{"id":102,"date":"2021-07-30T17:13:49","date_gmt":"2021-07-30T17:13:49","guid":{"rendered":"https:\/\/courses.lumenlearning.com\/calculus3\/?post_type=chapter&#038;p=102"},"modified":"2022-11-01T05:07:34","modified_gmt":"2022-11-01T05:07:34","slug":"summary-of-line-integrals","status":"publish","type":"chapter","link":"https:\/\/courses.lumenlearning.com\/calculus3\/chapter\/summary-of-line-integrals\/","title":{"raw":"Summary of Line Integrals","rendered":"Summary of Line Integrals"},"content":{"raw":"<div class=\"textbox learning-objectives\">\r\n<h3>Essential Concepts<\/h3>\r\n<ul>\r\n \t<li>Line integrals generalize the notion of a single-variable integral to higher dimensions. The domain of integration in a single-variable integral is a line segment along the\u00a0<em data-effect=\"italics\">x<\/em>-axis, but the domain of integration in a line integral is a curve in a plane or in space.<\/li>\r\n \t<li>If [latex]C[\/latex]\u00a0is a curve, then the length of\u00a0[latex]C[\/latex] is [latex]\\displaystyle\\int_{C} ds[\/latex].<\/li>\r\n \t<li>There are two kinds of line integral: scalar line integrals and vector line integrals. Scalar line integrals can be used to calculate the mass of a wire; vector line integrals can be used to calculate the work done on a particle traveling through a field.<\/li>\r\n \t<li>Scalar line integrals can be calculated using[latex]\\displaystyle\\int_{C} f(x,y,z)ds=\\displaystyle\\int_{a}^{b} f({\\bf{r}}(t))\\sqrt{(x^{\\prime}(t))^{2}+(y^{\\prime}(t))^{2}+(z^{\\prime}(t))^{2}}dt[\/latex]; vector line integrals can be calculated using\u00a0[latex]\\displaystyle\\int_{C} {\\bf{F}}\\cdot{ds}=\\displaystyle\\int_{C}{\\bf{F}}\\cdot{\\bf{T}}ds=\\displaystyle\\int_{a}^{b}{\\bf{F}}({\\bf{r}}(t))\\cdot{\\bf{r}}^{\\prime}(t)dt[\/latex].<\/li>\r\n \t<li>Two key concepts expressed in terms of line integrals are flux and circulation. Flux measures the rate that a field crosses a given line; circulation measures the tendency of a field to move in the same direction as a given closed curve.<\/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\">Calculating a scalar line integral<\/strong>\r\n[latex]\\displaystyle\\int_{C} f(x,y,z)ds=\\displaystyle\\int_{a}^{b} f({\\bf{r}}(t))\\sqrt{(x^{\\prime}(t))^{2}+(y^{\\prime}(t))^{2}+(z^{\\prime}(t))^{2}}dt[\/latex]<\/li>\r\n \t<li><strong>Calculating a vector line integral <\/strong>\r\n[latex]\\displaystyle\\int_{C} {\\bf{F}}\\cdot{ds}=\\displaystyle\\int_{C}{\\bf{F}}\\cdot{\\bf{T}}ds=\\displaystyle\\int_{a}^{b}{\\bf{F}}({\\bf{r}}(t))\\cdot{\\bf{r}}^{\\prime}(t)dt[\/latex]\r\nor\r\n[latex]{\\displaystyle\\int_{C}} Pdx+Qdy+Rdz= {\\displaystyle\\int_{a}^{b}}\\left(P({\\bf{r}}(t))\\frac{dx}{dt}+Q({\\bf{r}}(t))\\frac{dy}{dt}+R({\\bf{r}}(t))\\frac{dz}{dt}\\right)dt[\/latex]<\/li>\r\n \t<li><strong data-effect=\"bold\">Calculating flux<\/strong>\r\n[latex]\\displaystyle\\int_{C} {\\bf{F}}\\cdot{\\frac{{\\bf{n}}(t)}{\\Arrowvert{\\bf{n}}(t)\\Arrowvert}}ds=\\displaystyle\\int_{a}^{b}{\\bf{F}}({\\bf{r}}(t))\\cdot{\\bf{n}}(t)dt[\/latex]<\/li>\r\n<\/ul>\r\n<h2>Glossary<\/h2>\r\n<dl