Commit
·
1ad7d1f
1
Parent(s):
40456df
added old_scans math data
Browse files- bench_data/old_scans_math.jsonl +315 -0
- bench_data/pdfs/old_scans_math/1_pg10.pdf +3 -0
- bench_data/pdfs/old_scans_math/1_pg113.pdf +3 -0
- bench_data/pdfs/old_scans_math/1_pg125.pdf +3 -0
- bench_data/pdfs/old_scans_math/1_pg131.pdf +3 -0
- bench_data/pdfs/old_scans_math/1_pg19.pdf +3 -0
- bench_data/pdfs/old_scans_math/1_pg40.pdf +3 -0
- bench_data/pdfs/old_scans_math/1_pg61.pdf +3 -0
- bench_data/pdfs/old_scans_math/1_pg63.pdf +3 -0
- bench_data/pdfs/old_scans_math/1_pg72.pdf +3 -0
- bench_data/pdfs/old_scans_math/2_pg189.pdf +3 -0
- bench_data/pdfs/old_scans_math/2_pg238.pdf +3 -0
- bench_data/pdfs/old_scans_math/2_pg349.pdf +3 -0
- bench_data/pdfs/old_scans_math/2_pg39.pdf +3 -0
- bench_data/pdfs/old_scans_math/2_pg65.pdf +3 -0
- bench_data/pdfs/old_scans_math/3_pg101.pdf +3 -0
- bench_data/pdfs/old_scans_math/3_pg148.pdf +3 -0
- bench_data/pdfs/old_scans_math/3_pg199.pdf +3 -0
- bench_data/pdfs/old_scans_math/3_pg204.pdf +3 -0
- bench_data/pdfs/old_scans_math/3_pg251.pdf +3 -0
- bench_data/pdfs/old_scans_math/3_pg270.pdf +3 -0
- bench_data/pdfs/old_scans_math/3_pg39.pdf +3 -0
- bench_data/pdfs/old_scans_math/3_pg634.pdf +3 -0
- bench_data/pdfs/old_scans_math/3_pg94.pdf +3 -0
bench_data/old_scans_math.jsonl
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| 1 |
+
{"pdf": "1_pg72.pdf", "page": 1, "id": "1_pg72_equation_00", "type": "present", "max_diffs": 0, "checked": "verified", "url": null, "text": "\\therefore \\int \\frac{d x}{(a^{2}+x^{2})^{n}}=\\frac{1}{a^{2}} \\frac{x}{(2 n-2)(a^{2}+x^{2})^{n-1}}", "case_sensitive": true, "first_n": null, "last_n": null}
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| 2 |
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{"pdf": "1_pg72.pdf", "page": 1, "id": "1_pg72_equation_01", "type": "present", "max_diffs": 0, "checked": "verified", "url": null, "text": "-\\frac{1}{a^{2}.(2 n-2)} \\int \\frac{d x}{(a^{2}+x^{2})^{n-1}}+\\frac{1}{a^{2}} \\int \\frac{d x}{(a^{2}+x^{2})^{n-1}}", "case_sensitive": true, "first_n": null, "last_n": null}
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| 3 |
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{"pdf": "1_pg72.pdf", "page": 1, "id": "1_pg72_equation_02", "type": "present", "max_diffs": 0, "checked": "verified", "url": null, "text": "=\\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{d x}{(a^{2}+x^{2})^{n-1}}", "case_sensitive": true, "first_n": null, "last_n": null}
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| 4 |
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{"pdf": "1_pg72.pdf", "page": 1, "id": "1_pg72_equation_03", "type": "present", "max_diffs": 0, "checked": "verified", "url": null, "text": "\\int \\frac{x^{m} d x}{(a^{2}+x^{2})^{n}}=\\int x^{m-1} d x \\frac{x}{(a^{2}+x^{2})^{n}}", "case_sensitive": true, "first_n": null, "last_n": null}
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| 5 |
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{"pdf": "1_pg72.pdf", "page": 1, "id": "1_pg72_equation_04", "type": "present", "max_diffs": 0, "checked": "verified", "url": null, "text": "p=x^{m-1}, d q=x(a^{2}+x^{2})^{-n} d x", "case_sensitive": true, "first_n": null, "last_n": null}
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| 6 |
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{"pdf": "1_pg72.pdf", "page": 1, "id": "1_pg72_equation_05", "type": "present", "max_diffs": 0, "checked": "verified", "url": null, "text": "\\int \\frac{x^{m} d x}{(a^{2}+x^{2})^{n}}", "case_sensitive": true, "first_n": null, "last_n": null}
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| 7 |
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{"pdf": "1_pg72.pdf", "page": 1, "id": "1_pg72_equation_06", "type": "present", "max_diffs": 0, "checked": "verified", "url": null, "text": "=\\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} d x}{(a^{2}+x^{2})^{n-1}}", "case_sensitive": true, "first_n": null, "last_n": null}
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| 8 |
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{"pdf": "1_pg72.pdf", "page": 1, "id": "1_pg72_equation_07", "type": "present", "max_diffs": 0, "checked": "verified", "url": null, "text": "d u=(a^{2}-x^{2})^{\\frac{n}{2}} d x", "case_sensitive": true, "first_n": null, "last_n": null}
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| 9 |
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{"pdf": "1_pg72.pdf", "page": 1, "id": "1_pg72_equation_08", "type": "present", "max_diffs": 0, "checked": "verified", "url": null, "text": "(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}}", "case_sensitive": true, "first_n": null, "last_n": null}
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| 10 |
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{"pdf": "1_pg72.pdf", "page": 1, "id": "1_pg72_equation_09", "type": "present", "max_diffs": 0, "checked": "verified", "url": null, "text": "\\therefore u=a^{2} \\int(a^{2}-x^{2})^{\\frac{n-2}{2}} d x-\\int x . x(a^{2}-x^{2})^{\\frac{n-2}{2}} d x", "case_sensitive": true, "first_n": null, "last_n": null}
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| 11 |
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{"pdf": "1_pg72.pdf", "page": 1, "id": "1_pg72_equation_10", "type": "present", "max_diffs": 0, "checked": "verified", "url": null, "text": "=a^{2} \\int(a^{2}-x^{2})^{\\frac{n-2}{2}} d x+\\frac{x(a^{2}-x^{2})^{\\frac{n}{2}}}{n}-\\frac{u}{n}", "case_sensitive": true, "first_n": null, "last_n": null}
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| 12 |
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{"pdf": "1_pg72.pdf", "page": 1, "id": "1_pg72_equation_11", "type": "present", "max_diffs": 0, "checked": "verified", "url": null, "text": "\\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}} d x", "case_sensitive": true, "first_n": null, "last_n": null}
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| 13 |
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{"pdf": "1_pg40.pdf", "page": 1, "id": "1_pg40_equation_00", "type": "present", "max_diffs": 0, "checked": "rejected", "url": null, "text": "log\u2061x2+1+1x2+1\u221214(x2+1)2", "case_sensitive": true, "first_n": null, "last_n": null}
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| 14 |
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{"pdf": "1_pg40.pdf", "page": 1, "id": "1_pg40_equation_01", "type": "present", "max_diffs": 0, "checked": "rejected", "url": null, "text": "=4x2+34(x2+1)2+log\u2061x2+1.", "case_sensitive": true, "first_n": null, "last_n": null}
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| 15 |
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{"pdf": "1_pg40.pdf", "page": 1, "id": "1_pg40_equation_02", "type": "present", "max_diffs": 0, "checked": "verified", "url": null, "text": "(22.)\\int \\frac{x^2 dx}{x^4+1}1", "case_sensitive": true, "first_n": null, "last_n": null}
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| 16 |
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{"pdf": "1_pg40.pdf", "page": 1, "id": "1_pg40_equation_03", "type": "present", "max_diffs": 0, "checked": "verified", "url": null, "text": "(x^m+1) = (x^2-2x \\cos \\frac{\\pi}{m}+1)(x^2-2x \\cos \\frac{3\\pi}{m}+1)\\dots", "case_sensitive": true, "first_n": null, "last_n": null}
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| 17 |
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{"pdf": "1_pg40.pdf", "page": 1, "id": "1_pg40_equation_04", "type": "present", "max_diffs": 0, "checked": "verified", "url": null, "text": "(x^2-2x \\cos \\frac{m-1}{m}\\pi+1)", "case_sensitive": true, "first_n": null, "last_n": null}
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| 18 |
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{"pdf": "1_pg40.pdf", "page": 1, "id": "1_pg40_equation_05", "type": "present", "max_diffs": 0, "checked": "verified", "url": null, "text": "(x^4+1) = (x^2-2x \\cos \\frac{\\pi}{4}+1)(x^2-2x \\cos \\frac{3\\pi}{4}+1)", "case_sensitive": true, "first_n": null, "last_n": null}
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| 19 |
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{"pdf": "1_pg40.pdf", "page": 1, "id": "1_pg40_equation_06", "type": "present", "max_diffs": 0, "checked": "verified", "url": null, "text": "(x^4+1) = (x^2-x\\sqrt{2}+1)(x^2+x\\sqrt{2}+1)", "case_sensitive": true, "first_n": null, "last_n": null}
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| 20 |
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{"pdf": "1_pg40.pdf", "page": 1, "id": "1_pg40_equation_07", "type": "present", "max_diffs": 0, "checked": "verified", "url": null, "text": "\\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}", "case_sensitive": true, "first_n": null, "last_n": null}
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| 21 |
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{"pdf": "1_pg40.pdf", "page": 1, "id": "1_pg40_equation_08", "type": "present", "max_diffs": 0, "checked": "verified", "url": null, "text": "\\frac{x^2}{x^4+1} = \\frac{Ax+B}{x^2-x\\sqrt{2}+1} + \\frac{Cx+D}{x^2+x\\sqrt{2}+1}", "case_sensitive": true, "first_n": null, "last_n": null}
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| 22 |
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{"pdf": "1_pg40.pdf", "page": 1, "id": "1_pg40_equation_09", "type": "present", "max_diffs": 0, "checked": "verified", "url": null, "text": "\\therefore x^2= (Ax+B)(x^2+x\\sqrt{2}+1) + (Cx+D)(x^2-x\\sqrt{2}+1)", "case_sensitive": true, "first_n": null, "last_n": null}
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| 23 |
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{"pdf": "1_pg40.pdf", "page": 1, "id": "1_pg40_equation_10", "type": "present", "max_diffs": 0, "checked": "verified", "url": null, "text": "x^2 + x\\sqrt{2} + 1 = 0, x = \\frac{\\sqrt{-1}-1}{\\sqrt{2}}", "case_sensitive": true, "first_n": null, "last_n": null}
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| 24 |
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{"pdf": "1_pg40.pdf", "page": 1, "id": "1_pg40_equation_11", "type": "present", "max_diffs": 0, "checked": "verified", "url": null, "text": "x^2 - x\\sqrt{2} + 1 = 0, x = \\frac{\\sqrt{-1}+1}{\\sqrt{2}}", "case_sensitive": true, "first_n": null, "last_n": null}
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| 25 |
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{"pdf": "1_pg10.pdf", "page": 1, "id": "1_pg10_equation_00", "type": "present", "max_diffs": 0, "checked": "verified", "url": null, "text": "\\int \\frac{dx}{x} = \\log x", "case_sensitive": true, "first_n": null, "last_n": null}
