mathmlben
Version:
Visualization for goldstandard
52 lines • 11.8 kB
JSON
{
"definitions": {
"\\mathbb{R}": [
{
"Q12916": "real number (quantity along a continuous line)"
}
],
"n": [
{
"Q4440864": "dimension (minimum number of coordinates within a space needed to specify any point)"
}
],
"f": [
"integrable function",
{}
],
"x": [
"variable",
{}
],
"B": [
{
"Q838611": "ball (in mathematics, space inside a sphere)"
}
],
"x_{0}": [
{
"Q44946": "point (fundamental object of Euclidean geometry)"
}
],
"r": [
{
"Q173817": "radius (segment in a circle and its length)"
}
],
"S": [
"surface",
{}
]
},
"constraints": [],
"math_inputtex": "\\int_{\\mathbb{R}^n}f\\,dx = \\int_0^\\infty\\left\\{\\int_{\\partial B(x_0;r)} f\\,dS\\right\\}\\,dr.",
"math_inputtex_semantic": "\\int_{\\w{Q12916}{\\mathbb{R}}^{\\w{Q4440864}{n}}} \\wf{Q853831}{f} \\,d \\w{Q50701}{x} = \\int_0^\\infty\\left\\{\\int_{\\partial \\wf{Q838611}{B}(\\w{Q44946}{x_0}; \\w{Q173817}{r})} f\\,d \\w{Q484298}{S} \\right\\}\\,dr",
"correct_tex": "\\int_{\\mathbb{R}^n}f\\,dx = \\int_0^\\infty\\left\\{\\int_{\\partial B(x_0;r)} f\\,dS\\right\\}\\,dr",
"correct_mml": "<math xmlns=\"http://www.w3.org/1998/Math/MathML\" id=\"p1.1.m1.1\" class=\"ltx_Math\" alttext=\"\\int_{\\mathbb{R}^{n}}f\\,dx=\\int_{0}^{\\infty}\\left\\{\\int_{\\partial B(x_{0};r)}f% \\,dS\\right\\}\\,dr\" display=\"inline\">\n <semantics id=\"p1.1.m1.1a\">\n <mrow id=\"p1.1.m1.1.22\" xref=\"p1.1.m1.1.22.cmml\">\n <mrow id=\"p1.1.m1.1.22.1\" xref=\"p1.1.m1.1.22.1.cmml\">\n <msub id=\"p1.1.m1.1.22.1.1\" xref=\"p1.1.m1.1.22.1.1.cmml\">\n <mo largeop=\"true\" symmetric=\"true\" id=\"p1.1.m1.1.1\" xref=\"p1.1.m1.1.1.cmml\">∫</mo>\n <msup id=\"p1.1.m1.1.2.1\" xref=\"p1.1.m1.1.2.1.cmml\">\n <mi id=\"p1.1.m1.1.2.1.1\" xref=\"p1.1.m1.1.2.1.1.cmml\">ℝ</mi>\n <mi id=\"p1.1.m1.1.2.1.2.1\" xref=\"p1.1.m1.1.2.1.2.1.cmml\">n</mi>\n </msup>\n </msub>\n <mrow id=\"p1.1.m1.1.22.1.2\" xref=\"p1.1.m1.1.22.1.2.cmml\">\n <mpadded width=\"+1.7pt\" id=\"p1.1.m1.1.3\" xref=\"p1.1.m1.1.3.cmml\">\n <mi id=\"p1.1.m1.1.3a\" xref=\"p1.1.m1.1.3.cmml\">f</mi>\n </mpadded>\n <mo id=\"p1.1.m1.1.22.1.2.1\" xref=\"p1.1.m1.1.22.1.2.1.cmml\"></mo>\n <mrow id=\"p1.1.m1.1.22.1.2.2\" xref=\"p1.1.m1.1.22.1.2.2.cmml\">\n <mo id=\"p1.1.m1.1.5\" xref=\"p1.1.m1.1.5.cmml\">𝑑</mo>\n <mi id=\"p1.1.m1.1.6\" xref=\"p1.1.m1.1.6.cmml\">x</mi>\n </mrow>\n </mrow>\n </mrow>\n <mo id=\"p1.1.m1.1.7\" xref=\"p1.1.m1.1.7.cmml\">=</mo>\n <mrow