{"id":104,"date":"2021-07-30T17:14:07","date_gmt":"2021-07-30T17:14:07","guid":{"rendered":"https:\/\/courses.lumenlearning.com\/calculus3\/?post_type=chapter&#038;p=104"},"modified":"2022-11-01T05:15:42","modified_gmt":"2022-11-01T05:15:42","slug":"summary-of-conservative-vector-fields","status":"publish","type":"chapter","link":"https:\/\/courses.lumenlearning.com\/calculus3\/chapter\/summary-of-conservative-vector-fields\/","title":{"raw":"Summary of Conservative Vector Fields","rendered":"Summary of Conservative Vector Fields"},"content":{"raw":"<div class=\"textbox learning-objectives\">\r\n<h3>Essential Concepts<\/h3>\r\n<ul>\r\n \t<li>The theorems in this section require curves that are closed, simple, or both, and regions that are connected or simply connected.<\/li>\r\n \t<li>The line integral of a conservative vector field can be calculated using the Fundamental Theorem for Line Integrals. This theorem is a generalization of the Fundamental Theorem of Calculus in higher dimensions. Using this theorem usually makes the calculation of the line integral easier.<\/li>\r\n \t<li>Conservative fields are independent of path. The line integral of a conservative field depends only on the value of the potential function at the endpoints of the domain curve.<\/li>\r\n \t<li>Given vector field\u00a0[latex]{\\bf{F}}[\/latex], we can test whether\u00a0[latex]{\\bf{F}}[\/latex] is conservative by using the cross-partial property. If\u00a0[latex]{\\bf{F}}[\/latex]\u00a0has the cross-partial property and the domain is simply connected, then\u00a0[latex]{\\bf{F}}[\/latex]\u00a0is conservative (and thus has a potential function). If\u00a0[latex]{\\bf{F}}[\/latex]\u00a0is conservative, we can find a potential function by using the Problem-Solving Strategy.<\/li>\r\n \t<li>The circulation of a conservative vector field on a simply connected domain over a closed curve is zero<\/li>\r\n<\/ul>\r\n<\/div>\r\n<h2>Key Equations<\/h2>\r\n<ul id=\"fs-id1169736654976\">\r\n \t<li><strong data-effect=\"bold\">Fundamental Theorem for Line Integrals<\/strong>\r\n[latex]\\displaystyle\\int_{C}\\nabla {f}\\cdot {d{\\bf{r}}}=f({\\bf{r}}(b))-f({\\bf{r}}(a))[\/latex]<\/li>\r\n \t<li><strong>Circulation of a conservative field over curve\u00a0[latex]C[\/latex]\u00a0that encloses a simply connected region\u00a0<\/strong>\r\n[latex]\\displaystyle\\oint_{C}\\nabla {f}\\cdot {d{\\bf{r}}}=0[\/latex]<\/li>\r\n<\/ul>\r\n<h2>Glossary<\/h2>\r\n<dl class=\"definition\">\r\n \t<dt>closed curve<\/dt>\r\n \t<dd>a curve for which there exists a parameterization [latex]{\\bf{r}}(t),a\\le{t}\\le{b}[\/latex], such that [latex]{\\bf{r}}(a)={\\bf{r}}(b)[\/latex],\u00a0and the curve is traversed exactly once<\/dd>\r\n<\/dl>\r\n<dl class=\"definition\">\r\n \t<dt>connected region<\/dt>\r\n \t<dd><span style=\"font-size: 1em;\">a region in which any two points can be connected by a path with a trace contained entirely inside the region<\/span><\/dd>\r\n<\/dl>\r\n<dl class=\"definition\">\r\n \t<dt>Fundamental Theorem for Line Integrals<\/dt>\r\n \t<dd>the value of the line integral [latex]\\displaystyle\\int_{C} {\\nabla}{f}\\cdot{d{\\bf{r}}}[\/latex]\u00a0depends only on the value of [latex]f[\/latex]\u00a0at the endpoints of [latex]C[\/latex]:\u00a0[latex]\\displaystyle\\int_{C} {\\nabla}{f}\\cdot{d{\\bf{r}}}=f({\\bf{r}}(b)))-f({\\bf{r}}(a))[\/latex]<\/dd>\r\n<\/dl>\r\n<dl class=\"definition\">\r\n \t<dt>independent of path (path independent)<\/dt>\r\n \t<dd>a vector field [latex]{\\bf{F}}[\/latex] has path independence if [latex]\\displaystyle\\int_{C_{1}} {\\bf{F}}\\cdot{d{\\bf{r}}}=\\displaystyle\\int_{C_{2}} {\\bf{F}}\\cdot{d{\\bf{r}}}[\/latex]\u00a0for any curves [latex]C_{1}[\/latex]\u00a0and [latex]C_{2}[\/latex]\u00a0in the domain of [latex]{\\bf{F}}[\/latex] with the same initial points and terminal points<\/dd>\r\n<\/dl>\r\n<dl class=\"definition\">\r\n \t<dt>simple curve<\/dt>\r\n \t<dd><span style=\"font-size: 1em;\">a curve that does not cross itself<\/span><\/dd>\r\n<\/dl>\r\n<dl class=\"definition\">\r\n \t<dt>simply connected region<\/dt>\r\n \t<dd><span style=\"font-size: 1em;\">a region that is connected and has the property that any closed curve that lies entirely inside the region encompasses points that are entirely inside the region<\/span><\/dd>\r\n<\/dl>","rendered":"<div class=\"textbox learning-objectives\">\n<h3>Essential Concepts<\/h3>\n<ul>\n<li>The theorems in this section require curves that are closed, simple, or both, and regions that are connected or simply connected.<\/li>\n<li>The line integral of a conservative vector field can be calculated using the Fundamental Theorem for Line Integrals. This theorem is a generalization of the Fundamental Theorem of Calculus in higher dimensions. Using this theorem usually makes the calculation of the line integral easier.