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| 26 |
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{"pdf": "1_pg10.pdf", "page": 1, "id": "1_pg10_equation_01", "type": "present", "max_diffs": 0, "checked": "verified", "url": null, "text": "\\int \\frac{dx}{a+bx^2} = \\frac{1}{\\sqrt{ab}} \\tan^{-1} (x \\sqrt{\\frac{b}{a}})", "case_sensitive": true, "first_n": null, "last_n": null}
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| 27 |
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{"pdf": "1_pg10.pdf", "page": 1, "id": "1_pg10_equation_02", "type": "present", "max_diffs": 0, "checked": "verified", "url": null, "text": "= \\frac{1}{a} \\tan^{-1} \\frac{x}{a}", "case_sensitive": true, "first_n": null, "last_n": null}
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| 28 |
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{"pdf": "1_pg10.pdf", "page": 1, "id": "1_pg10_equation_03", "type": "present", "max_diffs": 0, "checked": "verified", "url": null, "text": "\\int \\frac{dx}{\\sqrt{a^2-x^2}} = \\sin^{-1} \\frac{x}{a}", "case_sensitive": true, "first_n": null, "last_n": null}
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| 29 |
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{"pdf": "1_pg10.pdf", "page": 1, "id": "1_pg10_equation_04", "type": "present", "max_diffs": 0, "checked": "verified", "url": null, "text": "\\int \\frac{-dx}{\\sqrt{a^2-x^2}} = \\cos^{-1} \\frac{x}{a}", "case_sensitive": true, "first_n": null, "last_n": null}
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| 30 |
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{"pdf": "1_pg10.pdf", "page": 1, "id": "1_pg10_equation_05", "type": "present", "max_diffs": 0, "checked": "verified", "url": null, "text": "\\int \\frac{dx}{\\sqrt{x^2\\pm a^2}} = \\log \\frac{(x+\\sqrt{x^2\\pm a^2})}{a}", "case_sensitive": true, "first_n": null, "last_n": null}
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| 31 |
+
{"pdf": "1_pg10.pdf", "page": 1, "id": "1_pg10_equation_06", "type": "present", "max_diffs": 0, "checked": "verified", "url": null, "text": "\\int \\frac{dx}{\\sqrt{2ax-x^2}} = \\operatorname{vers}^{-1} \\frac{x}{a}", "case_sensitive": true, "first_n": null, "last_n": null}
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| 32 |
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{"pdf": "1_pg10.pdf", "page": 1, "id": "1_pg10_equation_07", "type": "present", "max_diffs": 0, "checked": "verified", "url": null, "text": "\\int \\frac{dx}{\\sqrt{x^2\\pm 2ax}} = \\log (x\\pm a + \\sqrt{x^2\\pm 2ax}", "case_sensitive": true, "first_n": null, "last_n": null}
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| 33 |
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{"pdf": "1_pg10.pdf", "page": 1, "id": "1_pg10_equation_08", "type": "present", "max_diffs": 0, "checked": "verified", "url": null, "text": "\\int \\frac{dx}{x\\sqrt{x^2-a^2}} = \\frac{1}{a} \\sec^{-1} \\frac{x}{a}", "case_sensitive": true, "first_n": null, "last_n": null}
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| 34 |
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{"pdf": "1_pg125.pdf", "page": 1, "id": "1_pg125_equation_00", "type": "present", "max_diffs": 0, "checked": "verified", "url": null, "text": "\\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}}", "case_sensitive": true, "first_n": null, "last_n": null}
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| 35 |
+
{"pdf": "1_pg125.pdf", "page": 1, "id": "1_pg125_equation_01", "type": "present", "max_diffs": 0, "checked": "verified", "url": null, "text": "= \\frac{c (c^2 - a^2)}{4 a} \\frac{\\pi}{2}", "case_sensitive": true, "first_n": null, "last_n": null}
|
| 36 |
+
{"pdf": "1_pg125.pdf", "page": 1, "id": "1_pg125_equation_02", "type": "present", "max_diffs": 0, "checked": "verified", "url": null, "text": "\\text{area circle} = \\frac{a}{2} 2\\pi b = \\pi ab", "case_sensitive": true, "first_n": null, "last_n": null}
|
| 37 |
+
{"pdf": "1_pg125.pdf", "page": 1, "id": "1_pg125_equation_03", "type": "present", "max_diffs": 0, "checked": "verified", "url": null, "text": "= \\frac{c (c^2 - a^2)}{4 a} \\pi - \\pi ab", "case_sensitive": true, "first_n": null, "last_n": null}
|
| 38 |
+
{"pdf": "1_pg125.pdf", "page": 1, "id": "1_pg125_equation_04", "type": "present", "max_diffs": 0, "checked": "verified", "url": null, "text": "= \\frac{(a +2b) 4b(a + b)}{4 a} \\pi - \\pi ab", "case_sensitive": true, "first_n": null, "last_n": null}
|
| 39 |
+
{"pdf": "1_pg125.pdf", "page": 1, "id": "1_pg125_equation_05", "type": "present", "max_diffs": 0, "checked": "verified", "url": null, "text": "= \\frac{\\pi}{a}(a^2b + 3ab + 2b^3 - a^2b)", "case_sensitive": true, "first_n": null, "last_n": null}
|
| 40 |
+
{"pdf": "1_pg125.pdf", "page": 1, "id": "1_pg125_equation_06", "type": "present", "max_diffs": 0, "checked": "verified", "url": null, "text": "= \\pi b^2 \\left(3 + \\frac{2 b}{a}\\right)", "case_sensitive": true, "first_n": null, "last_n": null}
|
| 41 |
+
{"pdf": "1_pg125.pdf", "page": 1, "id": "1_pg125_equation_07", "type": "present", "max_diffs": 0, "checked": "verified", "url": null, "text": "x^{\\frac{2}{3}} + y^{\\frac{2}{3}} = a^{\\frac{2}{3}}", "case_sensitive": true, "first_n": null, "last_n": null}
|
| 42 |
+
{"pdf": "1_pg125.pdf", "page": 1, "id": "1_pg125_equation_08", "type": "present", "max_diffs": 0, "checked": "verified", "url": null, "text": "s = \\int \\sqrt{1 + \\frac{d y^2}{d x^2}} = \\int \\sqrt{1 + \\frac{y^{\\frac{2}{3}}}{x^{\\frac{2}{3}} }}", "case_sensitive": true, "first_n": null, "last_n": null}
|
| 43 |
+
{"pdf": "1_pg125.pdf", "page": 1, "id": "1_pg125_equation_09", "type": "present", "max_diffs": 0, "checked": "verified", "url": null, "text": "= \\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}}", "case_sensitive": true, "first_n": null, "last_n": null}
|
| 44 |
+
{"pdf": "1_pg125.pdf", "page": 1, "id": "1_pg125_equation_10", "type": "present", "max_diffs": 0, "checked": "verified", "url": null, "text": "s = \\frac{3}{2} a", "case_sensitive": true, "first_n": null, "last_n": null}
|
| 45 |
+
{"pdf": "1_pg125.pdf", "page": 1, "id": "1_pg125_equation_11", "type": "present", "max_diffs": 0, "checked": "verified", "url": null, "text": "4 \\times \\frac{3}{2} a = 6 a", "case_sensitive": true, "first_n": null, "last_n": null}
|
| 46 |
+
{"pdf": "1_pg19.pdf", "page": 1, "id": "1_pg19_equation_00", "type": "present", "max_diffs": 0, "checked": "verified", "url": null, "text": "\\frac{x^{2}}{(x^{2}+1)(x^{2}+4)}=\\frac{A}{(x^{2}+1)}+\\frac{B}{(x^{2}+4)}", "case_sensitive": true, "first_n": null, "last_n": null}
|
| 47 |
+
{"pdf": "1_pg19.pdf", "page": 1, "id": "1_pg19_equation_01", "type": "present", "max_diffs": 0, "checked": "verified", "url": null, "text": "x^{2}=A(x^{2}+4)+B(x^{2}+1)", "case_sensitive": true, "first_n": null, "last_n": null}
|
| 48 |
+
{"pdf": "1_pg19.pdf", "page": 1, "id": "1_pg19_equation_02", "type": "present", "max_diffs": 0, "checked": "verified", "url": null, "text": "x=\\sqrt{-1}", "case_sensitive": true, "first_n": null, "last_n": null}
|
| 49 |
+
{"pdf": "1_pg19.pdf", "page": 1, "id": "1_pg19_equation_03", "type": "present", "max_diffs": 0, "checked": "verified", "url": null, "text": "\\therefore -1=3A, \\therefore A=-\\frac{1}{3}", "case_sensitive": true, "first_n": null, "last_n": null}
|
| 50 |
+
{"pdf": "1_pg19.pdf", "page": 1, "id": "1_pg19_equation_04", "type": "present", "max_diffs": 0, "checked": "verified", "url": null, "text": "\\therefore x^{2}+\\frac{1}{3}(x^{2}+4)=B(x^{2}+1)", "case_sensitive": true, "first_n": null, "last_n": null}
|
| 51 |
+
{"pdf": "1_pg19.pdf", "page": 1, "id": "1_pg19_equation_05", "type": "present", "max_diffs": 0, "checked": "verified", "url": null, "text": "\\therefore \\frac{4(x^{2}+1)}{3}=B(x^{2}+1), \\therefore B=\\frac{4}{3}", "case_sensitive": true, "first_n": null, "last_n": null}
|
| 52 |
+
{"pdf": "1_pg19.pdf", "page": 1, "id": "1_pg19_equation_06", "type": "present", "max_diffs": 0, "checked": "verified", "url": null, "text": "\\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)}\\}", "case_sensitive": true, "first_n": null, "last_n": null}
|
| 53 |
+
{"pdf": "1_pg19.pdf", "page": 1, "id": "1_pg19_equation_07", "type": "present", "max_diffs": 0, "checked": "verified", "url": null, "text": "=\\frac{1}{3}\\int\\{\\frac{dx}{(1+x^{2})}-\\frac{dx}{(1+\\frac{x^{2}}{4})}\\}", "case_sensitive": true, "first_n": null, "last_n": null}
|
| 54 |
+
{"pdf": "1_pg19.pdf", "page": 1, "id": "1_pg19_equation_08", "type": "present", "max_diffs": 0, "checked": "verified", "url": null, "text": "=\\frac{1}{3}\\left\\{2\\tan^{-1}\\frac{x}{2}-\\tan^{-1}x\\right\\}", "case_sensitive": true, "first_n": null, "last_n": null}
|
| 55 |
+
{"pdf": "1_pg19.pdf", "page": 1, "id": "1_pg19_equation_09", "type": "present", "max_diffs": 0, "checked": "verified", "url": null, "text": "\\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)}", "case_sensitive": true, "first_n": null, "last_n": null}
|
| 56 |
+
{"pdf": "1_pg19.pdf", "page": 1, "id": "1_pg19_equation_10", "type": "present", "max_diffs": 0, "checked": "verified", "url": null, "text": "=\\frac{1}{3}\\int\\frac{\\{4x^{2}+4-(x^{2}+4)\\}dx}{(x^{2}+1)(x^{2}+4)}", "case_sensitive": true, "first_n": null, "last_n": null}
|
| 57 |
+
{"pdf": "1_pg19.pdf", "page": 1, "id": "1_pg19_equation_11", "type": "present", "max_diffs": 0, "checked": "verified", "url": null, "text": "=\\frac{1}{3}\\int\\left(\\frac{4dx}{(x^{2}+4)}-\\frac{dx}{x^{2}+1}\\right)", "case_sensitive": true, "first_n": null, "last_n": null}
|
| 58 |
+
{"pdf": "1_pg19.pdf", "page": 1, "id": "1_pg19_equation_12", "type": "present", "max_diffs": 0, "checked": "verified", "url": null, "text": "=\\frac{1}{3}\\left(2\\tan^{-1}\\frac{x}{2}-\\tan^{-1}x\\right)", "case_sensitive": true, "first_n": null, "last_n": null}
|
| 59 |
+
{"pdf": "1_pg19.pdf", "page": 1, "id": "1_pg19_equation_13", "type": "present", "max_diffs": 0, "checked": "verified", "url": null, "text": "du=\\frac{(3x^{2}+x-2)dx}{(x-1)^{3}(x^{2}+1)}", "case_sensitive": true, "first_n": null, "last_n": null}
|
| 60 |
+
{"pdf": "1_pg61.pdf", "page": 1, "id": "1_pg61_equation_00", "type": "present", "max_diffs": 0, "checked": "verified", "url": null, "text": "\\therefore \\int \\frac{dx}{(1 + x)\\sqrt{1 - x - x^2}} = -\\int \\frac{dz}{z^{\\frac{1}{2}}\\sqrt{1 + \\frac{1}{z} - \\frac{1}{z^2}}}", "case_sensitive": true, "first_n": null, "last_n": null}
|
| 61 |
+
