{"id":70,"date":"2021-07-30T17:08:38","date_gmt":"2021-07-30T17:08:38","guid":{"rendered":"https:\/\/courses.lumenlearning.com\/calculus3\/?post_type=chapter&#038;p=70"},"modified":"2022-10-29T00:38:01","modified_gmt":"2022-10-29T00:38:01","slug":"summary-of-calculus-of-vector-valued-functions","status":"publish","type":"chapter","link":"https:\/\/courses.lumenlearning.com\/calculus3\/chapter\/summary-of-calculus-of-vector-valued-functions\/","title":{"raw":"Summary of Calculus of Vector-Valued Functions","rendered":"Summary of Calculus of Vector-Valued Functions"},"content":{"raw":"<div class=\"textbox learning-objectives\">\r\n<h3>Essential Concepts<\/h3>\r\n<ul>\r\n \t<li>To calculate the derivative of a vector-valued function, calculate the derivatives of the component functions, then put them back into a new vector-valued function.<\/li>\r\n \t<li>Many of the properties of differentiation from the <em>Calculus I: Derivatives<\/em>\u00a0also apply to vector-valued functions.<\/li>\r\n \t<li>The derivative of a vector-valued function [latex]{\\bf{r}}(t)[\/latex] is also a tangent vector to the curve. The unit tangent vector [latex]{\\bf{T}}(t)[\/latex] is calculated by dividing the derivative of a vector-valued function by its magnitude.<\/li>\r\n \t<li>The antiderivative of a vector-valued function is found by finding the antiderivatives of the component functions, then putting them back together in a vector-valued function.<\/li>\r\n \t<li>The definite integral of a vector-valued function is found by finding the definite integrals of the component functions, then putting them back together in a vector-valued function.<\/li>\r\n<\/ul>\r\n<\/div>\r\n<h2>Key Equations<\/h2>\r\n<ul id=\"fs-id1169736654976\">\r\n \t<li><strong>Derivative of a vector-valued function\r\n<\/strong>[latex]{\\bf{r}}^{\\prime}(t)=\\underset{\\Delta{t}\\to{0}}{\\lim}\\frac{{\\bf{r}}(t+\\Delta{t})-{\\bf{r}}(t)}{\\Delta{t}}[\/latex]<\/li>\r\n \t<li><strong>Principal unit tangent vector<\/strong>\r\n[latex]{\\bf{T}}(t)=\\dfrac{{\\bf{r}}^{\\prime}(t)}{\\parallel{\\bf{r}}^{\\prime}(t)\\parallel}[\/latex]<\/li>\r\n \t<li><strong>Indefinite integral of a vector-valued function<\/strong>\r\n[latex]\\displaystyle\\int [f(t){\\bf{i}}+g(t){\\bf{j}}+h(t){\\bf{k}}]dt=\\left[\\displaystyle\\int f(t)dt\\right]{\\bf{i}}+\\left[\\displaystyle\\int g(t)dt\\right]{\\bf{j}}+\\left[\\displaystyle\\int h(t)dt\\right]{\\bf{k}}[\/latex]<\/li>\r\n \t<li><strong>Definite integral of a vector-valued function<\/strong>\r\n[latex]\\displaystyle\\int_{a}^{b} [f(t){\\bf{i}}+g(t){\\bf{j}}+h(t){\\bf{k}}]dt=\\left[\\displaystyle\\int_{a}^{b} f(t)dt\\right]{\\bf{i}}+\\left[\\displaystyle\\int_{a}^{b} g(t)dt\\right]{\\bf{j}}+\\left[\\displaystyle\\int_{a}^{b} h(t)dt\\right]{\\bf{k}}[\/latex]<\/li>\r\n<\/ul>\r\n<h2>Glossary<\/h2>\r\n<dl class=\"definition\">\r\n \t<dt>definite integral of a vector-valued function<\/dt>\r\n \t<dd>the vector obtained by calculating the definite integral of each of the component functions of a given vector-valued function, then using