{"id":52,"date":"2021-07-30T17:06:10","date_gmt":"2021-07-30T17:06:10","guid":{"rendered":"https:\/\/courses.lumenlearning.com\/calculus3\/?post_type=chapter&#038;p=52"},"modified":"2022-10-29T00:30:20","modified_gmt":"2022-10-29T00:30:20","slug":"summary-of-cylindrical-and-spherical-coordinates","status":"publish","type":"chapter","link":"https:\/\/courses.lumenlearning.com\/calculus3\/chapter\/summary-of-cylindrical-and-spherical-coordinates\/","title":{"raw":"Summary of Cylindrical and Spherical Coordinates","rendered":"Summary of Cylindrical and Spherical Coordinates"},"content":{"raw":"<div class=\"textbox learning-objectives\">\r\n<h3>Essential Concepts<\/h3>\r\n<ul>\r\n \t<li>In the cylindrical coordinate system, a point in space is represented by the ordered triple [latex](r,\\theta,z)[\/latex], where [latex](r,\\theta)[\/latex] represents the polar coordinates of the point\u2019s projection in the [latex]xy[\/latex]-plane and [latex]z[\/latex]\u00a0represents the point\u2019s projection onto the [latex]z[\/latex]-axis.<\/li>\r\n \t<li>To convert a point from cylindrical coordinates to Cartesian coordinates, use equations [latex]x=r\\cos\\theta[\/latex], [latex]y=r\\sin\\theta[\/latex], and [latex]z=z[\/latex].<\/li>\r\n \t<li>To convert a point from Cartesian coordinates to cylindrical coordinates, use equations [latex]r^2=x^2+y^2[\/latex], [latex]\\tan\\theta=\\frac{y}{x}[\/latex], and [latex]z=z[\/latex].<\/li>\r\n \t<li>In the spherical coordinate system, a point [latex]P[\/latex] in space is represented by the ordered triple [latex](\\rho,\\theta,\\varphi)[\/latex], where [latex]\\rho[\/latex]\u00a0is the distance between\u00a0[latex]P[\/latex] and the origin [latex](\\rho\\ne{0})[\/latex], [latex]\\theta[\/latex] is the same\u00a0angle used to describe the location in cylindrical coordinates, and [latex]\\varphi[\/latex] is the\u00a0angle formed by the positive [latex]z[\/latex]-axis and line segment [latex]\\overline{OP}[\/latex], where [latex]O[\/latex] is the origin and [latex]0\\le\\varphi\\le\\pi[\/latex].<\/li>\r\n \t<li>To convert a point from spherical coordinates to Cartesian coordinates, use equations\u00a0[latex]x=\\rho\\sin\\varphi\\cos\\theta[\/latex], [latex]y=\\rho\\sin\\varphi\\sin\\theta[\/latex], and [latex]z=\\rho\\cos\\varphi[\/latex].<\/li>\r\n \t<li>To convert a point from Cartesian coordinates to spherical coordinates, use equations [latex]\\rho^{2}=x^2+y^2+z^2[\/latex], [latex]\\tan\\theta=\\frac{y}{x}[\/latex], and [latex]\\varphi=\\arccos\\left(\\frac{z}{\\sqrt{x^2+y^2+z^2}}\\right)[\/latex]<\/li>\r\n \t<li>To convert a point from spherical coordinates to cylindrical coordinates, use equations [latex]r=\\rho\\sin\\varphi[\/latex], [latex]\\theta=\\theta[\/latex], and [latex]z=\\rho\\cos\\varphi[\/latex].<\/li>\r\n \t<li>To convert a point from cylindrical coordinates to spherical coordinates, use equations [latex]\\rho=\\sqrt{r^2+z^2}[\/latex], [latex]\\theta=\\theta[\/latex], and [latex]\\varphi=\\arccos\\left(\\frac{z}{\\sqrt{r^2+z^2}}\\right)[\/latex]<\/li>\r\n<\/ul>\r\n<\/div>\r\n<h2>Glossary<\/h2>\r\n<dl class=\"definition\">\r\n \t<dt>cylindrical coordinate system<\/dt>\r\n \t<dd>a way to describe a location in space with an ordered triple [latex](r,\\theta,z)[\/latex],<strong>\u00a0<\/strong>where\u00a0[latex](r,\\theta)[\/latex] represents the polar coordinates of the point\u2019s projection in the [latex]xy[\/latex]-plane, and [latex]z[\/latex] represents the point's projection onto the [latex]z[\/latex]-axis<\/dd>\r\n<\/dl>\r\n<dl class=\"definition\">\r\n \t<dt>spherical coordinate system<\/dt>\r\n \t<dd>a way to describe a location in space with an ordered triple\u00a0[latex](\\rho,\\theta,\\varphi)[\/latex],<strong>\u00a0<\/strong>where\u00a0[latex]\\rho[\/latex]\u00a0is the distance between\u00a0[latex]P[\/latex]\u00a0and the origin [latex]\\rho \\ne\r\n{0}[\/latex],\u00a0[latex]\\theta[\/latex] is the same angle used to describe the location in cylindrical coordinates, and [latex]\\varphi[\/latex] is the angle formed by the positive [latex]z[\/latex]-axis and line segment [latex]\\overline{OP}[\/latex]<strong>\u00a0<\/strong>where\u00a0[latex]O[\/latex]\u00a0is the origin and [latex]0\\le\\varphi\\le\\pi[\/latex]<\/dd>\r\n<\/dl>","rendered":"<div class=\"textbox learning-objectives\">\n<h3>Essential Concepts<\/h3>\n<ul>\n<li>In the cylindrical coordinate system, a point in space is represented by the ordered triple [latex](r,\\theta,z)[\/latex], where [latex](r,\\theta)[\/latex] represents the polar coordinates of the point\u2019s projection in the [latex]xy[\/latex]-plane and [latex]z[\/latex]\u00a0represents the point\u2019s projection onto the [latex]z[\/latex]-axis.