{"pdf": "1_pg61.pdf", "page": 1, "id": "1_pg61_equation_01", "type": "present", "max_diffs": 0, "checked": "verified", "url": null, "text": "=-\\int \\frac{dz}{\\sqrt{z^2 + z - 1}} = -\\int \\frac{dz}{\\sqrt{(z + \\frac{1}{2})^2 - \\frac{5}{4}}}", "case_sensitive": true, "first_n": null, "last_n": null}
|
| 62 |
+
{"pdf": "1_pg61.pdf", "page": 1, "id": "1_pg61_equation_02", "type": "present", "max_diffs": 0, "checked": "verified", "url": null, "text": "=-\\log \\{\\sqrt{z^2 + z - 1} + z + \\frac{1}{2}\\}", "case_sensitive": true, "first_n": null, "last_n": null}
|
| 63 |
+
{"pdf": "1_pg61.pdf", "page": 1, "id": "1_pg61_equation_03", "type": "present", "max_diffs": 0, "checked": "verified", "url": null, "text": "=-\\log \\{\\frac{\\sqrt{1 - x - x^2}}{1 + x} + \\frac{1}{1 + x} + \\frac{1}{2}\\}", "case_sensitive": true, "first_n": null, "last_n": null}
|
| 64 |
+
{"pdf": "1_pg61.pdf", "page": 1, "id": "1_pg61_equation_04", "type": "present", "max_diffs": 0, "checked": "verified", "url": null, "text": "=-\\log \\{\\frac{2\\sqrt{1 - x - x^2} + x + 3}{2(1 + x)}\\}", "case_sensitive": true, "first_n": null, "last_n": null}
|
| 65 |
+
{"pdf": "1_pg61.pdf", "page": 1, "id": "1_pg61_equation_05", "type": "present", "max_diffs": 0, "checked": "verified", "url": null, "text": "=\\log \\{\\frac{2(1 + x)}{2\\sqrt{1 - x - x^2} + x + 3}\\}", "case_sensitive": true, "first_n": null, "last_n": null}
|
| 66 |
+
{"pdf": "1_pg61.pdf", "page": 1, "id": "1_pg61_equation_06", "type": "present", "max_diffs": 0, "checked": "verified", "url": null, "text": "=\\log \\{\\frac{2(1 + x)(x + 3) - 2\\sqrt{1 - x - x^2}}{x^2 + 6x + 9 - 4 + 4x + 4x^2}\\}", "case_sensitive": true, "first_n": null, "last_n": null}
|
| 67 |
+
{"pdf": "1_pg61.pdf", "page": 1, "id": "1_pg61_equation_07", "type": "present", "max_diffs": 0, "checked": "verified", "url": null, "text": "=\\log \\frac{2(1 + x)(x + 3 - 2\\sqrt{1 - x - x^2})}{5 (1 + x)^2}", "case_sensitive": true, "first_n": null, "last_n": null}
|
| 68 |
+
{"pdf": "1_pg61.pdf", "page": 1, "id": "1_pg61_equation_08", "type": "present", "max_diffs": 0, "checked": "verified", "url": null, "text": "=\\log \\frac{x + 3 - 2\\sqrt{1 - x - x^2}}{1 + x} + c", "case_sensitive": true, "first_n": null, "last_n": null}
|
| 69 |
+
{"pdf": "1_pg61.pdf", "page": 1, "id": "1_pg61_equation_09", "type": "present", "max_diffs": 0, "checked": "verified", "url": null, "text": "\\int \\frac{dx}{\\sqrt{1 + 2x - x^2}} = \\int \\frac{dx}{\\sqrt{2 - (1 - x)^2}}", "case_sensitive": true, "first_n": null, "last_n": null}
|
| 70 |
+
{"pdf": "1_pg61.pdf", "page": 1, "id": "1_pg61_equation_10", "type": "present", "max_diffs": 0, "checked": "verified", "url": null, "text": "=-\\int \\frac{d(1 - x)}{\\sqrt{2 - (1 - x)^2}} = \\cos^{-1}\\frac{1 - x}{\\sqrt{2}}", "case_sensitive": true, "first_n": null, "last_n": null}
|
| 71 |
+
{"pdf": "1_pg63.pdf", "page": 1, "id": "1_pg63_equation_00", "type": "present", "max_diffs": 0, "checked": "verified", "url": null, "text": "\\int \\frac{\\sqrt{x} dx}{\\sqrt{a^3-x^3}}", "case_sensitive": true, "first_n": null, "last_n": null}
|
| 72 |
+
{"pdf": "1_pg63.pdf", "page": 1, "id": "1_pg63_equation_01", "type": "present", "max_diffs": 0, "checked": "verified", "url": null, "text": "x^3 = z^2", "case_sensitive": true, "first_n": null, "last_n": null}
|
| 73 |
+
{"pdf": "1_pg63.pdf", "page": 1, "id": "1_pg63_equation_02", "type": "present", "max_diffs": 0, "checked": "verified", "url": null, "text": "x = z^{\\frac{2}{3}}, dx = \\frac{2}{3} z^{-\\frac{1}{3}} dz, \\sqrt{x} = z^{\\frac{1}{3}}", "case_sensitive": true, "first_n": null, "last_n": null}
|
| 74 |
+
{"pdf": "1_pg63.pdf", "page": 1, "id": "1_pg63_equation_03", "type": "present", "max_diffs": 0, "checked": "verified", "url": null, "text": "\\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}}}", "case_sensitive": true, "first_n": null, "last_n": null}
|
| 75 |
+
{"pdf": "1_pg63.pdf", "page": 1, "id": "1_pg63_equation_04", "type": "present", "max_diffs": 0, "checked": "verified", "url": null, "text": "= \\frac{2}{3} \\sin^{-1} \\frac{x^{\\frac{3}{2}}}{a^{\\frac{3}{2}}}", "case_sensitive": true, "first_n": null, "last_n": null}
|
| 76 |
+
{"pdf": "1_pg63.pdf", "page": 1, "id": "1_pg63_equation_05", "type": "present", "max_diffs": 0, "checked": "verified", "url": null, "text": "\\theta = \\sin^{-1} \\frac{x^{\\frac{3}{2}}}{a^{\\frac{3}{2}}}", "case_sensitive": true, "first_n": null, "last_n": null}
|
| 77 |
+
{"pdf": "1_pg63.pdf", "page": 1, "id": "1_pg63_equation_06", "type": "present", "max_diffs": 0, "checked": "verified", "url": null, "text": "\\tan \\theta = \\frac{\\sin \\theta}{\\sqrt{1-\\sin^2 \\theta}} = \\frac{x^{\\frac{3}{2}}}{\\sqrt{a^3-x^3}}", "case_sensitive": true, "first_n": null, "last_n": null}
|
| 78 |
+
{"pdf": "1_pg63.pdf", "page": 1, "id": "1_pg63_equation_07", "type": "present", "max_diffs": 0, "checked": "verified", "url": null, "text": "\\therefore \\int \\frac{\\sqrt{x} dx}{\\sqrt{a^3-x^3}} = \\frac{2}{3} \\tan^{-1} \\sqrt{\\frac{x^3}{a^3-x^3}}", "case_sensitive": true, "first_n": null, "last_n": null}
|
| 79 |
+
{"pdf": "1_pg63.pdf", "page": 1, "id": "1_pg63_equation_08", "type": "present", "max_diffs": 0, "checked": "verified", "url": null, "text": "\\int \\frac{dx}{(2ax+x^2)^{\\frac{3}{2}}} = \\int \\frac{dx}{((x+a)^2-a^2)^{\\frac{3}{2}}}", "case_sensitive": true, "first_n": null, "last_n": null}
|
| 80 |
+
{"pdf": "1_pg63.pdf", "page": 1, "id": "1_pg63_equation_09", "type": "present", "max_diffs": 0, "checked": "verified", "url": null, "text": "x + a = \\frac{1}{z}, dx = -\\frac{dz}{z^2}", "case_sensitive": true, "first_n": null, "last_n": null}
|
| 81 |
+
{"pdf": "1_pg63.pdf", "page": 1, "id": "1_pg63_equation_10", "type": "present", "max_diffs": 0, "checked": "verified", "url": null, "text": "=-\\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}}}", "case_sensitive": true, "first_n": null, "last_n": null}
|
| 82 |
+
{"pdf": "1_pg131.pdf", "page": 1, "id": "1_pg131_equation_00", "type": "present", "max_diffs": 0, "checked": "verified", "url": null, "text": "\\int \\frac{d x}{\\sqrt{2 a x+x^{2}}}=\\log \\{x+a+\\sqrt{2 a x+x^{2}}\\}", "case_sensitive": true, "first_n": null, "last_n": null}
|
| 83 |
+
{"pdf": "1_pg131.pdf", "page": 1, "id": "1_pg131_equation_01", "type": "present", "max_diffs": 0, "checked": "verified", "url": null, "text": "\\therefore \\int \\frac{x d x}{\\sqrt{2 a x+x^{2}}}", "case_sensitive": true, "first_n": null, "last_n": null}
|
| 84 |
+
{"pdf": "1_pg131.pdf", "page": 1, "id": "1_pg131_equation_02", "type": "present", "max_diffs": 0, "checked": "verified", "url": null, "text": "=\\sqrt{2 a x+x^{2}}-a \\log \\{a+a+\\sqrt{2 a x+x^{2}}\\}", "case_sensitive": true, "first_n": null, "last_n": null}
|
| 85 |
+
{"pdf": "1_pg131.pdf", "page": 1, "id": "1_pg131_equation_03", "type": "present", "max_diffs": 0, "checked": "verified", "url": null, "text": "\\therefore \\int \\frac{x^{2} d x}{\\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}}", "case_sensitive": true, "first_n": null, "last_n": null}
|
| 86 |
+
{"pdf": "1_pg131.pdf", "page": 1, "id": "1_pg131_equation_04", "type": "present", "max_diffs": 0, "checked": "verified", "url": null, "text": "+\\frac{3 a^{2}}{2} \\log \\{x+a+\\sqrt{2 a x+x^{2}}\\}", "case_sensitive": true, "first_n": null, "last_n": null}
|
| 87 |
+
{"pdf": "1_pg131.pdf", "page": 1, "id": "1_pg131_equation_05", "type": "present", "max_diffs": 0, "checked": "verified", "url": null, "text": "\\int_{a}^{0} - \\frac{x^{2} d x}{(2 a x-x^{2})^{\\frac{3}{2}}}", "case_sensitive": true, "first_n": null, "last_n": null}
|
| 88 |
+
{"pdf": "1_pg131.pdf", "page": 1, "id": "1_pg131_equation_06", "type": "present", "max_diffs": 0, "checked": "verified", "url": null, "text": "d q=\\frac{x d x}{(2 a x-x^{2})^{\\frac{1}{2}}}", "case_sensitive": true, "first_n": null, "last_n": null}
|
| 89 |
+
{"pdf": "1_pg131.pdf", "page": 1, "id": "1_pg131_equation_07", "type": "present", "max_diffs": 0, "checked": "verified", "url": null, "text": "\\therefore q=\\int \\frac{x d x}{(2 a x-x)^{\\frac{3}{2}}}=\\int(2 a-x)^{-\\frac{3}{2}} x^{\\frac{1}{2}} d x=", "case_sensitive": true, "first_n": null, "last_n": null}
|
| 90 |
+
{"pdf": "1_pg131.pdf", "page": 1, "id": "1_pg131_equation_08", "type": "present", "max_diffs": 0, "checked": "verified", "url": null, "text": "\\int (2 a x^{-1}-1)^{-\\frac{3}{2}} x^{-2} d x", "case_sensitive": true, "first_n": null, "last_n": null}
|
| 91 |
+
{"pdf": "1_pg131.pdf", "page": 1, "id": "1_pg131_equation_09", "type": "present", "max_diffs": 0, "checked": "verified", "url": null, "text": "=-\\frac{1}{2 a} \\int(2 a x^{-1}-1)^{-\\frac{3}{2}} \\times-2 a x^{-2} d x", "case_sensitive": true, "first_n": null, "last_n": null}
|
| 92 |
+
{"pdf": "1_pg131.pdf", "page": 1, "id": "1_pg131_equation_10", "type": "present", "max_diffs": 0, "checked": "verified", "url": null, "text": "=\\frac{(2 a x^{-1}-1)^{-\\frac{1}{2}}}{a}=\\frac{1}{a} \\sqrt{\\frac{x}{2 a-x}}", "case_sensitive": true, "first_n": null, "last_n": null}
|
| 93 |
+
{"pdf": "1_pg113.pdf", "page": 1, "id": "1_pg113_equation_00", "type": "present", "max_diffs": 0, "checked": "verified", "url": null, "text": "CD = h", "case_sensitive": true, "first_n": null, "last_n": null}
|
| 94 |
+
{"pdf": "1_pg113.pdf", "page": 1, "id": "1_pg113_equation_01", "type": "present", "max_diffs": 0, "checked": "verified", "url": null, "text": "AB = 2c", "case_sensitive": true, "first_n": null, "last_n": null}
|
| 95 |
+
{"pdf": "1_pg113.pdf", "page": 1, "id": "1_pg113_equation_02", "type": "present", "max_diffs": 0, "checked": "verified", "url": null, "text": "x = AG", "case_sensitive": true, "first_n": null, "last_n": null}
|
| 96 |
+
{"pdf": "1_pg113.pdf", "page": 1, "id": "1_pg113_equation_03", "type": "present", "max_diffs": 0, "checked": "verified", "url": null, "text": "y= FG", "case_sensitive": true, "first_n": null, "last_n": null}
|
| 97 |
+
{"pdf": "1_pg113.pdf", "page": 1, "id": "1_pg113_equation_04", "type": "present", "max_diffs": 0, "checked": "verified", "url": null, "text": "AD^{2}: AG.GB :: CD : EG", "case_sensitive": true, "first_n": null, "last_n": null}
|
| 98 |
+
{"pdf": "1_pg113.pdf", "page": 1, "id": "1_pg113_equation_05", "type": "present", "max_diffs": 0, "checked": "verified", "url": null, "text": "c^{2}: x (2c - x) :: h: y", "case_sensitive": true, "first_n": null, "last_n": null}
|
| 99 |
+
{"pdf": "1_pg113.pdf", "page": 1, "id": "1_pg113_equation_06", "type": "present", "max_diffs": 0, "checked": "verified", "url": null, "text": "y = \\frac{h(2cx - x^{2})}{c^{2}}", "case_sensitive": true, "first_n": null, "last_n": null}
|
| 100 |
+