<\/li>\n<li>Conservative fields are independent of path. The line integral of a conservative field depends only on the value of the potential function at the endpoints of the domain curve.<\/li>\n<li>Given vector field\u00a0[latex]{\\bf{F}}[\/latex], we can test whether\u00a0[latex]{\\bf{F}}[\/latex] is conservative by using the cross-partial property. If\u00a0[latex]{\\bf{F}}[\/latex]\u00a0has the cross-partial property and the domain is simply connected, then\u00a0[latex]{\\bf{F}}[\/latex]\u00a0is conservative (and thus has a potential function). If\u00a0[latex]{\\bf{F}}[\/latex]\u00a0is conservative, we can find a potential function by using the Problem-Solving Strategy.<\/li>\n<li>The circulation of a conservative vector field on a simply connected domain over a closed curve is zero<\/li>\n<\/ul>\n<\/div>\n<h2>Key Equations<\/h2>\n<ul id=\"fs-id1169736654976\">\n<li><strong data-effect=\"bold\">Fundamental Theorem for Line Integrals<\/strong><br \/>\n[latex]\\displaystyle\\int_{C}\\nabla {f}\\cdot {d{\\bf{r}}}=f({\\bf{r}}(b))-f({\\bf{r}}(a))[\/latex]<\/li>\n<li><strong>Circulation of a conservative field over curve\u00a0[latex]C[\/latex]\u00a0that encloses a simply connected region\u00a0<\/strong><br \/>\n[latex]\\displaystyle\\oint_{C}\\nabla {f}\\cdot {d{\\bf{r}}}=0[\/latex]<\/li>\n<\/ul>\n<h2>Glossary<\/h2>\n<dl class=\"definition\">\n<dt>closed curve<\/dt>\n<dd>a curve for which there exists a parameterization [latex]{\\bf{r}}(t),a\\le{t}\\le{b}[\/latex], such that [latex]{\\bf{r}}(a)={\\bf{r}}(b)[\/latex],\u00a0and the curve is traversed exactly once<\/dd>\n<\/dl>\n<dl class=\"definition\">\n<dt>connected region<\/dt>\n<dd><span style=\"font-size: 1em;\">a region in which any two points can be connected by a path with a trace contained entirely inside the region<\/span><\/dd>\n<\/dl>\n<dl class=\"definition\">\n<dt>Fundamental Theorem for Line Integrals<\/dt>\n<dd>the value of the line integral [latex]\\displaystyle\\int_{C} {\\nabla}{f}\\cdot{d{\\bf{r}}}[\/latex]\u00a0depends only on the value of [latex]f[\/latex]\u00a0at the endpoints of [latex]C[\/latex]:\u00a0[latex]\\displaystyle\\int_{C} {\\nabla}{f}\\cdot{d{\\bf{r}}}=f({\\bf{r}}(b)))-f({\\bf{r}}(a))[\/latex]<\/dd>\n<\/dl>\n<dl class=\"definition\">\n<dt>independent of path (path independent)<\/dt>\n<dd>a vector field [latex]{\\bf{F}}[\/latex] has path independence if [latex]\\displaystyle\\int_{C_{1}} {\\bf{F}}\\cdot{d{\\bf{r}}}=\\displaystyle\\int_{C_{2}} {\\bf{F}}\\cdot{d{\\bf{r}}}[\/latex]\u00a0for any curves [latex]C_{1}[\/latex]\u00a0and [latex]C_{2}[\/latex]\u00a0in the domain of [latex]{\\bf{F}}[\/latex] with the same initial points and terminal points<\/dd>\n<\/dl>\n<dl class=\"definition\">\n<dt>simple curve<\/dt>\n<dd><span style=\"font-size: 1em;\">a curve that does not cross itself<\/span><\/dd>\n<\/dl>\n<dl class=\"definition\">\n<dt>simply connected region<\/dt>\n<dd><span style=\"font-size: 1em;\">a region that is connected and has the property that any closed curve that lies entirely inside the region encompasses points that are entirely inside the region<\/span><\/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-104\">\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":14,"template":"","meta":{"_candela_citation":"[{\"type\":\"cc\",\"description\":\"Calculus Volume 3\",\"author\":\"Gilbert Strang, Edwin (Jed) 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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-104","chapter","type-chapter","status-publish","hentry"],"part":24,"_links":{"self":[{"href":"https:\/\/courses.lumenlearning.com\/calculus3\/wp-json\/pressbooks\/v2\/chapters\/104","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":15,"href":"https:\/\/courses.lumenlearning.com\/calculus3\/wp-json\/pressbooks\/v2\/chapters\/104\/revisions"}],"predecessor-version":[{"id":3788,"href":"https:\/\/courses.lumenlearning.com\/calculus3\/wp-json\/pressbooks\/v2\/chapters\/104\/revisions\/3788"}],"part":[{"href":"https:\/\/courses.lumenlearning.com\/calculus3\/wp-json\/pressbooks\/v2\/parts\/24"}],"metadata":[{"href":"https:\/\/courses.lumenlearning.com\/calculus3\/wp-json\/pressbooks\/v2\/chapters\/104\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/courses.lumenlearning.com\/calculus3\/wp-json\/wp\/v2\/media?parent=104"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/calculus3\/wp-json\/pressbooks\/v2\/chapter-type?post=104"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/calculus3\/wp-json\/wp\/v2\/contributor?post=104"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/calculus3\/wp-json\/wp\/v2\/license?post=104"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}