id=\"p1.1.m1.1.22.2\" xref=\"p1.1.m1.1.22.2.cmml\">\n <msubsup id=\"p1.1.m1.1.22.2.1\" xref=\"p1.1.m1.1.22.2.1.cmml\">\n <mo largeop=\"true\" symmetric=\"true\" id=\"p1.1.m1.1.8\" xref=\"p1.1.m1.1.8.cmml\">∫</mo>\n <mn id=\"p1.1.m1.1.9.1\" xref=\"p1.1.m1.1.9.1.cmml\">0</mn>\n <mi mathvariant=\"normal\" id=\"p1.1.m1.1.10.1\" xref=\"p1.1.m1.1.10.1.cmml\">∞</mi>\n </msubsup>\n <mrow id=\"p1.1.m1.1.22.2.2\" xref=\"p1.1.m1.1.22.2.2.cmml\">\n <mrow id=\"p1.1.m1.1.22.2.2.2\" xref=\"p1.1.m1.1.22.2.2.2.1.cmml\">\n <mo id=\"p1.1.m1.1.11\" xref=\"p1.1.m1.1.22.2.2.2.1.cmml\">{</mo>\n <mrow id=\"p1.1.m1.1.22.2.2.2.2\" xref=\"p1.1.m1.1.22.2.2.2.2.cmml\">\n <msub id=\"p1.1.m1.1.22.2.2.2.2.1\" xref=\"p1.1.m1.1.22.2.2.2.2.1.cmml\">\n <mo largeop=\"true\" symmetric=\"true\" id=\"p1.1.m1.1.12\" xref=\"p1.1.m1.1.12.cmml\">∫</mo>\n <mrow id=\"p1.1.m1.1.13.1.10\" xref=\"p1.1.m1.1.13.1.9.cmml\">\n <mrow id=\"p1.1.m1.1.13.1.10.1\" xref=\"p1.1.m1.1.13.1.10.1.cmml\">\n <mo id=\"p1.1.m1.1.13.1.1\" xref=\"p1.1.m1.1.13.1.1.cmml\">∂</mo>\n <mo id=\"p1.1.m1.1.13.1.10.1a\" xref=\"p1.1.m1.1.13.1.10.1.cmml\"></mo>\n <mi id=\"p1.1.m1.1.13.1.2\" xref=\"p1.1.m1.1.13.1.2.cmml\">B</mi>\n </mrow>\n <mo id=\"p1.1.m1.1.13.1.10a\" xref=\"p1.1.m1.1.13.1.9.cmml\"></mo>\n <mrow id=\"p1.1.m1.1.13.1.10.3\" xref=\"p1.1.m1.1.13.1.9.cmml\">\n <mo stretchy=\"false\" id=\"p1.1.m1.1.13.1.3\" xref=\"p1.1.m1.1.13.1.9.cmml\">(</mo>\n <msub id=\"p1.1.m1.1.13.1.10.2\" xref=\"p1.1.m1.1.13.1.10.2.cmml\">\n <mi id=\"p1.1.m1.1.13.1.4\" xref=\"p1.1.m1.1.13.1.4.cmml\">x</mi>\n <mn id=\"p1.1.m1.1.13.1.5.1\" xref=\"p1.1.m1.1.13.1.5.1.cmml\">0</mn>\n </msub>\n <mo id=\"p1.1.m1.1.13.1.6\" xref=\"p1.1.m1.1.13.1.9.cmml\">;</mo>\n <mi id=\"p1.1.m1.1.13.1.7\" xref=\"p1.1.m1.1.13.1.7.cmml\">r</mi>\n <mo stretchy=\"false\" id=\"p1.1.m1.1.13.1.8\" xref=\"p1.1.m1.1.13.1.9.cmml\">)</mo>\n </mrow>\n </mrow>\n </msub>\n <mrow id=\"p1.1.m1.1.22.2.2.2.2.2\" xref=\"p1.1.m1.1.22.2.2.2.2.2.cmml\">\n <mpadded width=\"+1.7pt\" id=\"p1.1.m1.1.14\" xref=\"p1.1.m1.1.14.cmml\">\n <mi id=\"p1.1.m1.1.14a\" xref=\"p1.1.m1.1.14.cmml\">f</mi>\n </mpadded>\n <mo id=\"p1.1.m1.1.22.2.2.2.2.2.1\" xref=\"p1.1.m1.1.22.2.2.2.2.2.1.cmml\"></mo>\n <mrow id=\"p1.1.m1.1.22.2.2.2.2.2.2\" xref=\"p1.1.m1.1.22.2.2.2.2.2.2.cmml\">\n <mo id=\"p1.1.m1.1.16\" xref=\"p1.1.m1.1.16.cmml\">𝑑</mo>\n <mi id=\"p1.1.m1.1.17\" xref=\"p1.1.m1.1.17.cmml\">S</mi>\n </mrow>\n </mrow>\n </mrow>\n <mo