{"pdf": "1_pg113.pdf", "page": 1, "id": "1_pg113_equation_07", "type": "present", "max_diffs": 0, "checked": "verified", "url": null, "text": "y^{2} = \\frac{h^{2}}{c^{4}}(4c^{2}x^{2} - 4 cx^{3} + x^{4})", "case_sensitive": true, "first_n": null, "last_n": null}
|
| 101 |
+
{"pdf": "1_pg113.pdf", "page": 1, "id": "1_pg113_equation_08", "type": "present", "max_diffs": 0, "checked": "verified", "url": null, "text": "V = \\pi \\int y^{2}dx", "case_sensitive": true, "first_n": null, "last_n": null}
|
| 102 |
+
{"pdf": "1_pg113.pdf", "page": 1, "id": "1_pg113_equation_09", "type": "present", "max_diffs": 0, "checked": "verified", "url": null, "text": "\\frac{\\pi 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})", "case_sensitive": true, "first_n": null, "last_n": null}
|
| 103 |
+
{"pdf": "1_pg113.pdf", "page": 1, "id": "1_pg113_equation_10", "type": "present", "max_diffs": 0, "checked": "verified", "url": null, "text": "\\frac{\\pi h^{2}}{c^{4}}(\\frac{4 c^{5}}{3} - c^{5} + \\frac{c^{5}}{5}) = \\frac{8}{15} \\pi h^{2} c =", "case_sensitive": true, "first_n": null, "last_n": null}
|
| 104 |
+
{"pdf": "1_pg113.pdf", "page": 1, "id": "1_pg113_equation_11", "type": "present", "max_diffs": 0, "checked": "verified", "url": null, "text": "ANP = \\int y dx", "case_sensitive": true, "first_n": null, "last_n": null}
|
| 105 |
+
{"pdf": "1_pg113.pdf", "page": 1, "id": "1_pg113_equation_12", "type": "present", "max_diffs": 0, "checked": "verified", "url": null, "text": "= 2 \\sqrt{a} \\int \\sqrt{x} dx = \\frac{4}{3}\\sqrt{a}x^{\\frac{3}{2}}", "case_sensitive": true, "first_n": null, "last_n": null}
|
| 106 |
+
{"pdf": "1_pg113.pdf", "page": 1, "id": "1_pg113_equation_13", "type": "present", "max_diffs": 0, "checked": "verified", "url": null, "text": "A N'P' = \\int \\beta d\\alpha = \\frac{2}{3\\sqrt{3} a}\\int (\\alpha - 2a)^{\\frac{3}{2}} d\\alpha", "case_sensitive": true, "first_n": null, "last_n": null}
|
| 107 |
+
{"pdf": "3_pg199.pdf", "page": 1, "id": "3_pg199_equation_00", "type": "present", "max_diffs": 0, "checked": "verified", "url": null, "text": "\\phi(x, y) = 0", "case_sensitive": true, "first_n": null, "last_n": null}
|
| 108 |
+
{"pdf": "3_pg199.pdf", "page": 1, "id": "3_pg199_equation_01", "type": "present", "max_diffs": 0, "checked": "verified", "url": null, "text": "f_x + \\lambda \\phi_x = 0", "case_sensitive": true, "first_n": null, "last_n": null}
|
| 109 |
+
{"pdf": "3_pg199.pdf", "page": 1, "id": "3_pg199_equation_02", "type": "present", "max_diffs": 0, "checked": "verified", "url": null, "text": "f_y + \\lambda \\phi_y = 0", "case_sensitive": true, "first_n": null, "last_n": null}
|
| 110 |
+
{"pdf": "3_pg199.pdf", "page": 1, "id": "3_pg199_equation_03", "type": "present", "max_diffs": 0, "checked": "verified", "url": null, "text": "F = f + \\lambda \\phi", "case_sensitive": true, "first_n": null, "last_n": null}
|
| 111 |
+
{"pdf": "3_pg199.pdf", "page": 1, "id": "3_pg199_equation_04", "type": "present", "max_diffs": 0, "checked": "verified", "url": null, "text": "\\phi = 0", "case_sensitive": true, "first_n": null, "last_n": null}
|
| 112 |
+
{"pdf": "3_pg199.pdf", "page": 1, "id": "3_pg199_equation_05", "type": "present", "max_diffs": 0, "checked": "verified", "url": null, "text": "u = xy", "case_sensitive": true, "first_n": null, "last_n": null}
|
| 113 |
+
{"pdf": "3_pg199.pdf", "page": 1, "id": "3_pg199_equation_06", "type": "present", "max_diffs": 0, "checked": "verified", "url": null, "text": "x^2 + y^2 - 1 = 0", "case_sensitive": true, "first_n": null, "last_n": null}
|
| 114 |
+
{"pdf": "3_pg199.pdf", "page": 1, "id": "3_pg199_equation_07", "type": "present", "max_diffs": 0, "checked": "verified", "url": null, "text": "y + 2\\lambda x = 0", "case_sensitive": true, "first_n": null, "last_n": null}
|
| 115 |
+
{"pdf": "3_pg199.pdf", "page": 1, "id": "3_pg199_equation_08", "type": "present", "max_diffs": 0, "checked": "verified", "url": null, "text": "x + 2\\lambda y = 0", "case_sensitive": true, "first_n": null, "last_n": null}
|
| 116 |
+
{"pdf": "3_pg199.pdf", "page": 1, "id": "3_pg199_equation_09", "type": "present", "max_diffs": 0, "checked": "verified", "url": null, "text": "x^2 + y^2 - 1 = 0", "case_sensitive": true, "first_n": null, "last_n": null}
|
| 117 |
+
{"pdf": "3_pg199.pdf", "page": 1, "id": "3_pg199_equation_10", "type": "present", "max_diffs": 0, "checked": "verified", "url": null, "text": "\\xi = \\frac{1}{2} \\sqrt{2}", "case_sensitive": true, "first_n": null, "last_n": null}
|
| 118 |
+
{"pdf": "3_pg199.pdf", "page": 1, "id": "3_pg199_equation_11", "type": "present", "max_diffs": 0, "checked": "verified", "url": null, "text": "\\eta = \\frac{1}{2} \\sqrt{2}", "case_sensitive": true, "first_n": null, "last_n": null}
|
| 119 |
+
{"pdf": "3_pg199.pdf", "page": 1, "id": "3_pg199_equation_12", "type": "present", "max_diffs": 0, "checked": "verified", "url": null, "text": "\\xi = -\\frac{1}{2} \\sqrt{2}", "case_sensitive": true, "first_n": null, "last_n": null}
|
| 120 |
+
{"pdf": "3_pg199.pdf", "page": 1, "id": "3_pg199_equation_13", "type": "present", "max_diffs": 0, "checked": "verified", "url": null, "text": "\\eta = -\\frac{1}{2} \\sqrt{2}", "case_sensitive": true, "first_n": null, "last_n": null}
|
| 121 |
+
{"pdf": "3_pg199.pdf", "page": 1, "id": "3_pg199_equation_14", "type": "present", "max_diffs": 0, "checked": "verified", "url": null, "text": "\\xi = \\frac{1}{2} \\sqrt{2}", "case_sensitive": true, "first_n": null, "last_n": null}
|
| 122 |
+
{"pdf": "3_pg199.pdf", "page": 1, "id": "3_pg199_equation_15", "type": "present", "max_diffs": 0, "checked": "verified", "url": null, "text": "\\eta = -\\frac{1}{2} \\sqrt{2}", "case_sensitive": true, "first_n": null, "last_n": null}
|
| 123 |
+
{"pdf": "3_pg199.pdf", "page": 1, "id": "3_pg199_equation_16", "type": "present", "max_diffs": 0, "checked": "verified", "url": null, "text": "\\xi = -\\frac{1}{2} \\sqrt{2}", "case_sensitive": true, "first_n": null, "last_n": null}
|
| 124 |
+
{"pdf": "3_pg199.pdf", "page": 1, "id": "3_pg199_equation_17", "type": "present", "max_diffs": 0, "checked": "verified", "url": null, "text": "\\eta = \\frac{1}{2} \\sqrt{2}", "case_sensitive": true, "first_n": null, "last_n": null}
|
| 125 |
+
{"pdf": "3_pg199.pdf", "page": 1, "id": "3_pg199_equation_18", "type": "present", "max_diffs": 0, "checked": "verified", "url": null, "text": "u = \\frac{1}{2}", "case_sensitive": true, "first_n": null, "last_n": null}
|
| 126 |
+
{"pdf": "3_pg199.pdf", "page": 1, "id": "3_pg199_equation_19", "type": "present", "max_diffs": 0, "checked": "verified", "url": null, "text": "u = -\\frac{1}{2}", "case_sensitive": true, "first_n": null, "last_n": null}
|
| 127 |
+
{"pdf": "3_pg199.pdf", "page": 1, "id": "3_pg199_equation_20", "type": "present", "max_diffs": 0, "checked": "verified", "url": null, "text": "u = xy", "case_sensitive": true, "first_n": null, "last_n": null}
|
| 128 |
+
{"pdf": "3_pg634.pdf", "page": 1, "id": "3_pg634_equation_00", "type": "present", "max_diffs": 0, "checked": "verified", "url": null, "text": "x^{2}=1", "case_sensitive": true, "first_n": null, "last_n": null}
|
| 129 |
+
{"pdf": "3_pg634.pdf", "page": 1, "id": "3_pg634_equation_01", "type": "present", "max_diffs": 0, "checked": "verified", "url": null, "text": "\\dot{x} x=0", "case_sensitive": true, "first_n": null, "last_n": null}
|
| 130 |
+
{"pdf": "3_pg634.pdf", "page": 1, "id": "3_pg634_equation_02", "type": "present", "max_diffs": 0, "checked": "verified", "url": null, "text": "(y-x) x=0", "case_sensitive": true, "first_n": null, "last_n": null}
|
| 131 |
+
{"pdf": "3_pg634.pdf", "page": 1, "id": "3_pg634_equation_03", "type": "present", "max_diffs": 0, "checked": "verified", "url": null, "text": "(y - x) x=1", "case_sensitive": true, "first_n": null, "last_n": null}
|
| 132 |
+
{"pdf": "3_pg634.pdf", "page": 1, "id": "3_pg634_equation_04", "type": "present", "max_diffs": 0, "checked": "verified", "url": null, "text": "(y-x) x=0", "case_sensitive": true, "first_n": null, "last_n": null}
|
| 133 |
+
{"pdf": "3_pg634.pdf", "page": 1, "id": "3_pg634_equation_05", "type": "present", "max_diffs": 0, "checked": "verified", "url": null, "text": "y-x=\\frac{[x x]}{[x x] x}", "case_sensitive": true, "first_n": null, "last_n": null}
|
| 134 |
+
{"pdf": "3_pg634.pdf", "page": 1, "id": "3_pg634_equation_06", "type": "present", "max_diffs": 0, "checked": "verified", "url": null, "text": "\\xi_{1}==\\dot{x}", "case_sensitive": true, "first_n": null, "last_n": null}
|
| 135 |
+
{"pdf": "3_pg634.pdf", "page": 1, "id": "3_pg634_equation_07", "type": "present", "max_diffs": 0, "checked": "verified", "url": null, "text": "\\dot{x}^{2}=1", "case_sensitive": true, "first_n": null, "last_n": null}
|
| 136 |
+
{"pdf": "3_pg634.pdf", "page": 1, "id": "3_pg634_equation_08", "type": "present", "max_diffs": 0, "checked": "verified", "url": null, "text": "\\xi_{2}=x / h", "case_sensitive": true, "first_n": null, "last_n": null}
|
| 137 |
+
{"pdf": "3_pg634.pdf", "page": 1, "id": "3_pg634_equation_09", "type": "present", "max_diffs": 0, "checked": "verified", "url": null, "text": "\\xi_{3}=\\left[\\xi_{1} \\xi_{2}\\right], \\pm \\sqrt{\\dot{\\xi}_{3}^{2}}=1 / \\tau", "case_sensitive": true, "first_n": null, "last_n": null}
|
| 138 |
+
{"pdf": "3_pg634.pdf", "page": 1, "id": "3_pg634_equation_10", "type": "present", "max_diffs": 0, "checked": "verified", "url": null, "text": "\\xi_{1}=k \\xi_{2}", "case_sensitive": true, "first_n": null, "last_n": null}
|
| 139 |
+
{"pdf": "3_pg634.pdf", "page": 1, "id": "3_pg634_equation_11", "type": "present", "max_diffs": 0, "checked": "verified", "url": null, "text": "\\xi_{2}^{2}=1", "case_sensitive": true, "first_n": null, "last_n": null}
|
| 140 |
+
{"pdf": "3_pg634.pdf", "page": 1, "id": "3_pg634_equation_12", "type": "present", "max_diffs": 0, "checked": "verified", "url": null, "text": "\\xi_{3}^{2}=1", "case_sensitive": true, "first_n": null, "last_n": null}
|
| 141 |
+
{"pdf": "3_pg634.pdf", "page": 1, "id": "3_pg634_equation_13", "type": "present", "max_diffs": 0, "checked": null, "url": null, "text": "\\xi_{1} \\xi_{2}=\\xi_{2} \\xi_{3}=\\xi_{3} \\xi_{1}=0", "case_sensitive": true, "first_n": null, "last_n": null}
|
| 142 |
+
{"pdf": "3_pg634.pdf", "page": 1, "id": "3_pg634_equation_14", "type": "present", "max_diffs": 0, "checked": "verified", "url": null, "text": "\\xi_{3} \\xi_{1}=-\\xi_{1} \\xi_{3}=0", "case_sensitive": true, "first_n": null, "last_n": null}
|
| 143 |
+
{"pdf": "3_pg634.pdf", "page": 1, "id": "3_pg634_equation_15", "type": "present", "max_diffs": 0, "checked": "verified", "url": null, "text": "\\xi_{2} \\xi_{3}=0", "case_sensitive": true, "first_n": null, "last_n": null}
|
| 144 |
+
