{"id":535,"date":"2025-10-09T15:18:33","date_gmt":"2025-10-09T15:18:33","guid":{"rendered":"https:\/\/content.one.lumenlearning.com\/calculus3\/?post_type=chapter&#038;p=535"},"modified":"2025-11-13T14:55:49","modified_gmt":"2025-11-13T14:55:49","slug":"alternative-coordinate-systems-and-surfaces-fresh-take","status":"publish","type":"chapter","link":"https:\/\/content.one.lumenlearning.com\/calculus3\/chapter\/alternative-coordinate-systems-and-surfaces-fresh-take\/","title":{"raw":"Alternative Coordinate Systems and Surfaces: Fresh Take","rendered":"Alternative Coordinate Systems and Surfaces: Fresh Take"},"content":{"raw":"<section class=\"textbox learningGoals\" aria-label=\"Learning Goals\">\r\n<ul>\r\n \t<li>Switch between cylindrical and rectangular coordinates<\/li>\r\n \t<li>Switch between spherical and rectangular coordinates<\/li>\r\n \t<li>Recognize and describe different types of quadric surfaces<\/li>\r\n \t<li>Analyze 3D surfaces by looking at their cross-sections and traces<\/li>\r\n<\/ul>\r\n<\/section>\r\n<h2>Cylindrical Coordinates<\/h2>\r\n<div class=\"textbox shaded\">\r\n\r\n<strong>The Main Idea\u00a0<\/strong>\r\n<p class=\"whitespace-normal break-words\">Just as polar coordinates simplify circular problems in 2D, <strong>cylindrical coordinates<\/strong> extend this advantage to 3D, making problems with cylindrical symmetry much easier to handle.<\/p>\r\n<p class=\"whitespace-normal break-words\"><strong>Cylindrical Coordinate System<\/strong><\/p>\r\n<p class=\"whitespace-normal break-words\">A point in space is represented by [latex](r, \\theta, z)[\/latex]:<\/p>\r\n\r\n<ul class=\"[&amp;:not(:last-child)_ul]:pb-1 [&amp;:not(:last-child)_ol]:pb-1 list-disc space-y-2.5 pl-7\">\r\n \t<li class=\"whitespace-normal break-words\">[latex](r, \\theta)[\/latex]: polar coordinates of the point's projection onto the [latex]xy[\/latex]-plane<\/li>\r\n \t<li class=\"whitespace-normal break-words\">[latex]z[\/latex]: height above or below the [latex]xy[\/latex]-plane<\/li>\r\n<\/ul>\r\n<p class=\"whitespace-normal break-words\"><strong>Tip<\/strong>: Find the point's \"shadow\" on the floor using polar coordinates, then specify how high\/low it sits.<\/p>\r\n<p class=\"whitespace-normal break-words\"><strong>Conversion Formulas<\/strong><\/p>\r\n<p class=\"whitespace-normal break-words\">Cylindrical to Rectangular:<\/p>\r\n\r\n<ul class=\"[&amp;:not(:last-child)_ul]:pb-1 [&amp;:not(:last-child)_ol]:pb-1 list-disc space-y-2.5 pl-7\">\r\n \t<li class=\"whitespace-normal break-words\">[latex]x = r\\cos\\theta[\/latex]<\/li>\r\n \t<li class=\"whitespace-normal break-words\">[latex]y = r\\sin\\theta[\/latex]<\/li>\r\n \t<li class=\"whitespace-normal break-words\">[latex]z = z[\/latex]<\/li>\r\n<\/ul>\r\n<p class=\"whitespace-normal break-words\">Rectangular to Cylindrical:<\/p>\r\n\r\n<ul class=\"[&amp;:not(:last-child)_ul]:pb-1 [&amp;:not(:last-child)_ol]:pb-1 list-disc space-y-2.5 pl-7\">\r\n \t<li class=\"whitespace-normal break-words\">[latex]r = \\sqrt{x^2 + y^2}[\/latex]<\/li>\r\n \t<li class=\"whitespace-normal break-words\">[latex]\\tan\\theta = \\frac{y}{x}[\/latex] (choose [latex]\\theta[\/latex] based on quadrant)<\/li>\r\n \t<li class=\"whitespace-normal