class=\"definition\">\r\n \t<dt>circulation<\/dt>\r\n \t<dd>the tendency of a fluid to move in the direction of curve [latex]C[\/latex]. If [latex]C[\/latex] is a closed curve, then the circulation of [latex]{\\bf{F}}[\/latex]\u00a0along [latex]C[\/latex] is line integral [latex]\\displaystyle\\int_{C} {\\bf{F}}\\cdot{\\bf{T}}ds[\/latex], which we also denote\u00a0[latex]\\displaystyle\\oint_{C} {\\bf{F}}\\cdot{\\bf{T}}ds[\/latex]<\/dd>\r\n<\/dl>\r\n<dl class=\"definition\">\r\n \t<dt>closed curve<\/dt>\r\n \t<dd><span style=\"font-size: 1em;\">a curve that begins and ends at the same point<\/span><\/dd>\r\n<\/dl>\r\n<dl class=\"definition\">\r\n \t<dt>flux<\/dt>\r\n \t<dd>the rate of a fluid flowing across a curve in a vector field; the flux of vector field [latex]{\\bf{F}}[\/latex]\u00a0across plane curve [latex]C[\/latex] is line integral [latex]\\displaystyle\\int_{C} {\\bf{F}}\\cdot{\\frac{{\\bf{n}}(t)}{\\Arrowvert{\\bf{n}}(t)\\Arrowvert}}ds[\/latex]<\/dd>\r\n<\/dl>\r\n<dl class=\"definition\">\r\n \t<dt>line integral<\/dt>\r\n \t<dd><span style=\"font-size: 1em;\">the integral of a function along a curve in a plane or in space<\/span><\/dd>\r\n<\/dl>\r\n<dl class=\"definition\">\r\n \t<dt>orientation of a curve<\/dt>\r\n \t<dd><span style=\"font-size: 1em;\">the orientation of a curve [latex]C[\/latex]<\/span><span style=\"font-size: 1em;\">\u00a0is a specified direction of [latex]C[\/latex]<\/span><\/dd>\r\n<\/dl>\r\n<dl class=\"definition\">\r\n \t<dt>piecewise smooth curve<\/dt>\r\n \t<dd><span style=\"font-size: 1em;\">an oriented curve that is not smooth, but can be written as the union of finitely many smooth curves<\/span><\/dd>\r\n<\/dl>\r\n<dl class=\"definition\">\r\n \t<dt>scalar line integral<\/dt>\r\n \t<dd>the scalar line integral of a function\u00a0[latex]f[\/latex]\u00a0along a curve [latex]C[\/latex] with respect to arc length is the integral [latex]\\displaystyle\\int_C \\! f\\, \\mathrm{d}s[\/latex],\u00a0it is the integral of a scalar function [latex]f[\/latex] along a curve in a plane or in space; such an integral is defined in terms of a Riemann sum, as is a single-variable integral<\/dd>\r\n<\/dl>\r\n<dl class=\"definition\">\r\n \t<dt>vector line integral<\/dt>\r\n \t<dd>the vector line integral of vector field [latex]{\\bf{F}}[\/latex] along curve [latex]C[\/latex] is the integral of the dot product of [latex]{\\bf{F}}[\/latex]\u00a0with unit tangent vector [latex]{\\bf{T}}[\/latex]\u00a0of [latex]C[\/latex] with respect to arc length, [latex]\\displaystyle\\int_{C} {\\bf{F}}\\cdot{\\bf{T}}ds[\/latex];\u00a0such an integral is defined in terms of a Riemann sum, similar to a single-variable integral<\/dd>\r\n<\/dl>","rendered":"<div class=\"textbox learning-objectives\">\n<h3>Essential Concepts<\/h3>\n<ul>\n<li>Line integrals generalize the notion of a single-variable integral to higher dimensions. The domain of integration in a single-variable integral is a line segment along the\u00a0<em data-effect=\"italics\">x<\/em>-axis, but the domain of integration in a line integral is a curve in a plane or in space.