{"pdf": "3_pg634.pdf", "page": 1, "id": "3_pg634_equation_16", "type": "present", "max_diffs": 0, "checked": "verified", "url": null, "text": "\\xi_{3}=\\pm \\sqrt{\\left({\\xi}_{3}^{2}\\right)} \\xi_{2}=\\pm \\frac{\\xi_{2}}{\\tau}", "case_sensitive": true, "first_n": null, "last_n": null}
|
| 145 |
+
{"pdf": "3_pg634.pdf", "page": 1, "id": "3_pg634_equation_17", "type": "present", "max_diffs": 0, "checked": "verified", "url": null, "text": "\\xi=-\\frac{\\xi_{2}}{\\tau}", "case_sensitive": true, "first_n": null, "last_n": null}
|
| 146 |
+
{"pdf": "3_pg634.pdf", "page": 1, "id": "3_pg634_equation_18", "type": "present", "max_diffs": 0, "checked": "verified", "url": null, "text": "\\xi_{2} \\xi_{1}=-\\xi_{1} \\xi_{2}=-k", "case_sensitive": true, "first_n": null, "last_n": null}
|
| 147 |
+
{"pdf": "3_pg634.pdf", "page": 1, "id": "3_pg634_equation_19", "type": "present", "max_diffs": 0, "checked": "verified", "url": null, "text": "\\xi_{2} \\xi_{2}=0", "case_sensitive": true, "first_n": null, "last_n": null}
|
| 148 |
+
{"pdf": "3_pg634.pdf", "page": 1, "id": "3_pg634_equation_20", "type": "present", "max_diffs": 0, "checked": "verified", "url": null, "text": "\\xi_{2} \\xi_{3}=-\\xi_{3} \\xi_{2}=1 / \\tau", "case_sensitive": true, "first_n": null, "last_n": null}
|
| 149 |
+
{"pdf": "3_pg634.pdf", "page": 1, "id": "3_pg634_equation_21", "type": "present", "max_diffs": 0, "checked": "verified", "url": null, "text": "k \\xi_{2}-l^{2} \\xi_{1}+\\frac{k}{\\tau} \\xi_{3}", "case_sensitive": true, "first_n": null, "last_n": null}
|
| 150 |
+
{"pdf": "3_pg634.pdf", "page": 1, "id": "3_pg634_equation_22", "type": "present", "max_diffs": 0, "checked": "verified", "url": null, "text": "\\frac{k}{k^{2} \\tau} \\xi_{3}+\\frac{\\xi_{2}}{\\tau}", "case_sensitive": true, "first_n": null, "last_n": null}
|
| 151 |
+
{"pdf": "3_pg634.pdf", "page": 1, "id": "3_pg634_equation_23", "type": "present", "max_diffs": 0, "checked": "verified", "url": null, "text": "1 /|\\tau|=\\sqrt{{\\xi}_{3}^{2}}=0", "case_sensitive": true, "first_n": null, "last_n": null}
|
| 152 |
+
{"pdf": "3_pg634.pdf", "page": 1, "id": "3_pg634_equation_24", "type": "present", "max_diffs": 0, "checked": "verified", "url": null, "text": "v \\eta=\\xi_{1} \\eta", "case_sensitive": true, "first_n": null, "last_n": null}
|
| 153 |
+
{"pdf": "3_pg634.pdf", "page": 1, "id": "3_pg634_equation_25", "type": "present", "max_diffs": 0, "checked": "verified", "url": null, "text": "=\\xi_{1} \\xi_{3}=0", "case_sensitive": true, "first_n": null, "last_n": null}
|
| 154 |
+
{"pdf": "3_pg634.pdf", "page": 1, "id": "3_pg634_equation_26", "type": "present", "max_diffs": 0, "checked": "verified", "url": null, "text": "x \\eta=\\text { const. }", "case_sensitive": true, "first_n": null, "last_n": null}
|
| 155 |
+
{"pdf": "3_pg634.pdf", "page": 1, "id": "3_pg634_equation_27", "type": "present", "max_diffs": 0, "checked": "verified", "url": null, "text": "x=f(t)+s f^{\\prime}(t)", "case_sensitive": true, "first_n": null, "last_n": null}
|
| 156 |
+
{"pdf": "3_pg634.pdf", "page": 1, "id": "3_pg634_equation_28", "type": "present", "max_diffs": 0, "checked": "verified", "url": null, "text": "y=g(t)+s g^{\\prime}(t)", "case_sensitive": true, "first_n": null, "last_n": null}
|
| 157 |
+
{"pdf": "3_pg634.pdf", "page": 1, "id": "3_pg634_equation_29", "type": "present", "max_diffs": 0, "checked": "verified", "url": null, "text": "z=h(t)+s h^{\\prime}(t)", "case_sensitive": true, "first_n": null, "last_n": null}
|
| 158 |
+
{"pdf": "3_pg634.pdf", "page": 1, "id": "3_pg634_equation_30", "type": "present", "max_diffs": 0, "checked": "verified", "url": null, "text": "\\frac{\\partial^{2} z}{\\partial x^{2}}, \\frac{\\partial^{2} z}{\\partial x \\partial y}, \\frac{\\partial^{2} z}{\\partial y^{2}}", "case_sensitive": true, "first_n": null, "last_n": null}
|
| 159 |
+
{"pdf": "3_pg39.pdf", "page": 1, "id": "3_pg39_equation_00", "type": "present", "max_diffs": 0, "checked": "verified", "url": null, "text": "x' = ax+by+cz", "case_sensitive": true, "first_n": null, "last_n": null}
|
| 160 |
+
{"pdf": "3_pg39.pdf", "page": 1, "id": "3_pg39_equation_01", "type": "present", "max_diffs": 0, "checked": "verified", "url": null, "text": "x' = ax + by", "case_sensitive": true, "first_n": null, "last_n": null}
|
| 161 |
+
{"pdf": "3_pg39.pdf", "page": 1, "id": "3_pg39_equation_02", "type": "present", "max_diffs": 0, "checked": "verified", "url": null, "text": "y'= dx + ey + fz", "case_sensitive": true, "first_n": null, "last_n": null}
|
| 162 |
+
{"pdf": "3_pg39.pdf", "page": 1, "id": "3_pg39_equation_03", "type": "present", "max_diffs": 0, "checked": "verified", "url": null, "text": "y' = cx + dy", "case_sensitive": true, "first_n": null, "last_n": null}
|
| 163 |
+
{"pdf": "3_pg39.pdf", "page": 1, "id": "3_pg39_equation_04", "type": "present", "max_diffs": 0, "checked": "verified", "url": null, "text": "z'=gx+hy+kz", "case_sensitive": true, "first_n": null, "last_n": null}
|
| 164 |
+
{"pdf": "3_pg39.pdf", "page": 1, "id": "3_pg39_equation_05", "type": "present", "max_diffs": 0, "checked": "verified", "url": null, "text": "x = a'x' + b'y' + c'z'", "case_sensitive": true, "first_n": null, "last_n": null}
|
| 165 |
+
{"pdf": "3_pg39.pdf", "page": 1, "id": "3_pg39_equation_06", "type": "present", "max_diffs": 0, "checked": "verified", "url": null, "text": "x = a'x' + b'y'", "case_sensitive": true, "first_n": null, "last_n": null}
|
| 166 |
+
{"pdf": "3_pg39.pdf", "page": 1, "id": "3_pg39_equation_07", "type": "present", "max_diffs": 0, "checked": "verified", "url": null, "text": "y = d'x' + e'y'+ f'z'", "case_sensitive": true, "first_n": null, "last_n": null}
|
| 167 |
+
{"pdf": "3_pg39.pdf", "page": 1, "id": "3_pg39_equation_08", "type": "present", "max_diffs": 0, "checked": "verified", "url": null, "text": "y = c'x' + d'y'", "case_sensitive": true, "first_n": null, "last_n": null}
|
| 168 |
+
{"pdf": "3_pg39.pdf", "page": 1, "id": "3_pg39_equation_09", "type": "present", "max_diffs": 0, "checked": "verified", "url": null, "text": "z = g'x' + h'y' + k'z'", "case_sensitive": true, "first_n": null, "last_n": null}
|
| 169 |
+
{"pdf": "3_pg39.pdf", "page": 1, "id": "3_pg39_equation_10", "type": "present", "max_diffs": 0, "checked": "verified", "url": null, "text": "x = y = z = 0", "case_sensitive": true, "first_n": null, "last_n": null}
|
| 170 |
+
{"pdf": "3_pg39.pdf", "page": 1, "id": "3_pg39_equation_11", "type": "present", "max_diffs": 0, "checked": "verified", "url": null, "text": "x' = y' = z' = 0", "case_sensitive": true, "first_n": null, "last_n": null}
|
| 171 |
+
{"pdf": "3_pg39.pdf", "page": 1, "id": "3_pg39_equation_12", "type": "present", "max_diffs": 0, "checked": "verified", "url": null, "text": "Ax + By + Cz + D = 0", "case_sensitive": true, "first_n": null, "last_n": null}
|
| 172 |
+
{"pdf": "3_pg39.pdf", "page": 1, "id": "3_pg39_equation_13", "type": "present", "max_diffs": 0, "checked": "verified", "url": null, "text": "Ax + By + D = 0", "case_sensitive": true, "first_n": null, "last_n": null}
|
| 173 |
+
{"pdf": "3_pg39.pdf", "page": 1, "id": "3_pg39_equation_14", "type": "present", "max_diffs": 0, "checked": "verified", "url": null, "text": "A(a'x' + b'y + c'z') + B(d'x' + e'y' + f'z')", "case_sensitive": true, "first_n": null, "last_n": null}
|
| 174 |
+
{"pdf": "3_pg39.pdf", "page": 1, "id": "3_pg39_equation_15", "type": "present", "max_diffs": 0, "checked": "verified", "url": null, "text": "+ C(g'x' + h'y' + k'x') + D = 0", "case_sensitive": true, "first_n": null, "last_n": null}
|
| 175 |
+
{"pdf": "3_pg39.pdf", "page": 1, "id": "3_pg39_equation_16", "type": "present", "max_diffs": 0, "checked": "verified", "url": null, "text": "A(a'x' + b'y') + B(c'x' + d'y') + D = 0", "case_sensitive": true, "first_n": null, "last_n": null}
|
| 176 |
+
{"pdf": "3_pg39.pdf", "page": 1, "id": "3_pg39_equation_17", "type": "present", "max_diffs": 0, "checked": "verified", "url": null, "text": "A' = a'A + d'B + g'O", "case_sensitive": true, "first_n": null, "last_n": null}
|
| 177 |
+
{"pdf": "3_pg39.pdf", "page": 1, "id": "3_pg39_equation_18", "type": "present", "max_diffs": 0, "checked": "verified", "url": null, "text": "B' = b'A + e'B + h'O", "case_sensitive": true, "first_n": null, "last_n": null}
|
| 178 |
+
{"pdf": "3_pg39.pdf", "page": 1, "id": "3_pg39_equation_19", "type": "present", "max_diffs": 0, "checked": "verified", "url": null, "text": "O'=c'A + f'B + k'O", "case_sensitive": true, "first_n": null, "last_n": null}
|
| 179 |
+
{"pdf": "3_pg39.pdf", "page": 1, "id": "3_pg39_equation_20", "type": "present", "max_diffs": 0, "checked": "verified", "url": null, "text": "A'=a'A+c'B", "case_sensitive": true, "first_n": null, "last_n": null}
|
| 180 |
+
{"pdf": "3_pg39.pdf", "page": 1, "id": "3_pg39_equation_21", "type": "present", "max_diffs": 0, "checked": "verified", "url": null, "text": "B'=b'A+d'B", "case_sensitive": true, "first_n": null, "last_n": null}
|
| 181 |
+
{"pdf": "3_pg39.pdf", "page": 1, "id": "3_pg39_equation_22", "type": "present", "max_diffs": 0, "checked": "verified", "url": null, "text": "b'A + e'B+h'O=0", "case_sensitive": true, "first_n": null, "last_n": null}
|
| 182 |
+
{"pdf": "3_pg39.pdf", "page": 1, "id": "3_pg39_equation_23", "type": "present", "max_diffs": 0, "checked": "verified", "url": null, "text": "a'A + c'B = 0", "case_sensitive": true, "first_n": null, "last_n": null}
|
| 183 |
+
{"pdf": "3_pg39.pdf", "page": 1, "id": "3_pg39_equation_24", "type": "present", "max_diffs": 0, "checked": "verified", "url": null, "text": "b'A + d'B = 0", "case_sensitive": true, "first_n": null, "last_n": null}
|
| 184 |
+
{"pdf": "3_pg39.pdf", "page": 1, "id": "3_pg39_equation_25", "type": "present", "max_diffs": 0, "checked": "verified", "url": null, "text": "c'A + f'B + h'O = 0", "case_sensitive": true, "first_n": null, "last_n": null}
|
| 185 |
+
{"pdf": "3_pg204.pdf", "page": 1, "id": "3_pg204_equation_00", "type": "present", "max_diffs": 0, "checked": "verified", "url": null, "text": "f_{x} + \\lambda \\phi_{x} + \\mu \\psi_{x} = 0", "case_sensitive": true, "first_n": null, "last_n": null}
|
| 186 |
+
{"pdf": "3_pg204.pdf", "page": 1, "id": "3_pg204_equation_01", "type": "present", "max_diffs": 0, "checked": "verified", "url": null, "text": "\\phi_{y} + \\phi_{z} \\frac{\\partial z}{\\partial y} + \\phi_{t} \\frac{\\partial t}{\\partial y} = 0", "case_sensitive": true, "first_n": null, "last_n": null}
|
| 187 |
+
{"pdf": "3_pg204.pdf", "page": 1, "id": "3_pg204_equation_02", "type": "present", "max_diffs": 0, "checked": "verified", "url": null, "text": "\\psi_{y} + \\psi_{z} \\frac{\\partial z}{\\partial y} + \\psi_{t} \\frac{\\partial t}{\\partial y} = 0", "case_sensitive": true, "first_n": null, "last_n": null}