rspace=\"4.2pt\" id=\"p1.1.m1.1.18\" xref=\"p1.1.m1.1.22.2.2.2.1.cmml\">}</mo>\n </mrow>\n <mo id=\"p1.1.m1.1.22.2.2.1\" xref=\"p1.1.m1.1.22.2.2.1.cmml\"></mo>\n <mrow id=\"p1.1.m1.1.22.2.2.3\" xref=\"p1.1.m1.1.22.2.2.3.cmml\">\n <mo id=\"p1.1.m1.1.20\" xref=\"p1.1.m1.1.20.cmml\">𝑑</mo>\n <mi id=\"p1.1.m1.1.21\" xref=\"p1.1.m1.1.21.cmml\">r</mi>\n </mrow>\n </mrow>\n </mrow>\n </mrow>\n <annotation-xml encoding=\"MathML-Content\" id=\"p1.1.m1.1b\">\n <apply id=\"p1.1.m1.1.22.cmml\" xref=\"p1.1.m1.1.22\">\n <eq id=\"p1.1.m1.1.7.cmml\" xref=\"p1.1.m1.1.7\"/>\n <apply id=\"p1.1.m1.1.22.1.cmml\" xref=\"p1.1.m1.1.22.1\">\n <apply id=\"p1.1.m1.1.22.1.1.cmml\" xref=\"p1.1.m1.1.22.1.1\">\n <csymbol cd=\"ambiguous\" id=\"p1.1.m1.1.22.1.1.1.cmml\" xref=\"p1.1.m1.1.22.1.1\">subscript</csymbol>\n <int id=\"p1.1.m1.1.1.cmml\" xref=\"p1.1.m1.1.1\"/>\n <apply id=\"p1.1.m1.1.2.1.cmml\" xref=\"p1.1.m1.1.2.1\">\n <power xmlns:m=\"http://www.w3.org/1998/Math/MathML\" xref=\"p1.1.m1.1.2.1\"/>\n <csymbol cd=\"latexml\" id=\"p1.1.m1.1.2.1.1.cmml\" xref=\"p1.1.m1.1.2.1.1\">Q12916</csymbol>\n <csymbol cd=\"latexml\" id=\"p1.1.m1.1.2.1.2.1.cmml\" xref=\"p1.1.m1.1.2.1.2.1\">Q4440864</csymbol>\n </apply>\n </apply>\n <apply id=\"p1.1.m1.1.22.1.2.cmml\" xref=\"p1.1.m1.1.22.1.2\">\n <times id=\"p1.1.m1.1.22.1.2.1.cmml\" xref=\"p1.1.m1.1.22.1.2.1\"/>\n <csymbol cd=\"latexml\" id=\"p1.1.m1.1.3.cmml\" xref=\"p1.1.m1.1.3\">Q853831</csymbol>\n <apply id=\"p1.1.m1.1.22.1.2.2.cmml\" xref=\"p1.1.m1.1.22.1.2.2\">\n <csymbol cd=\"latexml\" id=\"p1.1.m1.1.5.cmml\" xref=\"p1.1.m1.1.5\">differential-d</csymbol>\n <csymbol cd=\"latexml\" id=\"p1.1.m1.1.6.cmml\" xref=\"p1.1.m1.1.6\">Q50701</csymbol>\n </apply>\n </apply>\n </apply>\n <apply id=\"p1.1.m1.1.22.2.cmml\" xref=\"p1.1.m1.1.22.2\">\n <apply id=\"p1.1.m1.1.22.2.1.cmml\" xref=\"p1.1.m1.1.22.2.1\">\n <power xmlns:m=\"http://www.w3.org/1998/Math/MathML\" xref=\"p1.1.m1.1.22.2.1\"/>\n <apply id=\"p1.1.m1.1.22.2.1.2.cmml\" xref=\"p1.1.m1.1.22.2.1\">\n <csymbol cd=\"ambiguous\" id=\"p1.1.m1.1.22.2.1.2.1.cmml\" xref=\"p1.1.m1.1.22.2.1\">subscript</csymbol>\n <int id=\"p1.1.m1.1.8.cmml\" xref=\"p1.1.m1.1.8\"/>\n <cn type=\"integer\" id=\"p1.1.m1.1.9.1.cmml\" xref=\"p1.1.m1.1.9.1\">0</cn>\n </apply>\n <infinity id=\"p1.1.m1.1.10.1.cmml\" xref=\"p1.1.m1.1.10.1\"/>\n </apply>\n <apply id=\"p1.1.m1.1.22.2.2.cmml\" xref=\"p1.1.m1.1.22.2.2\">\n <times id=\"p1.1.m1.1.22.2.2.1.cmml\" xref=\"p1.1.m1.1.22.2.2.1\"/>\n <set