = N-a^m"} | |
| {"pdf": "old_scans_math/5_pg558.pdf", "page": 1, "id": "5_pg558_equation_06", "type": "math", "max_diffs": 0, "checked": "verified", "url": "https://archive.org/details/doctrinepermuta00newtgoog/page/527/mode/2up", "math": "z (= \\frac{N-a^m}{ma^{m-1} +m \\times \\frac{m-1}{2} \\times a^{m-2} \\times \\sqrt{\\frac{N-a^m}{ma^{m-1}}}}"} | |
| {"pdf": "old_scans_math/5_pg558.pdf", "page": 1, "id": "5_pg558_equation_07", "type": "math", "max_diffs": 0, "checked": "verified", "url": "https://archive.org/details/doctrinepermuta00newtgoog/page/527/mode/2up", "math": "=\\frac{N-a^m}{ma^{m-1} + \\frac{m-1}{2} \\times a^{m-2} \\times \\sqrt{\\frac{N-a^m}{a^{m-1}}}}"} | |
| {"pdf": "old_scans_math/5_pg558.pdf", "page": 1, "id": "5_pg558_equation_08", "type": "math", "max_diffs": 0, "checked": "verified", "url": "https://archive.org/details/doctrinepermuta00newtgoog/page/527/mode/2up", "math": "=\\frac{2 \\times \\overline {N-a^m}}{2ma^{m-1} + \\overline{m-1} a^{m-2} \\times \\sqrt{\\frac{N-a^m}{a^{m-1}}}}"} | |
| {"pdf": "old_scans_math/5_pg558.pdf", "page": 1, "id": "5_pg558_equation_09", "type": "math", "max_diffs": 0, "checked": "verified", "url": "https://archive.org/details/doctrinepermuta00newtgoog/page/527/mode/2up", "math": "=\\frac{2 \\times \\overline{N-a^m}}{2ma^{2m-2} + \\overline{m-1} \\times a^{m-2} \\times \\overline{N-a^m}}"} | |
| {"pdf": "old_scans_math/5_pg281.pdf", "page": 1, "id": "5_pg281_equation_00", "type": "math", "max_diffs": 0, "checked": "verified", "url": "https://archive.org/details/doctrinepermuta00newtgoog/page/250/mode/2up", "math": "a^m+\\frac{m}{1}Aa^{m-1}b+\\frac{m-1}{2}Ba^{m-2}b^2+\\frac{m-2}{3}Ca^{m-3}b^3+\\frac{m-3}{4}Da^{m-4}b^4+\\frac{m-4}{5}Ea^{m-5}b^5+\\frac{m-5}{6}Fa^{m-6}b^6+\\&c"} | |
| {"pdf": "old_scans_math/5_pg281.pdf", "page": 1, "id": "5_pg281_equation_01", "type": "math", "max_diffs": 0, "checked": "verified", "url": "https://archive.org/details/doctrinepermuta00newtgoog/page/250/mode/2up", "math": "\\frac{m-1}{2}"} | |
| {"pdf": "old_scans_math/5_pg281.pdf", "page": 1, "id": "5_pg281_equation_02", "type": "math", "max_diffs": 0, "checked": "verified", "url": "https://archive.org/details/doctrinepermuta00newtgoog/page/250/mode/2up", "math": "\\frac{m-2}{3}"} | |
| {"pdf": "old_scans_math/5_pg281.pdf", "page": 1, "id": "5_pg281_equation_03", "type": "math", "max_diffs": 0, "checked": "verified", "url": "https://archive.org/details/doctrinepermuta00newtgoog/page/250/mode/2up", "math": "\\frac{m-3}{4}"} | |
| {"pdf": "old_scans_math/5_pg281.pdf", "page": 1, "id": "5_pg281_equation_04", "type": "math", "max_diffs": 0, "checked": "verified", "url": "https://archive.org/details/doctrinepermuta00newtgoog/page/250/mode/2up", "math": "\\frac{m-4}{5}"} | |
| {"pdf": "old_scans_math/5_pg174.pdf", "page": 1, "id": "5_pg174_equation_00", "type": "math", "max_diffs": 0, "checked": "verified", "url": "https://archive.org/details/doctrinepermuta00newtgoog/page/143/mode/2up", "math": "1-3x+6x^2-10x^3+15x^4-21x^3+28x^6-36x^7 +45x^8-55x^9+ 66x^{10}-78x^{11} + \\&c"} | |