break-words\">[latex]z = z[\/latex]<\/li>\r\n<\/ul>\r\n<p class=\"whitespace-normal break-words\"><strong>Constant Coordinate Surfaces<\/strong><\/p>\r\n\r\n<ul class=\"[&amp;:not(:last-child)_ul]:pb-1 [&amp;:not(:last-child)_ol]:pb-1 list-disc space-y-2.5 pl-7\">\r\n \t<li class=\"whitespace-normal break-words\">[latex]r = c[\/latex]: Vertical circular cylinder of radius [latex]c[\/latex] (fixed distance from [latex]z[\/latex]-axis)<\/li>\r\n \t<li class=\"whitespace-normal break-words\">[latex]\\theta = c[\/latex]: Half-plane starting at [latex]z[\/latex]-axis, making angle [latex]c[\/latex] with positive [latex]x[\/latex]-axis<\/li>\r\n \t<li class=\"whitespace-normal break-words\">[latex]z = c[\/latex]: Horizontal plane (same as rectangular coordinates)<\/li>\r\n<\/ul>\r\n<p class=\"whitespace-normal break-words\"><strong>Why Use Cylindrical Coordinates?<\/strong><\/p>\r\n<p class=\"whitespace-normal break-words\">They simplify equations with cylindrical symmetry. For example:<\/p>\r\n\r\n<ul class=\"[&amp;:not(:last-child)_ul]:pb-1 [&amp;:not(:last-child)_ol]:pb-1 list-disc space-y-2.5 pl-7\">\r\n \t<li class=\"whitespace-normal break-words\">Rectangular: [latex]x^2 + y^2 = 25[\/latex] (a cylinder)<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Cylindrical: [latex]r = 5[\/latex] (much simpler!)<\/li>\r\n<\/ul>\r\n<\/div>\r\n<section class=\"textbox example\" aria-label=\"Example\">Point [latex]R[\/latex] has cylindrical coordinates [latex]\\left(5,\\frac{\\pi}6,4\\right)[\/latex]. Plot [latex]R[\/latex] and describe its location in space using rectangular, or Cartesian, coordinates.[reveal-answer q=\"572615912\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"572615912\"]The rectangular coordinates of the point are [latex]\\left(\\frac{5\\sqrt3}2,\\frac52,4\\right)[\/latex].\u00a0<span data-type=\"newline\">\r\n<\/span>[caption id=\"attachment_5281\" align=\"aligncenter\" width=\"462\"]<img class=\"size-full wp-image-5281\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/5667\/2021\/08\/31230153\/2.55.jpg\" alt=\"This figure is the 3-dimensional coordinate system. There is a point labeled \u201c(5, pi\/6, 4).\u201d The point is located above a line segment in the x y-plane labeled r = 5 that is pi\/6 degrees from the x-axis. The distance from the x y-plane to the point is labeled \u201cz = 4.\u201d\" width=\"462\" height=\"370\" \/> Figure 6. The rectangular coordinates of the point are [latex]\\left(\\frac{5\\sqrt3}2,\\frac52,4\\right)[\/latex][\/caption][\/hidden-answer]<\/section><section class=\"textbox watchIt\" aria-label=\"Watch It\">Watch the following video to see the worked solution to the above example.<center><iframe src=\"\/\/plugin.3playmedia.com\/show?mf=7809468&amp;p3sdk_version=1.10.1&amp;p=20361&amp;pt=375&amp;video_id=glvfBsNTgRc&amp;video_target=tpm-plugin-tf4lbpbl-glvfBsNTgRc\" width=\"800px\" height=\"450px\" frameborder=\"0\" marginwidth=\"0px\" marginheight=\"0px\" data-mce-fragment=\"1\"><\/iframe><\/center><center>You can view the transcript for <a href=\"https:\/\/course-building.s3.us-west-2.amazonaws.com\/Calculus+3\/Calc+3+transcripts\/CP2.55_transcript.html\">\u201cCP 2.55\u201d here (opens in new window).