|
| 188 |
+
{"pdf": "3_pg204.pdf", "page": 1, "id": "3_pg204_equation_03", "type": "present", "max_diffs": 0, "checked": "verified", "url": null, "text": "f_{y} + f{z} \\frac{\\partial z}{\\partial y} + f_{t} \\frac{\\partial t}{\\partial y} = 0,", "case_sensitive": true, "first_n": null, "last_n": null}
|
| 189 |
+
{"pdf": "3_pg204.pdf", "page": 1, "id": "3_pg204_equation_04", "type": "present", "max_diffs": 0, "checked": "verified", "url": null, "text": "f_{y} + \\lambda \\phi_{y} + \\mu \\psi_{y} = 0", "case_sensitive": true, "first_n": null, "last_n": null}
|
| 190 |
+
{"pdf": "3_pg204.pdf", "page": 1, "id": "3_pg204_equation_05", "type": "present", "max_diffs": 0, "checked": "verified", "url": null, "text": "\\phi(x, y, z, t) = 0", "case_sensitive": true, "first_n": null, "last_n": null}
|
| 191 |
+
{"pdf": "3_pg204.pdf", "page": 1, "id": "3_pg204_equation_06", "type": "present", "max_diffs": 0, "checked": "verified", "url": null, "text": "\\psi(x, y, z, t) = 0", "case_sensitive": true, "first_n": null, "last_n": null}
|
| 192 |
+
{"pdf": "3_pg204.pdf", "page": 1, "id": "3_pg204_equation_07", "type": "present", "max_diffs": 0, "checked": "verified", "url": null, "text": "\\frac{\\partial(\\phi, \\psi)}{\\partial(z, t)}", "case_sensitive": true, "first_n": null, "last_n": null}
|
| 193 |
+
{"pdf": "3_pg204.pdf", "page": 1, "id": "3_pg204_equation_08", "type": "present", "max_diffs": 0, "checked": "verified", "url": null, "text": "f_{x} + \\lambda \\phi_{x} + \\mu \\psi_{x} = 0", "case_sensitive": true, "first_n": null, "last_n": null}
|
| 194 |
+
{"pdf": "3_pg204.pdf", "page": 1, "id": "3_pg204_equation_09", "type": "present", "max_diffs": 0, "checked": "verified", "url": null, "text": "f_{y} + \\lambda \\phi_{y} + \\mu \\psi_{y} = 0", "case_sensitive": true, "first_n": null, "last_n": null}
|
| 195 |
+
{"pdf": "3_pg204.pdf", "page": 1, "id": "3_pg204_equation_10", "type": "present", "max_diffs": 0, "checked": "verified", "url": null, "text": "f_{z} + \\lambda \\phi_{z} + \\mu \\psi_{z} = 0", "case_sensitive": true, "first_n": null, "last_n": null}
|
| 196 |
+
{"pdf": "3_pg204.pdf", "page": 1, "id": "3_pg204_equation_11", "type": "present", "max_diffs": 0, "checked": "verified", "url": null, "text": "f_{t} + \\lambda \\phi_{t} + \\mu \\psi_{t} = 0", "case_sensitive": true, "first_n": null, "last_n": null}
|
| 197 |
+
{"pdf": "3_pg204.pdf", "page": 1, "id": "3_pg204_equation_12", "type": "present", "max_diffs": 0, "checked": "verified", "url": null, "text": "\\frac{\\partial(\\phi, \\psi)}{\\partial(z, t)} \\neq 0", "case_sensitive": true, "first_n": null, "last_n": null}
|
| 198 |
+
{"pdf": "3_pg204.pdf", "page": 1, "id": "3_pg204_equation_13", "type": "present", "max_diffs": 0, "checked": "verified", "url": null, "text": "\\frac{\\partial(\\phi, \\psi)}{\\partial(x, y)}, \\frac{\\partial(\\phi, \\psi)}{\\partial(x, z)}, \\dots, \\frac{\\partial(\\phi, \\psi)}{\\partial(z, t)}", "case_sensitive": true, "first_n": null, "last_n": null}
|
| 199 |
+
{"pdf": "3_pg101.pdf", "page": 1, "id": "3_pg101_equation_00", "type": "present", "max_diffs": 0, "checked": "rejected", "url": null, "text": "\\sqrt{\\{(\\xi{1} - x{1})^{2} + (\\xi{2} - x{2})^{2} + (\\xi{3} - x{3})^{2}\\}", "case_sensitive": true, "first_n": null, "last_n": null}
|
| 200 |
+
{"pdf": "3_pg101.pdf", "page": 1, "id": "3_pg101_equation_01", "type": "present", "max_diffs": 0, "checked": "rejected", "url": null, "text": "\\frac{1}{r} = \\frac{1}{\\sqrt{\\{(\\xi{1} - x{1})^{2} + (\\xi{2} - x{2})^{2} + (\\xi{3} - x{3})^{2}\\}}", "case_sensitive": true, "first_n": null, "last_n": null}
|
| 201 |
+
{"pdf": "3_pg101.pdf", "page": 1, "id": "3_pg101_equation_02", "type": "present", "max_diffs": 0, "checked": "verified", "url": null, "text": "\\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}}", "case_sensitive": true, "first_n": null, "last_n": null}
|
| 202 |
+
{"pdf": "3_pg101.pdf", "page": 1, "id": "3_pg101_equation_03", "type": "present", "max_diffs": 0, "checked": "verified", "url": null, "text": "\\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}}", "case_sensitive": true, "first_n": null, "last_n": null}
|
| 203 |
+
{"pdf": "3_pg101.pdf", "page": 1, "id": "3_pg101_equation_04", "type": "present", "max_diffs": 0, "checked": "verified", "url": null, "text": "\\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}}", "case_sensitive": true, "first_n": null, "last_n": null}
|
| 204 |
+
{"pdf": "3_pg270.pdf", "page": 1, "id": "3_pg270_equation_00", "type": "present", "max_diffs": 0, "checked": "verified", "url": null, "text": "\\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", "case_sensitive": true, "first_n": null, "last_n": null}
|
| 205 |
+
{"pdf": "3_pg270.pdf", "page": 1, "id": "3_pg270_equation_01", "type": "present", "max_diffs": 0, "checked": "verified", "url": null, "text": "\\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}", "case_sensitive": true, "first_n": null, "last_n": null}
|
| 206 |
+
{"pdf": "3_pg270.pdf", "page": 1, "id": "3_pg270_equation_02", "type": "present", "max_diffs": 0, "checked": "verified", "url": null, "text": "\\left(2\\int_{0}^{\\infty}e^{-x^{2}}dx\\right)^{2}=\\pi", "case_sensitive": true, "first_n": null, "last_n": null}
|
| 207 |
+
{"pdf": "3_pg270.pdf", "page": 1, "id": "3_pg270_equation_03", "type": "present", "max_diffs": 0, "checked": "verified", "url": null, "text": "\\int_{0}^{\\infty}e^{-x^{2}}dx = \\frac{1}{2}\\sqrt{\\pi}", "case_sensitive": true, "first_n": null, "last_n": null}
|
| 208 |
+
{"pdf": "3_pg94.pdf", "page": 1, "id": "3_pg94_equation_00", "type": "present", "max_diffs": 0, "checked": "verified", "url": null, "text": "\\omega_1 = \\alpha_1u_1 + \\beta_1u_2 + \\gamma_1u_3", "case_sensitive": true, "first_n": null, "last_n": null}
|
| 209 |
+
{"pdf": "3_pg94.pdf", "page": 1, "id": "3_pg94_equation_01", "type": "present", "max_diffs": 0, "checked": "verified", "url": null, "text": "\\omega_2 = \\alpha_2u_1 + \\beta_2u_2 + \\gamma_2u_3", "case_sensitive": true, "first_n": null, "last_n": null}
|
| 210 |
+
{"pdf": "3_pg94.pdf", "page": 1, "id": "3_pg94_equation_02", "type": "present", "max_diffs": 0, "checked": "verified", "url": null, "text": "\\omega_3 = \\alpha_3u_1 + \\beta_3u_2 + \\gamma_3u_3", "case_sensitive": true, "first_n": null, "last_n": null}
|
| 211 |
+
{"pdf": "3_pg94.pdf", "page": 1, "id": "3_pg94_equation_03", "type": "present", "max_diffs": 0, "checked": "verified", "url": null, "text": "u_1 = \\alpha_1\\omega_1 + \\alpha_2\\omega_2 + \\alpha_3\\omega_3", "case_sensitive": true, "first_n": null, "last_n": null}
|
| 212 |
+
{"pdf": "3_pg94.pdf", "page": 1, "id": "3_pg94_equation_04", "type": "present", "max_diffs": 0, "checked": "verified", "url": null, "text": "u_2 = \\beta_1\\omega_1 + \\beta_2\\omega_2 + \\beta_3\\omega_3", "case_sensitive": true, "first_n": null, "last_n": null}
|
| 213 |
+
{"pdf": "3_pg94.pdf", "page": 1, "id": "3_pg94_equation_05", "type": "present", "max_diffs": 0, "checked": "verified", "url": null, "text": "u_3 = \\gamma_1\\omega_1 + \\gamma_2\\omega_2 + \\gamma_3\\omega_3", "case_sensitive": true, "first_n": null, "last_n": null}
|
| 214 |
+
{"pdf": "3_pg94.pdf", "page": 1, "id": "3_pg94_equation_06", "type": "present", "max_diffs": 0, "checked": "verified", "url": null, "text": "u = f(x_1, x_2, x_3)", "case_sensitive": true, "first_n": null, "last_n": null}
|
| 215 |
+
{"pdf": "3_pg94.pdf", "page": 1, "id": "3_pg94_equation_07", "type": "present", "max_diffs": 0, "checked": "verified", "url": null, "text": "|u|^2 = u_1^2 + u_2^2 + u_3^2", "case_sensitive": true, "first_n": null, "last_n": null}
|
| 216 |
+
{"pdf": "3_pg148.pdf", "page": 1, "id": "3_pg148_equation_00", "type": "present", "max_diffs": 0, "checked": "verified", "url": null, "text": "\\xi = r = \\sqrt{(x^2 + y^2)}", "case_sensitive": true, "first_n": null, "last_n": null}
|
| 217 |
+
{"pdf": "3_pg148.pdf", "page": 1, "id": "3_pg148_equation_01", "type": "present", "max_diffs": 0, "checked": "verified", "url": null, "text": "\\eta = \\theta = \\arctan (y/x)", "case_sensitive": true, "first_n": null, "last_n": null}
|
| 218 |
+
{"pdf": "3_pg148.pdf", "page": 1, "id": "3_pg148_equation_02", "type": "present", "max_diffs": 0, "checked": "verified", "url": null, "text": "(0\\leq\\theta <2\\pi)", "case_sensitive": true, "first_n": null, "last_n": null}
|
| 219 |
+
{"pdf": "3_pg148.pdf", "page": 1, "id": "3_pg148_equation_03", "type": "present", "max_diffs": 0, "checked": "verified", "url": null, "text": "\\xi = \\phi(x, y)", "case_sensitive": true, "first_n": null, "last_n": null}
|
| 220 |
+
{"pdf": "3_pg148.pdf", "page": 1, "id": "3_pg148_equation_04", "type": "present", "max_diffs": 0, "checked": "verified", "url": null, "text": "\\eta = \\psi(x, y)", "case_sensitive": true, "first_n": null, "last_n": null}
|
| 221 |
+
{"pdf": "3_pg148.pdf", "page": 1, "id": "3_pg148_equation_05", "type": "present", "max_diffs": 0, "checked": "verified", "url": null, "text": "\\phi(x, y) = const", "case_sensitive": true, "first_n": null, "last_n": null}
|
| 222 |
+
{"pdf": "3_pg148.pdf", "page": 1, "id": "3_pg148_equation_06", "type": "present", "max_diffs": 0, "checked": "verified", "url": null, "text": "\\psi(x, y) = const", "case_sensitive": true, "first_n": null, "last_n": null}
|
| 223 |
+
{"pdf": "3_pg148.pdf", "page": 1, "id": "3_pg148_equation_07", "type": "present", "max_diffs": 0, "checked": "verified", "url": null, "text": "x = g(\\xi, \\eta)", "case_sensitive": true, "first_n": null, "last_n": null}
|
| 224 |
+
{"pdf": "3_pg148.pdf", "page": 1, "id": "3_pg148_equation_08", "type": "present", "max_diffs": 0, "checked": "verified", "url": null, "text": "y = h(\\xi, \\eta)", "case_sensitive": true, "first_n": null, "last_n": null}
|
| 225 |
+
{"pdf": "3_pg148.pdf", "page": 1, "id": "3_pg148_equation_09", "type": "present", "max_diffs": 0, "checked": "verified", "url": null, "text": "\\xi = \\phi(x, y)", "case_sensitive": true, "first_n": null, "last_n": null}
|
| 226 |
+
{"pdf": "3_pg148.pdf", "page": 1, "id": "3_pg148_equation_10", "type": "present", "max_diffs": 0, "checked": "verified", "url": null, "text": "\\eta = \\psi(x, y)", "case_sensitive": true, "first_n": null, "last_n": null}
|
| 227 |
+
{"pdf": "3_pg251.pdf", "page": 1, "id": "3_pg251_equation_00", "type": "present", "max_diffs": 0, "checked": "verified", "url": null, "text": "\\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", "case_sensitive": true, "first_n": null, "last_n": null}
|
| 228 |
+