<\/li>\n<li>To convert a point from cylindrical coordinates to Cartesian coordinates, use equations [latex]x=r\\cos\\theta[\/latex], [latex]y=r\\sin\\theta[\/latex], and [latex]z=z[\/latex].<\/li>\n<li>To convert a point from Cartesian coordinates to cylindrical coordinates, use equations [latex]r^2=x^2+y^2[\/latex], [latex]\\tan\\theta=\\frac{y}{x}[\/latex], and [latex]z=z[\/latex].<\/li>\n<li>In the spherical coordinate system, a point [latex]P[\/latex] in space is represented by the ordered triple [latex](\\rho,\\theta,\\varphi)[\/latex], where [latex]\\rho[\/latex]\u00a0is the distance between\u00a0[latex]P[\/latex] and the origin [latex](\\rho\\ne{0})[\/latex], [latex]\\theta[\/latex] is the same\u00a0angle used to describe the location in cylindrical coordinates, and [latex]\\varphi[\/latex] is the\u00a0angle formed by the positive [latex]z[\/latex]-axis and line segment [latex]\\overline{OP}[\/latex], where [latex]O[\/latex] is the origin and [latex]0\\le\\varphi\\le\\pi[\/latex].<\/li>\n<li>To convert a point from spherical coordinates to Cartesian coordinates, use equations\u00a0[latex]x=\\rho\\sin\\varphi\\cos\\theta[\/latex], [latex]y=\\rho\\sin\\varphi\\sin\\theta[\/latex], and [latex]z=\\rho\\cos\\varphi[\/latex].<\/li>\n<li>To convert a point from Cartesian coordinates to spherical coordinates, use equations [latex]\\rho^{2}=x^2+y^2+z^2[\/latex], [latex]\\tan\\theta=\\frac{y}{x}[\/latex], and [latex]\\varphi=\\arccos\\left(\\frac{z}{\\sqrt{x^2+y^2+z^2}}\\right)[\/latex]<\/li>\n<li>To convert a point from spherical coordinates to cylindrical coordinates, use equations [latex]r=\\rho\\sin\\varphi[\/latex], [latex]\\theta=\\theta[\/latex], and [latex]z=\\rho\\cos\\varphi[\/latex].<\/li>\n<li>To convert a point from cylindrical coordinates to spherical coordinates, use equations [latex]\\rho=\\sqrt{r^2+z^2}[\/latex], [latex]\\theta=\\theta[\/latex], and [latex]\\varphi=\\arccos\\left(\\frac{z}{\\sqrt{r^2+z^2}}\\right)[\/latex]<\/li>\n<\/ul>\n<\/div>\n<h2>Glossary<\/h2>\n<dl class=\"definition\">\n<dt>cylindrical coordinate system<\/dt>\n<dd>a way to describe a location in space with an ordered triple [latex](r,\\theta,z)[\/latex],<strong>\u00a0<\/strong>where\u00a0[latex](r,\\theta)[\/latex] represents the polar coordinates of the point\u2019s projection in the [latex]xy[\/latex]-plane, and [latex]z[\/latex] represents the point&#8217;s projection onto the [latex]z[\/latex]-axis<\/dd>\n<\/dl>\n<dl class=\"definition\">\n<dt>spherical coordinate system<\/dt>\n<dd>a way to describe a location in space with an ordered triple\u00a0[latex](\\rho,\\theta,\\varphi)[\/latex],<strong>\u00a0<\/strong>where\u00a0[latex]\\rho[\/latex]\u00a0is the distance between\u00a0[latex]P[\/latex]\u00a0and the origin [latex]\\rho \\ne  {0}[\/latex],\u00a0[latex]\\theta[\/latex] is the same angle used to describe the location in cylindrical coordinates, and [latex]\\varphi[\/latex] is the angle formed by the positive [latex]z[\/latex]-axis and line segment [latex]\\overline{OP}[\/latex]<strong>\u00a0<\/strong>where\u00a0[latex]O[\/latex]\u00a0is the origin and [latex]0\\le\\varphi\\le\\pi[\/latex]<\/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-52\">\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-52","chapter","type-chapter","status-publish","hentry"],"part":20,"_links":{"self":[{"href":"https:\/\/courses.lumenlearning.com\/calculus3\/wp-json\/pressbooks\/v2\/chapters\/52","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":9,"href":"https:\/\/courses.lumenlearning.com\/calculus3\/wp-json\/pressbooks\/v2\/chapters\/52\/revisions"}],"predecessor-version":[{"id":3692,"href":"https:\/\/courses.lumenlearning.com\/calculus3\/wp-json\/pressbooks\/v2\/chapters\/52\/revisions\/3692"}],"part":[{"href":"https:\/\/courses.lumenlearning.com\/calculus3\/wp-json\/pressbooks\/v2\/parts\/20"}],"metadata":[{"href":"https:\/\/courses.lumenlearning.com\/calculus3\/wp-json\/pressbooks\/v2\/chapters\/52\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/courses.lumenlearning.com\/calculus3\/wp-json\/wp\/v2\/media?parent=52"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/calculus3\/wp-json\/pressbooks\/v2\/chapter-type?post=52"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/calculus3\/wp-json\/wp\/v2\/contributor?post=52"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/calculus3\/wp-json\/wp\/v2\/license?post=52"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}