<\/li>\n<li>If [latex]C[\/latex]\u00a0is a curve, then the length of\u00a0[latex]C[\/latex] is [latex]\\displaystyle\\int_{C} ds[\/latex].<\/li>\n<li>There are two kinds of line integral: scalar line integrals and vector line integrals. Scalar line integrals can be used to calculate the mass of a wire; vector line integrals can be used to calculate the work done on a particle traveling through a field.<\/li>\n<li>Scalar line integrals can be calculated using[latex]\\displaystyle\\int_{C} f(x,y,z)ds=\\displaystyle\\int_{a}^{b} f({\\bf{r}}(t))\\sqrt{(x^{\\prime}(t))^{2}+(y^{\\prime}(t))^{2}+(z^{\\prime}(t))^{2}}dt[\/latex]; vector line integrals can be calculated using\u00a0[latex]\\displaystyle\\int_{C} {\\bf{F}}\\cdot{ds}=\\displaystyle\\int_{C}{\\bf{F}}\\cdot{\\bf{T}}ds=\\displaystyle\\int_{a}^{b}{\\bf{F}}({\\bf{r}}(t))\\cdot{\\bf{r}}^{\\prime}(t)dt[\/latex].<\/li>\n<li>Two key concepts expressed in terms of line integrals are flux and circulation. Flux measures the rate that a field crosses a given line; circulation measures the tendency of a field to move in the same direction as a given closed curve.<\/li>\n<\/ul>\n<\/div>\n<h2>Key Equations<\/h2>\n<ul id=\"fs-id1169736654976\">\n<li><strong data-effect=\"bold\">Calculating a scalar line integral<\/strong><br \/>\n[latex]\\displaystyle\\int_{C} f(x,y,z)ds=\\displaystyle\\int_{a}^{b} f({\\bf{r}}(t))\\sqrt{(x^{\\prime}(t))^{2}+(y^{\\prime}(t))^{2}+(z^{\\prime}(t))^{2}}dt[\/latex]<\/li>\n<li><strong>Calculating a vector line integral <\/strong><br \/>\n[latex]\\displaystyle\\int_{C} {\\bf{F}}\\cdot{ds}=\\displaystyle\\int_{C}{\\bf{F}}\\cdot{\\bf{T}}ds=\\displaystyle\\int_{a}^{b}{\\bf{F}}({\\bf{r}}(t))\\cdot{\\bf{r}}^{\\prime}(t)dt[\/latex]<br \/>\nor<br \/>\n[latex]{\\displaystyle\\int_{C}} Pdx+Qdy+Rdz= {\\displaystyle\\int_{a}^{b}}\\left(P({\\bf{r}}(t))\\frac{dx}{dt}+Q({\\bf{r}}(t))\\frac{dy}{dt}+R({\\bf{r}}(t))\\frac{dz}{dt}\\right)dt[\/latex]<\/li>\n<li><strong data-effect=\"bold\">Calculating flux<\/strong><br \/>\n[latex]\\displaystyle\\int_{C} {\\bf{F}}\\cdot{\\frac{{\\bf{n}}(t)}{\\Arrowvert{\\bf{n}}(t)\\Arrowvert}}ds=\\displaystyle\\int_{a}^{b}{\\bf{F}}({\\bf{r}}(t))\\cdot{\\bf{n}}(t)dt[\/latex]<\/li>\n<\/ul>\n<h2>Glossary<\/h2>\n<dl class=\"definition\">\n<dt>circulation<\/dt>\n<dd>the tendency of a fluid to move in the direction of curve [latex]C[\/latex]. If [latex]C[\/latex] is a closed curve, then the circulation of [latex]{\\bf{F}}[\/latex]\u00a0along [latex]C[\/latex] is line integral [latex]\\displaystyle\\int_{C} {\\bf{F}}\\cdot{\\bf{T}}ds[\/latex], which we also denote\u00a0[latex]\\displaystyle\\oint_{C} {\\bf{F}}\\cdot{\\bf{T}}ds[\/latex]<\/dd>\n<\/dl>\n<dl class=\"definition\">\n<dt>closed curve<\/dt>\n<dd><span style=\"font-size: 1em;\">a curve that begins and ends at the same point<\/span><\/dd>\n<\/dl>\n<dl class=\"definition\">\n<dt>flux<\/dt>\n<dd>the rate of a fluid flowing across a curve in a vector field; the flux of vector field [latex]{\\bf{F}}[\/latex]\u00a0across plane curve [latex]C[\/latex] is line integral [latex]\\displaystyle\\int_{C} {\\bf{F}}\\cdot{\\frac{{\\bf{n}}(t)}{\\Arrowvert{\\bf{n}}(t)\\Arrowvert}}ds[\/latex]<\/dd>\n<\/dl>\n<dl class=\"definition\">\n<dt>line integral<\/dt>\n<dd><span style=\"font-size: 1em;\">the integral of a function along a curve in a plane or in space<\/span><\/dd>\n<\/dl>\n<dl class=\"definition\">\n<dt>orientation of a curve<\/dt>\n<dd><span style=\"font-size: 1em;\">the orientation of a curve [latex]C[\/latex]<\/span><span style=\"font-size: 1em;\">\u00a0is a specified direction of [latex]C[\/latex]<\/span><\/dd>\n<\/dl>\n<dl class=\"definition\">\n<dt>piecewise smooth curve<\/dt>\n<dd><span style=\"font-size: 1em;\">an oriented curve that is not smooth, but can be written as the union of finitely many smooth curves<\/span><\/dd>\n<\/dl>\n<dl class=\"definition\">\n<dt>scalar line integral<\/dt>\n<dd>the scalar line integral of a function\u00a0[latex]f[\/latex]\u00a0along a curve [latex]C[\/latex] with respect to arc length is the integral [latex]\\displaystyle\\int_C \\! f\\, \\mathrm{d}s[\/latex],\u00a0it is the integral of a scalar function [latex]f[\/latex] along a curve in a plane or in space; such an integral is defined in terms of a Riemann sum, as is a single-variable integral<\/dd>\n<\/dl>\n<dl class=\"definition\">\n<dt>vector line integral<\/dt>\n<dd>the vector line integral of vector field [latex]{\\bf{F}}[\/latex] along curve [latex]C[\/latex] is the integral of the dot product of [latex]{\\bf{F}}[\/latex]\u00a0with unit tangent vector [latex]{\\bf{T}}[\/latex]\u00a0of [latex]C[\/latex] with respect to arc length, [latex]\\displaystyle\\int_{C} {\\bf{F}}\\cdot{\\bf{T}}ds[\/latex];\u00a0such an integral is defined in terms of a Riemann sum, similar to a single-variable integral<\/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-102\">\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":10,"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-102","chapter","type-chapter","status-publish","hentry"],"part":24,"_links":{"self":[{"href":"https:\/\/courses.lumenlearning.com\/calculus3\/wp-json\/pressbooks\/v2\/chapters\/102","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":16,"href":"https:\/\/courses.lumenlearning.com\/calculus3\/wp-json\/pressbooks\/v2\/chapters\/102\/revisions"}],"predecessor-version":[{"id":5707,"href":"https:\/\/courses.lumenlearning.com\/calculus3\/wp-json\/pressbooks\/v2\/chapters\/102\/revisions\/5707"}],"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\/102\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/courses.lumenlearning.com\/calculus3\/wp-json\/wp\/v2\/media?parent=102"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/calculus3\/wp-json\/pressbooks\/v2\/chapter-type?post=102"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/calculus3\/wp-json\/wp\/v2\/contributor?post=102"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/calculus3\/wp-json\/wp\/v2\/license?post=102"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}