the results as the components of the resulting function<\/dd>\r\n<\/dl>\r\n<dl class=\"definition\">\r\n \t<dt>derivative of a vector-valued function<\/dt>\r\n \t<dd>the derivative of a vector-valued function\u00a0[latex]{\\bf{r}}(t)[\/latex]<strong>\u00a0<\/strong>is\u00a0[latex]{\\bf{r}}^{\\prime}(t)=\\underset{\\Delta{t}\\to{0}}{\\lim}\\frac{{\\bf{r}}(t+\\Delta{t})-{\\bf{r}}(t)}{\\Delta{t}}[\/latex],<strong>\u00a0<\/strong>provided the limit exists<\/dd>\r\n<\/dl>\r\n<dl class=\"definition\">\r\n \t<dt>indefinite integral of a vector-valued function<\/dt>\r\n \t<dd>a vector-valued function with a derivative that is equal to a given vector-valued function<\/dd>\r\n<\/dl>\r\n<dl class=\"definition\">\r\n \t<dt>principal unit tangent vector<\/dt>\r\n \t<dd>a unit vector tangent to a curve [latex]C[\/latex]<\/dd>\r\n<\/dl>\r\n<dl class=\"definition\">\r\n \t<dt>tangent vector<\/dt>\r\n \t<dd>to [latex]{\\bf{r}}(t)[\/latex] at [latex]t=t_{0}[\/latex] any vector [latex]{\\bf{v}}[\/latex] such that, when the\u00a0tail of the vector is placed at point\u00a0[latex]{\\bf{r}}(t_{0})[\/latex]\u00a0on the graph, vector [latex]{\\bf{v}}[\/latex] is tangent to curve\u00a0[latex]C[\/latex]<\/dd>\r\n \t<dt><\/dt>\r\n<\/dl>","rendered":"<div class=\"textbox learning-objectives\">\n<h3>Essential Concepts<\/h3>\n<ul>\n<li>To calculate the derivative of a vector-valued function, calculate the derivatives of the component functions, then put them back into a new vector-valued function.<\/li>\n<li>Many of the properties of differentiation from the <em>Calculus I: Derivatives<\/em>\u00a0also apply to vector-valued functions.<\/li>\n<li>The derivative of a vector-valued function [latex]{\\bf{r}}(t)[\/latex] is also a tangent vector to the curve. The unit tangent vector [latex]{\\bf{T}}(t)[\/latex] is calculated by dividing the derivative of a vector-valued function by its magnitude.<\/li>\n<li>The antiderivative of a vector-valued function is found by finding the antiderivatives of the component functions, then putting them back together in a vector-valued function.<\/li>\n<li>The definite integral of a vector-valued function is found by finding the definite integrals of the component functions, then putting them back together in a vector-valued function.<\/li>\n<\/ul>\n<\/div>\n<h2>Key Equations<\/h2>\n<ul id=\"fs-id1169736654976\">\n<li><strong>Derivative of a vector-valued function<br \/>\n<\/strong>[latex]{\\bf{r}}^{\\prime}(t)=\\underset{\\Delta{t}\\to{0}}{\\lim}\\frac{{\\bf{r}}(t+\\Delta{t})-{\\bf{r}}(t)}{\\Delta{t}}[\/latex]<\/li>\n<li><strong>Principal unit tangent vector<\/strong><br \/>\n[latex]{\\bf{T}}(t)=\\dfrac{{\\bf{r}}^{\\prime}(t)}{\\parallel{\\bf{r}}^{\\prime}(t)\\parallel}[\/latex]<\/li>\n<li><strong>Indefinite integral of a vector-valued function<\/strong><br \/>\n[latex]\\displaystyle\\int [f(t){\\bf{i}}+g(t){\\bf{j}}+h(t){\\bf{k}}]dt=\\left[\\displaystyle\\int f(t)dt\\right]{\\bf{i}}+\\left[\\displaystyle\\int g(t)dt\\right]{\\bf{j}}+\\left[\\displaystyle\\int h(t)dt\\right]{\\bf{k}}[\/latex]<\/li>\n<li><strong>Definite