{"pdf": "3_pg251.pdf", "page": 1, "id": "3_pg251_equation_01", "type": "present", "max_diffs": 0, "checked": "verified", "url": null, "text": "\\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", "case_sensitive": true, "first_n": null, "last_n": null}
|
| 229 |
+
{"pdf": "3_pg251.pdf", "page": 1, "id": "3_pg251_equation_02", "type": "present", "max_diffs": 0, "checked": "verified", "url": null, "text": "\\lim_{\\delta \\to 0} \\iint_{R} f(x, y) dS = \\iint_{R} f(x, y) dS", "case_sensitive": true, "first_n": null, "last_n": null}
|
| 230 |
+
{"pdf": "3_pg251.pdf", "page": 1, "id": "3_pg251_equation_03", "type": "present", "max_diffs": 0, "checked": "verified", "url": null, "text": "\\iint_{R} f(x, y) dS = \\int_{x_{0}}^{x_{1}} dx \\int_{\\psi{1}(x)}^{\\psi_{2}(x)} f(x, y) dy", "case_sensitive": true, "first_n": null, "last_n": null}
|
| 231 |
+
{"pdf": "3_pg251.pdf", "page": 1, "id": "3_pg251_equation_04", "type": "present", "max_diffs": 0, "checked": "verified", "url": null, "text": "\\iint f(x, y) dy", "case_sensitive": true, "first_n": null, "last_n": null}
|
| 232 |
+
{"pdf": "2_pg39.pdf", "page": 1, "id": "2_pg39_equation_00", "type": "present", "max_diffs": 0, "checked": "verified", "url": null, "text": "dy = \\frac{(x^2 - 1) d (x^2 + 1) - (x^2 + 1) d (x^2 - 1)}{(x^2 - 1)^2}", "case_sensitive": true, "first_n": null, "last_n": null}
|
| 233 |
+
{"pdf": "2_pg39.pdf", "page": 1, "id": "2_pg39_equation_01", "type": "present", "max_diffs": 0, "checked": "verified", "url": null, "text": " = \\frac{(x^2 - 1) 2 x dx - (x^2 + 1) 2 x dx}{(x^2 - 1)^2}", "case_sensitive": true, "first_n": null, "last_n": null}
|
| 234 |
+
{"pdf": "2_pg39.pdf", "page": 1, "id": "2_pg39_equation_02", "type": "present", "max_diffs": 0, "checked": "verified", "url": null, "text": "= -\\frac{4 x dx}{(x^2 - 1)^2}", "case_sensitive": true, "first_n": null, "last_n": null}
|
| 235 |
+
{"pdf": "2_pg39.pdf", "page": 1, "id": "2_pg39_equation_03", "type": "present", "max_diffs": 0, "checked": "verified", "url": null, "text": "y = \\sqrt{x^2 - 1}", "case_sensitive": true, "first_n": null, "last_n": null}
|
| 236 |
+
{"pdf": "2_pg39.pdf", "page": 1, "id": "2_pg39_equation_04", "type": "present", "max_diffs": 0, "checked": "verified", "url": null, "text": "u = x^2 - 1", "case_sensitive": true, "first_n": null, "last_n": null}
|
| 237 |
+
{"pdf": "2_pg39.pdf", "page": 1, "id": "2_pg39_equation_05", "type": "present", "max_diffs": 0, "checked": "verified", "url": null, "text": "\\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)", "case_sensitive": true, "first_n": null, "last_n": null}
|
| 238 |
+
{"pdf": "2_pg39.pdf", "page": 1, "id": "2_pg39_equation_06", "type": "present", "max_diffs": 0, "checked": "verified", "url": null, "text": "= \\frac{1}{2} (x^2 - 1)^{-\\frac{1}{2}} (2 x) = \\frac{x}{\\sqrt{x^2 - 1}}", "case_sensitive": true, "first_n": null, "last_n": null}
|
| 239 |
+
{"pdf": "2_pg39.pdf", "page": 1, "id": "2_pg39_equation_07", "type": "present", "max_diffs": 0, "checked": "verified", "url": null, "text": "x^2 + xy - y^2 = 1", "case_sensitive": true, "first_n": null, "last_n": null}
|
| 240 |
+
{"pdf": "2_pg39.pdf", "page": 1, "id": "2_pg39_equation_08", "type": "present", "max_diffs": 0, "checked": "verified", "url": null, "text": "d (x^2) + d (xy) - d (y^2) = d (1) = 0", "case_sensitive": true, "first_n": null, "last_n": null}
|
| 241 |
+
{"pdf": "2_pg39.pdf", "page": 1, "id": "2_pg39_equation_09", "type": "present", "max_diffs": 0, "checked": "verified", "url": null, "text": "2 x dx + x dy + y dx - 2 y dy = 0", "case_sensitive": true, "first_n": null, "last_n": null}
|
| 242 |
+
{"pdf": "2_pg39.pdf", "page": 1, "id": "2_pg39_equation_10", "type": "present", "max_diffs": 0, "checked": "verified", "url": null, "text": "(2 x + y) dx + (x - 2 y) dy = 0", "case_sensitive": true, "first_n": null, "last_n": null}
|
| 243 |
+
{"pdf": "2_pg39.pdf", "page": 1, "id": "2_pg39_equation_11", "type": "present", "max_diffs": 0, "checked": "verified", "url": null, "text": "\\frac{dy}{dx} = \\frac{2 x + y}{2 y - x}", "case_sensitive": true, "first_n": null, "last_n": null}
|
| 244 |
+
{"pdf": "2_pg39.pdf", "page": 1, "id": "2_pg39_equation_12", "type": "present", "max_diffs": 0, "checked": "verified", "url": null, "text": "x = t + \\frac{1}{t}", "case_sensitive": true, "first_n": null, "last_n": null}
|
| 245 |
+
{"pdf": "2_pg39.pdf", "page": 1, "id": "2_pg39_equation_13", "type": "present", "max_diffs": 0, "checked": "verified", "url": null, "text": "dx = dt - \\frac{dt}{t^2}", "case_sensitive": true, "first_n": null, "last_n": null}
|
| 246 |
+
{"pdf": "2_pg39.pdf", "page": 1, "id": "2_pg39_equation_14", "type": "present", "max_diffs": 0, "checked": "verified", "url": null, "text": "\\frac{dy}{dx} = \\frac{1 + \\frac{1}{t^2}}{1 - \\frac{1}{t^2}} = \\frac{t^2 + 1}{t^2 - 1}", "case_sensitive": true, "first_n": null, "last_n": null}
|
| 247 |
+
{"pdf": "2_pg39.pdf", "page": 1, "id": "2_pg39_equation_15", "type": "present", "max_diffs": 0, "checked": "verified", "url": null, "text": "y = (\\frac{1 - x}{1 + x})^{\\frac{1}{3}}", "case_sensitive": true, "first_n": null, "last_n": null}
|
| 248 |
+
{"pdf": "2_pg39.pdf", "page": 1, "id": "2_pg39_equation_16", "type": "present", "max_diffs": 0, "checked": "verified", "url": null, "text": "x = 0.2", "case_sensitive": true, "first_n": null, "last_n": null}
|
| 249 |
+
{"pdf": "2_pg65.pdf", "page": 1, "id": "2_pg65_equation_00", "type": "present", "max_diffs": 0, "checked": "verified", "url": null, "text": "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", "case_sensitive": true, "first_n": null, "last_n": null}
|
| 250 |
+
{"pdf": "2_pg65.pdf", "page": 1, "id": "2_pg65_equation_01", "type": "present", "max_diffs": 0, "checked": "verified", "url": null, "text": "tan u = \\frac{sin u}{cos u}", "case_sensitive": true, "first_n": null, "last_n": null}
|
| 251 |
+
{"pdf": "2_pg65.pdf", "page": 1, "id": "2_pg65_equation_02", "type": "present", "max_diffs": 0, "checked": "verified", "url": null, "text": "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}", "case_sensitive": true, "first_n": null, "last_n": null}
|
| 252 |
+
{"pdf": "2_pg65.pdf", "page": 1, "id": "2_pg65_equation_03", "type": "present", "max_diffs": 0, "checked": "verified", "url": null, "text": "= sec^2 u du", "case_sensitive": true, "first_n": null, "last_n": null}
|
| 253 |
+
{"pdf": "2_pg65.pdf", "page": 1, "id": "2_pg65_equation_04", "type": "present", "max_diffs": 0, "checked": "verified", "url": null, "text": "cot u = \\frac{cos u}{sin u}", "case_sensitive": true, "first_n": null, "last_n": null}
|
| 254 |
+
{"pdf": "2_pg65.pdf", "page": 1, "id": "2_pg65_equation_05", "type": "present", "max_diffs": 0, "checked": "verified", "url": null, "text": "y = sin^2 (x^2 + 3)", "case_sensitive": true, "first_n": null, "last_n": null}
|
| 255 |
+
{"pdf": "2_pg65.pdf", "page": 1, "id": "2_pg65_equation_06", "type": "present", "max_diffs": 0, "checked": "verified", "url": null, "text": "sin^2 (x^2 + 3) = [sin (x^2 +3)]^2", "case_sensitive": true, "first_n": null, "last_n": null}
|
| 256 |
+
{"pdf": "2_pg65.pdf", "page": 1, "id": "2_pg65_equation_07", "type": "present", "max_diffs": 0, "checked": "verified", "url": null, "text": "dy = 2 sin (x^2 + 3) d sin (x^2 + 3)", "case_sensitive": true, "first_n": null, "last_n": null}
|
| 257 |
+
{"pdf": "2_pg65.pdf", "page": 1, "id": "2_pg65_equation_08", "type": "present", "max_diffs": 0, "checked": "verified", "url": null, "text": "= 2 sin (x^2 + 3) cos (x^2 + 3) d (x^2 + 3)", "case_sensitive": true, "first_n": null, "last_n": null}
|
| 258 |
+
{"pdf": "2_pg65.pdf", "page": 1, "id": "2_pg65_equation_09", "type": "present", "max_diffs": 0, "checked": "verified", "url": null, "text": "= 4 x sin (x^2 + 3) cos (x^2 + 3) dx", "case_sensitive": true, "first_n": null, "last_n": null}
|
| 259 |
+
{"pdf": "2_pg65.pdf", "page": 1, "id": "2_pg65_equation_10", "type": "present", "max_diffs": 0, "checked": "verified", "url": null, "text": "y = sec 2 x tan 2 x", "case_sensitive": true, "first_n": null, "last_n": null}
|
| 260 |
+
{"pdf": "2_pg65.pdf", "page": 1, "id": "2_pg65_equation_11", "type": "present", "max_diffs": 0, "checked": "verified", "url": null, "text": "\\frac{dy}{dx} = sec 2 x \\frac{d}{dx} tan 2 x + tan 2 x \\frac{d}{dx} sec 2 x", "case_sensitive": true, "first_n": null, "last_n": null}
|
| 261 |
+
{"pdf": "2_pg65.pdf", "page": 1, "id": "2_pg65_equation_12", "type": "present", "max_diffs": 0, "checked": "verified", "url": null, "text": "= sec 2 x sec^2 2 x (2) + tan-2 x sec 2 x tan 2 x (2)", "case_sensitive": true, "first_n": null, "last_n": null}
|
| 262 |
+
{"pdf": "2_pg65.pdf", "page": 1, "id": "2_pg65_equation_13", "type": "present", "max_diffs": 0, "checked": "verified", "url": null, "text": "= 2 sec 2 x (sec^2 2 x + tan^2 2 x)", "case_sensitive": true, "first_n": null, "last_n": null}
|
| 263 |
+
{"pdf": "2_pg65.pdf", "page": 1, "id": "2_pg65_equation_14", "type": "present", "max_diffs": 0, "checked": "verified", "url": null, "text": " \\quad \\frac{dy}{dx} = 6 (cos 3 x - sin 2 x).", "case_sensitive": true, "first_n": null, "last_n": null}
|
| 264 |
+
{"pdf": "2_pg189.pdf", "page": 1, "id": "2_pg189_equation_00", "type": "present", "max_diffs": 0, "checked": "verified", "url": null, "text": "\\frac{dx}{dt} = C_1", "case_sensitive": true, "first_n": null, "last_n": null}
|
| 265 |
+
{"pdf": "2_pg189.pdf", "page": 1, "id": "2_pg189_equation_01", "type": "present", "max_diffs": 0, "checked": "verified", "url": null, "text": "\\frac{dy}{dt} = -gt + C_2", "case_sensitive": true, "first_n": null, "last_n": null}
|
| 266 |
+
{"pdf": "2_pg189.pdf", "page": 1, "id": "2_pg189_equation_02", "type": "present", "max_diffs": 0, "checked": "verified", "url": null, "text": "\\frac{dx}{dt}", "case_sensitive": true, "first_n": null, "last_n": null}
|
| 267 |
+
{"pdf": "2_pg189.pdf", "page": 1, "id": "2_pg189_equation_03", "type": "present", "max_diffs": 0, "checked": "verified", "url": null, "text": "C_2 = v_0 \\sin{\\alpha}", "case_sensitive": true, "first_n": null, "last_n": null}
|
| 268 |
+
{"pdf": "2_pg189.pdf", "page": 1, "id": "2_pg189_equation_04", "type": "present", "max_diffs": 0, "checked": "verified", "url": null, "text": "C_1=v_0 \\cos{\\alpha}", "case_sensitive": true, "first_n": null, "last_n": null}
|
| 269 |
+
{"pdf": "2_pg189.pdf", "page": 1, "id": "2_pg189_equation_05", "type": "present", "max_diffs": 0, "checked": "verified", "url": null, "text": "\\frac{dx}{dt} = v_0 \\cos{\\alpha}", "case_sensitive": true, "first_n": null, "last_n": null}
|
| 270 |
+
