{"id":144,"date":"2021-07-30T17:20:23","date_gmt":"2021-07-30T17:20:23","guid":{"rendered":"https:\/\/courses.lumenlearning.com\/calculus3\/?post_type=chapter&#038;p=144"},"modified":"2022-11-01T04:43:30","modified_gmt":"2022-11-01T04:43:30","slug":"summary-of-change-of-variables-in-multiple-integrals","status":"publish","type":"chapter","link":"https:\/\/courses.lumenlearning.com\/calculus3\/chapter\/summary-of-change-of-variables-in-multiple-integrals\/","title":{"raw":"Summary of Change of Variables in Multiple Integrals","rendered":"Summary of Change of Variables in Multiple Integrals"},"content":{"raw":"<div class=\"textbox learning-objectives\">\r\n<h3>Essential Concepts<\/h3>\r\n<ul>\r\n \t<li>A transformation\u00a0[latex]T[\/latex]\u00a0is a function that transforms a region\u00a0[latex]G[\/latex]\u00a0in one plane (space) into a region\u00a0[latex]R[\/latex]\u00a0in another plane (space) by a change of variables.<\/li>\r\n \t<li>A transformation\u00a0[latex]T:G\\rightarrow{R}[\/latex] defined as\u00a0[latex]T(u,v)=(x,y)[\/latex] [latex](\\text{or }T(u,v,w)=(x,y,z))([\/latex]\u00a0is said to be a one-to-one transformation if no two points map to the same image point.<\/li>\r\n \t<li>If [latex]f[\/latex]\u00a0is continuous on\u00a0[latex]R[\/latex], then\u00a0[latex]\\underset{R}{\\displaystyle\\iint} f(x,y) dA =\\underset{S}{\\displaystyle\\iint} f\\left(g(u,v),h(u,v)\\right)\\left\\arrowvert\\frac{\\partial(x,y)}{\\partial(u,v)}\\right\\arrowvert du dv[\/latex]<\/li>\r\n \t<li>If [latex]F[\/latex]\u00a0is continuous on\u00a0[latex]R[\/latex], then\u00a0[latex]R[\/latex], then\u00a0[latex]\\underset{R}{\\displaystyle\\iiint} F(x,y,z) dV =\\underset{G}{\\displaystyle\\iiint} F\\left(g(u,v,w),h(u,v,w),k(u,v,w)\\right)\\left\\arrowvert\\frac{\\partial(x,y,z)}{\\partial(u,v,w)}\\right\\arrowvert du dv dw = \\underset{G}{\\displaystyle\\iiint}H(u,v,w)|J(u,v,w)| du dv dw[\/latex]<\/li>\r\n<\/ul>\r\n<\/div>\r\n<h2>Glossary<\/h2>\r\n<dl class=\"definition\">\r\n \t<dt>Jacobian<\/dt>\r\n \t<dd>the Jacobian [latex]J(u ,v)[\/latex] in two variables is a [latex]2{\\times}2[\/latex] determinant:<\/dd>\r\n \t<dd>[latex]J(u,v) = \\begin{vmatrix}\\frac{dx}{du} &amp; \\frac{dy}{du}\\\\\\frac{dx}{dv} &amp; \\frac{dy}{dv}\\end{vmatrix}[\/latex]<\/dd>\r\n \t<dd>the Jacobian [latex]J(u ,v, w)[\/latex]\u00a0in three variables is a [latex]3{\\times}3[\/latex]\u00a0determinant:<\/dd>\r\n \t<dd>[latex]J(u,v,w)=\\begin{vmatrix}\\frac{dx}{du} &amp; \\frac{dy}{du} &amp; \\frac{dz}{du}\\\\\\frac{dx}{dv} &amp; \\frac{dy}{dv} &amp; \\frac{dz}{dv}\\\\\\frac{dx}{dw} &amp; \\frac{dy}{dw} &amp; \\frac{dz}{dw}\\end{vmatrix}[\/latex]<\/dd>\r\n<\/dl>\r\n<dl class=\"definition\">\r\n \t<dt>one-to-one transformation<\/dt>\r\n \t<dd>a transformation [latex]T : G {\\rightarrow} R[\/latex]\u00a0defined as [latex]T(u, v) = (x, y)[\/latex]\u00a0is said to be one-to-one if no two points map to the same image point<\/dd>\r\n<\/dl>\r\n<dl class=\"definition\">\r\n \t<dt>planar transformation<\/dt>\r\n \t<dd>a function [latex]T[\/latex]\u00a0that transforms a region [latex]G[\/latex]\u00a0in one plane into a region [latex]R[\/latex]\u00a0in another plane by a change of variables<\/dd>\r\n<\/dl>\r\n<dl class=\"definition\">\r\n \t<dt>transformation<\/dt>\r\n \t<dd>a function that transforms a region [latex]G[\/latex] in one plane into a region [latex]R[\/latex]\u00a0in another plane by a change of variables<\/dd>\r\n<\/dl>","rendered":"<div class=\"textbox learning-objectives\">\n<h3>Essential Concepts<\/h3>\n<ul>\n<li>A transformation\u00a0[latex]T[\/latex]\u00a0is a function that transforms a region\u00a0[latex]G[\/latex]\u00a0in one plane (space) into a region\u00a0[latex]R[\/latex]\u00a0in another plane (space) by a change of variables.<\/li>\n<li>A transformation\u00a0[latex]T:G\\rightarrow{R}[\/latex] defined as\u00a0[latex]T(u,v)=(x,y)[\/latex] [latex](\\text{or }T(u,v,w)=(x,y,z))([\/latex]\u00a0is said to be a one-to-one transformation if no two points map to the same image point.<\/li>\n<li>If [latex]f[\/latex]\u00a0is continuous on\u00a0[latex]R[\/latex], then\u00a0[latex]\\underset{R}{\\displaystyle\\iint} f(x,y) dA =\\underset{S}{\\displaystyle\\iint} f\\left(g(u,v),h(u,v)\\right)\\left\\arrowvert\\frac{\\partial(x,y)}{\\partial(u,v)}\\right\\arrowvert du dv[\/latex]<\/li>\n<li>If [latex]F[\/latex]\u00a0is continuous on\u00a0[latex]R[\/latex], then\u00a0[latex]R[\/latex], then\u00a0[latex]\\underset{R}{\\displaystyle\\iiint} F(x,y,z) dV =\\underset{G}{\\displaystyle\\iiint} F\\left(g(u,v,w),h(u,v,w),k(u,v,w)\\right)\\left\\arrowvert\\frac{\\partial(x,y,z)}{\\partial(u,v,w)}\\right\\arrowvert du dv dw = \\underset{G}{\\displaystyle\\iiint}H(u,v,w)|J(u,v,w)| du dv dw[\/latex]<\/li>\n<\/ul>\n<\/div>\n<h2>Glossary<\/h2>\n<dl class=\"definition\">\n<dt>Jacobian<\/dt>\n<dd>the Jacobian [latex]J(u ,v)[\/latex] in two variables is a [latex]2{\\times}2[\/latex] determinant:<\/dd>\n<dd>[latex]J(u,v) = \\begin{vmatrix}\\frac{dx}{du} & \\frac{dy}{du}\\\\\\frac{dx}{dv} & \\frac{dy}{dv}\\end{vmatrix}[\/latex]<\/dd>\n<dd>the Jacobian [latex]J(u ,v, w)[\/latex]\u00a0in three variables is a [latex]3{\\times}3[\/latex]\u00a0determinant:<\/dd>\n<dd>[latex]J(u,v,w)=\\begin{vmatrix}\\frac{dx}{du} & \\frac{dy}{du} & \\frac{dz}{du}\\\\\\frac{dx}{dv} & \\frac{dy}{dv} & \\frac{dz}{dv}\\\\\\frac{dx}{dw} & \\frac{dy}{dw} & \\frac{dz}{dw}\\end{vmatrix}[\/latex]<\/dd>\n<\/dl>\n<dl class=\"definition\">\n<dt>one-to-one transformation<\/dt>\n<dd>a transformation [latex]T : G {\\rightarrow} R[\/latex]\u00a0defined as [latex]T(u, v) = (x, y)[\/latex]\u00a0is said to be one-to-one if no two points map to the same image point<\/dd>\n<\/dl>\n<dl class=\"definition\">\n<dt>planar transformation<\/dt>\n<dd>a function [latex]T[\/latex]\u00a0that transforms a region [latex]G[\/latex]\u00a0in one plane into a region [latex]R[\/latex]\u00a0in another plane by a change of variables<\/dd>\n<\/dl>\n<dl class=\"definition\">\n<dt>transformation<\/dt>\n<dd>a function that transforms a region [latex]G[\/latex] in one plane into a region [latex]R[\/latex]\u00a0in another plane by a change of variables<\/dd>\n<\/dl>\n\n\t\t\t <section class=\"citations-section\" role=\"contentinfo\">\n\t\t\t <h3>Candela Citations<\/h3>\n\t\t\t\t\t <div>\n\t\t\t\t\t\t <div id=\"citation-list-144\">\n\t\t\t\t\t\t\t <div class=\"licensing\"><div class=\"license-attribution-dropdown-subheading\">CC licensed content, Shared previously<\/div><ul class=\"citation-list\"><li>Calculus Volume 3. <strong>Authored by<\/strong>: Gilbert Strang, Edwin (Jed) Herman. <strong>Provided by<\/strong>: OpenStax. <strong>Located at<\/strong>: <a target=\"_blank\" href=\"https:\/\/openstax.org\/books\/calculus-volume-3\/pages\/1-introduction\">https:\/\/openstax.org\/books\/calculus-volume-3\/pages\/1-introduction<\/a>. <strong>License<\/strong>: <em><a target=\"_blank\" rel=\"license\" href=\"https:\/\/creativecommons.org\/licenses\/by-nc-sa\/4.0\/\">CC BY-NC-SA: Attribution-NonCommercial-ShareAlike<\/a><\/em>. <strong>License Terms<\/strong>: Access for free at https:\/\/openstax.org\/books\/calculus-volume-3\/pages\/1-introduction<\/li><\/ul><\/div>\n\t\t\t\t\t\t <\/div>\n\t\t\t\t\t <\/div>\n\t\t\t <\/section>","protected":false},"author":17533,"menu_order":31,"template":"","meta":{"_candela_citation":"[{\"type\":\"cc\",\"description\":\"Calculus Volume 3\",\"author\":\"Gilbert Strang, Edwin (Jed) Herman\",\"organization\":\"OpenStax\",\"url\":\"https:\/\/openstax.org\/books\/calculus-volume-3\/pages\/1-introduction\",\"project\":\"\",\"license\":\"cc-by-nc-sa\",\"license_terms\":\"Access for free at https:\/\/openstax.org\/books\/calculus-volume-3\/pages\/1-introduction\"}]","CANDELA_OUTCOMES_GUID":"","pb_show_title":"on","pb_short_title":"","pb_subtitle":"","pb_authors":[],"pb_section_license":""},"chapter-type":[],"contributor":[],"license":[],"class_list":["post-144","chapter","type-chapter","status-publish","hentry"],"part":23,"_links":{"self":[{"href":"https:\/\/courses.lumenlearning.com\/calculus3\/wp-json\/pressbooks\/v2\/chapters\/144","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/courses.lumenlearning.com\/calculus3\/wp-json\/pressbooks\/v2\/chapters"}],"about":[{"href":"https:\/\/courses.lumenlearning.com\/calculus3\/wp-json\/wp\/v2\/types\/chapter"}],"author":[{"embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/calculus3\/wp-json\/wp\/v2\/users\/17533"}],"version-history":[{"count":22,"href":"https:\/\/courses.lumenlearning.com\/calculus3\/wp-json\/pressbooks\/v2\/chapters\/144\/revisions"}],"predecessor-version":[{"id":3781,"href":"https:\/\/courses.lumenlearning.com\/calculus3\/wp-json\/pressbooks\/v2\/chapters\/144\/revisions\/3781"}],"part":[{"href":"https:\/\/courses.lumenlearning.com\/calculus3\/wp-json\/pressbooks\/v2\/parts\/23"}],"metadata":[{"href":"https:\/\/courses.lumenlearning.com\/calculus3\/wp-json\/pressbooks\/v2\/chapters\/144\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/courses.lumenlearning.com\/calculus3\/wp-json\/wp\/v2\/media?parent=144"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/calculus3\/wp-json\/pressbooks\/v2\/chapter-type?post=144"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/calculus3\/wp-json\/wp\/v2\/contributor?post=144"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/calculus3\/wp-json\/wp\/v2\/license?post=144"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}