| {"pdf": "old_scans_math/5_pg174.pdf", "page": 1, "id": "5_pg174_equation_01", "type": "math", "max_diffs": 0, "checked": "verified", "url": "https://archive.org/details/doctrinepermuta00newtgoog/page/143/mode/2up", "math": "1-2x+3x^2-4x^3+5x^4-6x^5+7x^6-8x^7+9x^8-10x^9+11x^{10}-12x^{11}+\\&c"} | |
| {"pdf": "old_scans_math/5_pg174.pdf", "page": 1, "id": "5_pg174_equation_02", "type": "math", "max_diffs": 0, "checked": "verified", "url": "https://archive.org/details/doctrinepermuta00newtgoog/page/143/mode/2up", "math": "*-3x+3x^2"} | |
| {"pdf": "old_scans_math/5_pg174.pdf", "page": 1, "id": "5_pg174_equation_03", "type": "math", "max_diffs": 0, "checked": "verified", "url": "https://archive.org/details/doctrinepermuta00newtgoog/page/143/mode/2up", "math": "-3x-3x^2"} | |
| {"pdf": "old_scans_math/5_pg174.pdf", "page": 1, "id": "5_pg174_equation_04", "type": "math", "max_diffs": 0, "checked": "verified", "url": "https://archive.org/details/doctrinepermuta00newtgoog/page/143/mode/2up", "math": "*+6x^2-4x^3"} | |
| {"pdf": "old_scans_math/5_pg174.pdf", "page": 1, "id": "5_pg174_equation_05", "type": "math", "max_diffs": 0, "checked": "verified", "url": "https://archive.org/details/doctrinepermuta00newtgoog/page/143/mode/2up", "math": "+6x^2+6x^3"} | |
| {"pdf": "old_scans_math/5_pg174.pdf", "page": 1, "id": "5_pg174_equation_06", "type": "math", "max_diffs": 0, "checked": "verified", "url": "https://archive.org/details/doctrinepermuta00newtgoog/page/143/mode/2up", "math": "*-10x^3+5x^4"} | |
| {"pdf": "old_scans_math/5_pg174.pdf", "page": 1, "id": "5_pg174_equation_07", "type": "math", "max_diffs": 0, "checked": "verified", "url": "https://archive.org/details/doctrinepermuta00newtgoog/page/143/mode/2up", "math": "-10x^3-10x^4"} | |
| {"pdf": "old_scans_math/5_pg174.pdf", "page": 1, "id": "5_pg174_equation_08", "type": "math", "max_diffs": 0, "checked": "verified", "url": "https://archive.org/details/doctrinepermuta00newtgoog/page/143/mode/2up", "math": "*+15x^4-6x^5"} | |
| {"pdf": "old_scans_math/5_pg174.pdf", "page": 1, "id": "5_pg174_equation_09", "type": "math", "max_diffs": 0, "checked": "verified", "url": "https://archive.org/details/doctrinepermuta00newtgoog/page/143/mode/2up", "math": "+15x^4+15x^5"} | |
| {"pdf": "old_scans_math/5_pg174.pdf", "page": 1, "id": "5_pg174_equation_10", "type": "math", "max_diffs": 0, "checked": "verified", "url": "https://archive.org/details/doctrinepermuta00newtgoog/page/143/mode/2up", "math": "*-21x^5+7x^6"} | |
| {"pdf": "old_scans_math/5_pg174.pdf", "page": 1, "id": "5_pg174_equation_11", "type": "math", "max_diffs": 0, "checked": "verified", "url": "https://archive.org/details/doctrinepermuta00newtgoog/page/143/mode/2up", "math": "-21x^5-21x^6"} | |