{"pdf": "2_pg189.pdf", "page": 1, "id": "2_pg189_equation_06", "type": "present", "max_diffs": 0, "checked": "verified", "url": null, "text": "\\frac{dy}{dt} = v_0 \\sin{\\alpha} - gt", "case_sensitive": true, "first_n": null, "last_n": null}
|
| 271 |
+
{"pdf": "2_pg189.pdf", "page": 1, "id": "2_pg189_equation_07", "type": "present", "max_diffs": 0, "checked": "verified", "url": null, "text": "x = v_0 t \\cos{\\alpha}", "case_sensitive": true, "first_n": null, "last_n": null}
|
| 272 |
+
{"pdf": "2_pg189.pdf", "page": 1, "id": "2_pg189_equation_08", "type": "present", "max_diffs": 0, "checked": "verified", "url": null, "text": "y = v_0 t \\sin{\\alpha} - \\frac{1}{2} gt^2", "case_sensitive": true, "first_n": null, "last_n": null}
|
| 273 |
+
{"pdf": "2_pg189.pdf", "page": 1, "id": "2_pg189_equation_09", "type": "present", "max_diffs": 0, "checked": "verified", "url": null, "text": "\\frac{dy}{dx} = f(x)", "case_sensitive": true, "first_n": null, "last_n": null}
|
| 274 |
+
{"pdf": "2_pg189.pdf", "page": 1, "id": "2_pg189_equation_10", "type": "present", "max_diffs": 0, "checked": "verified", "url": null, "text": "dy = f(x) dx", "case_sensitive": true, "first_n": null, "last_n": null}
|
| 275 |
+
{"pdf": "2_pg189.pdf", "page": 1, "id": "2_pg189_equation_11", "type": "present", "max_diffs": 0, "checked": "verified", "url": null, "text": "y = \\int f(x) dx + C", "case_sensitive": true, "first_n": null, "last_n": null}
|
| 276 |
+
{"pdf": "2_pg349.pdf", "page": 1, "id": "2_pg349_equation_00", "type": "present", "max_diffs": 0, "checked": "verified", "url": null, "text": "x^{2} + y^{2} = 2 ax", "case_sensitive": true, "first_n": null, "last_n": null}
|
| 277 |
+
{"pdf": "2_pg349.pdf", "page": 1, "id": "2_pg349_equation_01", "type": "present", "max_diffs": 0, "checked": "verified", "url": null, "text": "z^{2} = x^{2} + y^{2}", "case_sensitive": true, "first_n": null, "last_n": null}
|
| 278 |
+
{"pdf": "2_pg349.pdf", "page": 1, "id": "2_pg349_equation_02", "type": "present", "max_diffs": 0, "checked": "verified", "url": null, "text": "y dx + (x - xy) dy = 0", "case_sensitive": true, "first_n": null, "last_n": null}
|
| 279 |
+
{"pdf": "2_pg349.pdf", "page": 1, "id": "2_pg349_equation_03", "type": "present", "max_diffs": 0, "checked": "verified", "url": null, "text": "sin x sin y dx + cos x cos y dy = 0", "case_sensitive": true, "first_n": null, "last_n": null}
|
| 280 |
+
{"pdf": "2_pg349.pdf", "page": 1, "id": "2_pg349_equation_04", "type": "present", "max_diffs": 0, "checked": "verified", "url": null, "text": "(2 xy - y^{2} + 6 x^{2}) dx + (3y^{2} + x^{2} - 2 xy) dy = 0", "case_sensitive": true, "first_n": null, "last_n": null}
|
| 281 |
+
{"pdf": "2_pg349.pdf", "page": 1, "id": "2_pg349_equation_05", "type": "present", "max_diffs": 0, "checked": "verified", "url": null, "text": "x \\frac{dy}{dx} + y = x^{3}y", "case_sensitive": true, "first_n": null, "last_n": null}
|
| 282 |
+
{"pdf": "2_pg349.pdf", "page": 1, "id": "2_pg349_equation_06", "type": "present", "max_diffs": 0, "checked": "verified", "url": null, "text": "x \\frac{dy}{dx} + y = cot x", "case_sensitive": true, "first_n": null, "last_n": null}
|
| 283 |
+
{"pdf": "2_pg349.pdf", "page": 1, "id": "2_pg349_equation_07", "type": "present", "max_diffs": 0, "checked": "verified", "url": null, "text": "x dy - (y + e^{\\frac{1}{x}}) dx = 0", "case_sensitive": true, "first_n": null, "last_n": null}
|
| 284 |
+
{"pdf": "2_pg349.pdf", "page": 1, "id": "2_pg349_equation_08", "type": "present", "max_diffs": 0, "checked": "verified", "url": null, "text": "(1 + x^{2}) dy + (xy + x) dx = 0", "case_sensitive": true, "first_n": null, "last_n": null}
|
| 285 |
+
{"pdf": "2_pg349.pdf", "page": 1, "id": "2_pg349_equation_09", "type": "present", "max_diffs": 0, "checked": "verified", "url": null, "text": "x dx + y dy = x dy - y dx", "case_sensitive": true, "first_n": null, "last_n": null}
|
| 286 |
+
{"pdf": "2_pg349.pdf", "page": 1, "id": "2_pg349_equation_10", "type": "present", "max_diffs": 0, "checked": "verified", "url": null, "text": "(sin x + y) dy + (y cos x - x^{2}) dx = 0", "case_sensitive": true, "first_n": null, "last_n": null}
|
| 287 |
+
{"pdf": "2_pg349.pdf", "page": 1, "id": "2_pg349_equation_11", "type": "present", "max_diffs": 0, "checked": "verified", "url": null, "text": "y (e^{x} + 2) dx + (e^{x} + 2x) dy = 0", "case_sensitive": true, "first_n": null, "last_n": null}
|
| 288 |
+
{"pdf": "2_pg349.pdf", "page": 1, "id": "2_pg349_equation_12", "type": "present", "max_diffs": 0, "checked": "verified", "url": null, "text": "(\u0445\u0443^{2} - x) dx + (y + xy) dy = 0", "case_sensitive": true, "first_n": null, "last_n": null}
|
| 289 |
+
{"pdf": "2_pg349.pdf", "page": 1, "id": "2_pg349_equation_13", "type": "present", "max_diffs": 0, "checked": "verified", "url": null, "text": "(1 + x^{2}) \\frac{dy}{dx} + xy = 2 y", "case_sensitive": true, "first_n": null, "last_n": null}
|
| 290 |
+
{"pdf": "2_pg349.pdf", "page": 1, "id": "2_pg349_equation_14", "type": "present", "max_diffs": 0, "checked": "verified", "url": null, "text": "x dy - y dx = \\sqrt{x^{2} + y^{2}} dx", "case_sensitive": true, "first_n": null, "last_n": null}
|
| 291 |
+
{"pdf": "2_pg349.pdf", "page": 1, "id": "2_pg349_equation_15", "type": "present", "max_diffs": 0, "checked": "verified", "url": null, "text": "(x - y) dx + x dy = 0", "case_sensitive": true, "first_n": null, "last_n": null}
|
| 292 |
+
{"pdf": "2_pg349.pdf", "page": 1, "id": "2_pg349_equation_16", "type": "present", "max_diffs": 0, "checked": "verified", "url": null, "text": "x dy - y dx = x \\sqrt{x^{2} + y^{2}} dx", "case_sensitive": true, "first_n": null, "last_n": null}
|
| 293 |
+
{"pdf": "2_pg349.pdf", "page": 1, "id": "2_pg349_equation_17", "type": "present", "max_diffs": 0, "checked": "verified", "url": null, "text": "e^{x+y} dy + (1+e^{y}) dx = 0", "case_sensitive": true, "first_n": null, "last_n": null}
|
| 294 |
+
{"pdf": "2_pg349.pdf", "page": 1, "id": "2_pg349_equation_18", "type": "present", "max_diffs": 0, "checked": "verified", "url": null, "text": "(2x+3y - 1) dx + (4x+6y-5) dy = 0", "case_sensitive": true, "first_n": null, "last_n": null}
|
| 295 |
+
{"pdf": "2_pg349.pdf", "page": 1, "id": "2_pg349_equation_19", "type": "present", "max_diffs": 0, "checked": "verified", "url": null, "text": "(3y^{2} + 3 xy + x^{2}) dx = (x^{2} + 2 xy) dy", "case_sensitive": true, "first_n": null, "last_n": null}
|
| 296 |
+
{"pdf": "2_pg349.pdf", "page": 1, "id": "2_pg349_equation_20", "type": "present", "max_diffs": 0, "checked": "verified", "url": null, "text": "(1 + x^{2}) dy + (xy - x^{2}) dx = 0", "case_sensitive": true, "first_n": null, "last_n": null}
|
| 297 |
+
{"pdf": "2_pg238.pdf", "page": 1, "id": "2_pg238_equation_00", "type": "present", "max_diffs": 0, "checked": "verified", "url": null, "text": "x^{\\frac{1}{2}} + y^{\\frac{1}{2}} = a^{\\frac{1}{2}}", "case_sensitive": true, "first_n": null, "last_n": null}
|
| 298 |
+
{"pdf": "2_pg238.pdf", "page": 1, "id": "2_pg238_equation_01", "type": "present", "max_diffs": 0, "checked": "verified", "url": null, "text": "y = \\frac{a}{2} (e^{\\frac{x}{a}} + e^{-\\frac{x}{a}})", "case_sensitive": true, "first_n": null, "last_n": null}
|
| 299 |
+
{"pdf": "2_pg238.pdf", "page": 1, "id": "2_pg238_equation_02", "type": "present", "max_diffs": 0, "checked": "verified", "url": null, "text": "y = sin x", "case_sensitive": true, "first_n": null, "last_n": null}
|
| 300 |
+
{"pdf": "2_pg238.pdf", "page": 1, "id": "2_pg238_equation_03", "type": "present", "max_diffs": 0, "checked": "verified", "url": null, "text": "y^2 = 4 ax", "case_sensitive": true, "first_n": null, "last_n": null}
|
| 301 |
+
{"pdf": "2_pg238.pdf", "page": 1, "id": "2_pg238_equation_04", "type": "present", "max_diffs": 0, "checked": "verified", "url": null, "text": "y^2 = 4 ax", "case_sensitive": true, "first_n": null, "last_n": null}
|
| 302 |
+
{"pdf": "2_pg238.pdf", "page": 1, "id": "2_pg238_equation_05", "type": "present", "max_diffs": 0, "checked": "verified", "url": null, "text": "y = -2 a", "case_sensitive": true, "first_n": null, "last_n": null}
|
| 303 |
+
{"pdf": "2_pg238.pdf", "page": 1, "id": "2_pg238_equation_06", "type": "present", "max_diffs": 0, "checked": "verified", "url": null, "text": "y^2 = 4 ax", "case_sensitive": true, "first_n": null, "last_n": null}
|
| 304 |
+
{"pdf": "2_pg238.pdf", "page": 1, "id": "2_pg238_equation_07", "type": "present", "max_diffs": 0, "checked": "verified", "url": null, "text": "x = a (\\phi - sin \\phi)", "case_sensitive": true, "first_n": null, "last_n": null}
|
| 305 |
+
{"pdf": "2_pg238.pdf", "page": 1, "id": "2_pg238_equation_08", "type": "present", "max_diffs": 0, "checked": "verified", "url": null, "text": "y = a (1 - cos \\phi)", "case_sensitive": true, "first_n": null, "last_n": null}
|
| 306 |
+
{"pdf": "2_pg238.pdf", "page": 1, "id": "2_pg238_equation_09", "type": "present", "max_diffs": 0, "checked": "verified", "url": null, "text": "x = a cos^3 \\phi", "case_sensitive": true, "first_n": null, "last_n": null}
|
| 307 |
+
{"pdf": "2_pg238.pdf", "page": 1, "id": "2_pg238_equation_10", "type": "present", "max_diffs": 0, "checked": "verified", "url": null, "text": "y = a sin^3 \\phi", "case_sensitive": true, "first_n": null, "last_n": null}
|
| 308 |
+
{"pdf": "2_pg238.pdf", "page": 1, "id": "2_pg238_equation_11", "type": "present", "max_diffs": 0, "checked": "verified", "url": null, "text": "r = a (1 + cos \\theta)", "case_sensitive": true, "first_n": null, "last_n": null}
|
| 309 |
+
{"pdf": "2_pg238.pdf", "page": 1, "id": "2_pg238_equation_12", "type": "present", "max_diffs": 0, "checked": "verified", "url": null, "text": "a (1 + cos \\theta)", "case_sensitive": true, "first_n": null, "last_n": null}
|
| 310 |
+
{"pdf": "2_pg238.pdf", "page": 1, "id": "2_pg238_equation_13", "type": "present", "max_diffs": 0, "checked": "verified", "url": null, "text": "x = -\\frac{a}{4}", "case_sensitive": true, "first_n": null, "last_n": null}
|
| 311 |
+
{"pdf": "2_pg238.pdf", "page": 1, "id": "2_pg238_equation_14", "type": "present", "max_diffs": 0, "checked": "verified", "url": null, "text": "x^2 + xy + y^2 = 3", "case_sensitive": true, "first_n": null, "last_n": null}
|
| 312 |
+
{"pdf": "2_pg238.pdf", "page": 1, "id": "2_pg238_equation_15", "type": "present", "max_diffs": 0, "checked": "verified", "url": null, "text": "y = x", "case_sensitive": true, "first_n": null, "last_n": null}
|
| 313 |
+
{"pdf": "2_pg238.pdf", "page": 1, "id": "2_pg238_equation_16", "type": "present", "max_diffs": 0, "checked": "verified", "url": null, "text": "x^{\\frac{1}{2}} + y^{\\frac{1}{2}} = a^{\\frac{1}{2}}", "case_sensitive": true, "first_n": null, "last_n": null}
|
| 314 |
+
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