id=\"p1.1.m1.1.22.2.2.2.1.cmml\" xref=\"p1.1.m1.1.22.2.2.2\">\n <apply id=\"p1.1.m1.1.22.2.2.2.2.cmml\" xref=\"p1.1.m1.1.22.2.2.2.2\">\n <apply id=\"p1.1.m1.1.22.2.2.2.2.1.cmml\" xref=\"p1.1.m1.1.22.2.2.2.2.1\">\n <csymbol cd=\"ambiguous\" id=\"p1.1.m1.1.22.2.2.2.2.1.1.cmml\" xref=\"p1.1.m1.1.22.2.2.2.2.1\">subscript</csymbol>\n <int id=\"p1.1.m1.1.12.cmml\" xref=\"p1.1.m1.1.12\"/>\n <apply id=\"p1.1.m1.1.13.1.9.cmml\" xref=\"p1.1.m1.1.13.1.10\">\n <apply id=\"p1.1.m1.1.13.1.10.1.cmml\" xref=\"p1.1.m1.1.13.1.10.1\">\n <partialdiff id=\"p1.1.m1.1.13.1.1.cmml\" xref=\"p1.1.m1.1.13.1.1\"/>\n <csymbol cd=\"latexml\" id=\"p1.1.m1.1.13.1.2.cmml\" xref=\"p1.1.m1.1.13.1.2\">Q838611</csymbol>\n </apply>\n <apply id=\"p1.1.m1.1.13.1.10.2.cmml\" xref=\"p1.1.m1.1.13.1.10.2\">\n <csymbol cd=\"ambiguous\" id=\"p1.1.m1.1.13.1.10.2.1.cmml\" xref=\"p1.1.m1.1.13.1.10.2\">subscript</csymbol>\n <csymbol cd=\"latexml\" id=\"p1.1.m1.1.13.1.4.cmml\" xref=\"p1.1.m1.1.13.1.4\">Q44946</csymbol>\n <cn type=\"integer\" id=\"p1.1.m1.1.13.1.5.1.cmml\" xref=\"p1.1.m1.1.13.1.5.1\">0</cn>\n </apply>\n <csymbol cd=\"latexml\" id=\"p1.1.m1.1.13.1.7.cmml\" xref=\"p1.1.m1.1.13.1.7\">Q173817</csymbol>\n </apply>\n </apply>\n <apply id=\"p1.1.m1.1.22.2.2.2.2.2.cmml\" xref=\"p1.1.m1.1.22.2.2.2.2.2\">\n <times id=\"p1.1.m1.1.22.2.2.2.2.2.1.cmml\" xref=\"p1.1.m1.1.22.2.2.2.2.2.1\"/>\n <csymbol cd=\"latexml\" id=\"p1.1.m1.1.14.cmml\" xref=\"p1.1.m1.1.14\">Q853831</csymbol>\n <apply id=\"p1.1.m1.1.22.2.2.2.2.2.2.cmml\" xref=\"p1.1.m1.1.22.2.2.2.2.2.2\">\n <csymbol cd=\"latexml\" id=\"p1.1.m1.1.16.cmml\" xref=\"p1.1.m1.1.16\">differential-d</csymbol>\n <csymbol cd=\"latexml\" id=\"p1.1.m1.1.17.cmml\" xref=\"p1.1.m1.1.17\">Q484298</csymbol>\n </apply>\n </apply>\n </apply>\n </set>\n <apply id=\"p1.1.m1.1.22.2.2.3.cmml\" xref=\"p1.1.m1.1.22.2.2.3\">\n <csymbol cd=\"latexml\" id=\"p1.1.m1.1.20.cmml\" xref=\"p1.1.m1.1.20\">differential-d</csymbol>\n <csymbol cd=\"latexml\" id=\"p1.1.m1.1.21.cmml\" xref=\"p1.1.m1.1.21\">Q173817</csymbol>\n </apply>\n </apply>\n </apply>\n </apply>\n </annotation-xml>\n <annotation encoding=\"application/x-tex\" id=\"p1.1.m1.1c\">\\int_{\\mathbb{R}^{n}}f\\,dx=\\int_{0}^{\\infty}\\left\\{\\int_{\\partial B(x_{0};r)}f%\n\\,dS\\right\\}\\,dr</annotation>\n </semantics>\n</math>",
"uri": "https://en.formulasearchengine.com/w/index.php?oldid=22382#math22382.3",
"title": "Coarea_formula",
"type": "equation",
"check": {
"tree": false
}
}