| {"pdf": "old_scans_math/5_pg174.pdf", "page": 1, "id": "5_pg174_equation_12", "type": "math", "max_diffs": 0, "checked": "verified", "url": "https://archive.org/details/doctrinepermuta00newtgoog/page/143/mode/2up", "math": "*+28x^6-8x^7"} | |
| {"pdf": "old_scans_math/5_pg174.pdf", "page": 1, "id": "5_pg174_equation_13", "type": "math", "max_diffs": 0, "checked": "verified", "url": "https://archive.org/details/doctrinepermuta00newtgoog/page/143/mode/2up", "math": "+28x^6+28x^7"} | |
| {"pdf": "old_scans_math/5_pg174.pdf", "page": 1, "id": "5_pg174_equation_14", "type": "math", "max_diffs": 0, "checked": "verified", "url": "https://archive.org/details/doctrinepermuta00newtgoog/page/143/mode/2up", "math": "*-36x^7+9x^8"} | |
| {"pdf": "old_scans_math/5_pg174.pdf", "page": 1, "id": "5_pg174_equation_15", "type": "math", "max_diffs": 0, "checked": "verified", "url": "https://archive.org/details/doctrinepermuta00newtgoog/page/143/mode/2up", "math": "-36x^7-36x^8"} | |
| {"pdf": "old_scans_math/5_pg174.pdf", "page": 1, "id": "5_pg174_equation_16", "type": "math", "max_diffs": 0, "checked": "verified", "url": "https://archive.org/details/doctrinepermuta00newtgoog/page/143/mode/2up", "math": "*+45x^8-10x^9"} | |
| {"pdf": "old_scans_math/5_pg174.pdf", "page": 1, "id": "5_pg174_equation_17", "type": "math", "max_diffs": 0, "checked": "verified", "url": "https://archive.org/details/doctrinepermuta00newtgoog/page/143/mode/2up", "math": "+45x^8+45x^9"} | |
| {"pdf": "old_scans_math/5_pg174.pdf", "page": 1, "id": "5_pg174_equation_18", "type": "math", "max_diffs": 0, "checked": "verified", "url": "https://archive.org/details/doctrinepermuta00newtgoog/page/143/mode/2up", "math": "*-55x^9+11x^{10}"} | |
| {"pdf": "old_scans_math/5_pg174.pdf", "page": 1, "id": "5_pg174_equation_19", "type": "math", "max_diffs": 0, "checked": "verified", "url": "https://archive.org/details/doctrinepermuta00newtgoog/page/143/mode/2up", "math": "-55x^9-55x^{10}"} | |
| {"pdf": "old_scans_math/5_pg174.pdf", "page": 1, "id": "5_pg174_equation_20", "type": "math", "max_diffs": 0, "checked": "verified", "url": "https://archive.org/details/doctrinepermuta00newtgoog/page/143/mode/2up", "math": "*+66x^{10}-12x^{11}"} | |
| {"pdf": "old_scans_math/5_pg174.pdf", "page": 1, "id": "5_pg174_equation_21", "type": "math", "max_diffs": 0, "checked": "verified", "url": "https://archive.org/details/doctrinepermuta00newtgoog/page/143/mode/2up", "math": "+66x^{10}+66x^{11}"} | |
| {"pdf": "old_scans_math/5_pg174.pdf", "page": 1, "id": "5_pg174_equation_22", "type": "math", "max_diffs": 0, "checked": "verified", "url": "https://archive.org/details/doctrinepermuta00newtgoog/page/143/mode/2up", "math": "*-78x^{11}+\\&c."} | |
| {"pdf": "old_scans_math/5_pg174.pdf", "page": 1, "id": "5_pg174_equation_23", "type": "math", "max_diffs": 0, "checked": "verified", "url": "https://archive.org/details/doctrinepermuta00newtgoog/page/143/mode/2up", "math": "-78x^{11}-\\&c."} | |