integral of a vector-valued function<\/strong><br \/>\n[latex]\\displaystyle\\int_{a}^{b} [f(t){\\bf{i}}+g(t){\\bf{j}}+h(t){\\bf{k}}]dt=\\left[\\displaystyle\\int_{a}^{b} f(t)dt\\right]{\\bf{i}}+\\left[\\displaystyle\\int_{a}^{b} g(t)dt\\right]{\\bf{j}}+\\left[\\displaystyle\\int_{a}^{b} h(t)dt\\right]{\\bf{k}}[\/latex]<\/li>\n<\/ul>\n<h2>Glossary<\/h2>\n<dl class=\"definition\">\n<dt>definite integral of a vector-valued function<\/dt>\n<dd>the vector obtained by calculating the definite integral of each of the component functions of a given vector-valued function, then using the results as the components of the resulting function<\/dd>\n<\/dl>\n<dl class=\"definition\">\n<dt>derivative of a vector-valued function<\/dt>\n<dd>the derivative of a vector-valued function\u00a0[latex]{\\bf{r}}(t)[\/latex]<strong>\u00a0<\/strong>is\u00a0[latex]{\\bf{r}}^{\\prime}(t)=\\underset{\\Delta{t}\\to{0}}{\\lim}\\frac{{\\bf{r}}(t+\\Delta{t})-{\\bf{r}}(t)}{\\Delta{t}}[\/latex],<strong>\u00a0<\/strong>provided the limit exists<\/dd>\n<\/dl>\n<dl class=\"definition\">\n<dt>indefinite integral of a vector-valued function<\/dt>\n<dd>a vector-valued function with a derivative that is equal to a given vector-valued function<\/dd>\n<\/dl>\n<dl class=\"definition\">\n<dt>principal unit tangent vector<\/dt>\n<dd>a unit vector tangent to a curve [latex]C[\/latex]<\/dd>\n<\/dl>\n<dl class=\"definition\">\n<dt>tangent vector<\/dt>\n<dd>to [latex]{\\bf{r}}(t)[\/latex] at [latex]t=t_{0}[\/latex] any vector [latex]{\\bf{v}}[\/latex] such that, when the\u00a0tail of the vector is placed at point\u00a0[latex]{\\bf{r}}(t_{0})[\/latex]\u00a0on the graph, vector [latex]{\\bf{v}}[\/latex] is tangent to curve\u00a0[latex]C[\/latex]<\/dd>\n<dt><\/dt>\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-70\">\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-70","chapter","type-chapter","status-publish","hentry"],"part":21,"_links":{"self":[{"href":"https:\/\/courses.lumenlearning.com\/calculus3\/wp-json\/pressbooks\/v2\/chapters\/70","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":13,"href":"https:\/\/courses.lumenlearning.com\/calculus3\/wp-json\/pressbooks\/v2\/chapters\/70\/revisions"}],"predecessor-version":[{"id":3696,"href":"https:\/\/courses.lumenlearning.com\/calculus3\/wp-json\/pressbooks\/v2\/chapters\/70\/revisions\/3696"}],"part":[{"href":"https:\/\/courses.lumenlearning.com\/calculus3\/wp-json\/pressbooks\/v2\/parts\/21"}],"metadata":[{"href":"https:\/\/courses.lumenlearning.com\/calculus3\/wp-json\/pressbooks\/v2\/chapters\/70\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/courses.lumenlearning.com\/calculus3\/wp-json\/wp\/v2\/media?parent=70"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/calculus3\/wp-json\/pressbooks\/v2\/chapter-type?post=70"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/calculus3\/wp-json\/wp\/v2\/contributor?post=70"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/calculus3\/wp-json\/wp\/v2\/license?post=70"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}