<\/a><\/center><\/section><section class=\"textbox example\" aria-label=\"Example\">Convert point [latex](-8, 8, -7)[\/latex] from Cartesian coordinates to cylindrical coordinates.[reveal-answer q=\"004357132\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"004357132\"][latex](8\\sqrt2,\\frac{3\\pi}4,-7)[\/latex][\/hidden-answer]<\/section><section class=\"textbox example\" aria-label=\"Example\">Describe the surface with cylindrical equation [latex]r=6[\/latex].[reveal-answer q=\"765298743\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"765298743\"]Describe the surface with cylindrical equation [latex]r=6[\/latex].[caption id=\"attachment_5291\" align=\"aligncenter\" width=\"276\"]<img class=\"size-full wp-image-5291\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/5667\/2021\/08\/31230631\/2.57.jpg\" alt=\"This figure is a right circular cylinder. It is upright with the z-axis through the center. It is on top of the x y plane.\" width=\"276\" height=\"380\" \/> Figure 10. The surface with cylindrical equation [latex]r=6[\/latex].[\/caption][\/hidden-answer]<\/section>\r\n<h2>Spherical Coordinates<\/h2>\r\n<div class=\"textbox shaded\">\r\n\r\n<strong>The Main Idea\u00a0<\/strong>\r\n<p class=\"whitespace-normal break-words\"><strong>Spherical coordinates<\/strong> use one distance and two angles to locate points in 3D space. They excel at problems with spherical symmetry\u2014like planetary fields, domed structures, or radiation from a point source.<\/p>\r\n<p class=\"whitespace-normal break-words\"><strong>Spherical Coordinate System<\/strong><\/p>\r\n<p class=\"whitespace-normal break-words\">A point is represented by [latex](\\rho, \\theta, \\varphi)[\/latex]:<\/p>\r\n\r\n<ul class=\"[&amp;:not(:last-child)_ul]:pb-1 [&amp;:not(:last-child)_ol]:pb-1 list-disc space-y-2.5 pl-7\">\r\n \t<li class=\"whitespace-normal break-words\">[latex]\\rho[\/latex] (rho): distance from the origin to the point<\/li>\r\n \t<li class=\"whitespace-normal break-words\">[latex]\\theta[\/latex]: angle in the [latex]xy[\/latex]-plane (same as cylindrical)<\/li>\r\n \t<li class=\"whitespace-normal break-words\">[latex]\\varphi[\/latex] (phi): angle down from the positive [latex]z[\/latex]-axis ([latex]0 \\leq \\varphi \\leq \\pi[\/latex])<\/li>\r\n<\/ul>\r\n<p class=\"whitespace-normal break-words\"><strong>Key point<\/strong>: [latex]\\varphi[\/latex] measures from the [latex]z[\/latex]-axis, not from the [latex]xy[\/latex]-plane. So [latex]\\varphi = 0[\/latex] points straight up, [latex]\\varphi = \\frac{\\pi}{2}[\/latex] lies in the [latex]xy[\/latex]-plane, and [latex]\\varphi = \\pi[\/latex] points straight down.<\/p>\r\n<p class=\"whitespace-normal break-words\"><strong>Conversion Formulas<\/strong><\/p>\r\n<p class=\"whitespace-normal break-words\">Spherical to Rectangular:<\/p>\r\n\r\n<ul class=\"[&amp;:not(:last-child)_ul]:pb-1 [&amp;:not(:last-child)_ol]:pb-1 list-disc space-y-2.5 pl-7\">\r\n \t<li class=\"whitespace-normal break-words\">[latex]x = \\rho \\sin \\varphi \\cos \\theta[\/latex]<\/li>\r\n \t<li class=\"whitespace-normal break-words\">[latex]y = \\rho \\sin \\varphi \\sin \\theta[\/latex]<\/li>\r\n \t<li class=\"whitespace-normal break-words\">[latex]z = \\rho \\cos \\varphi[\/latex]<\/li>\r\n<\/ul>\r\n<p class=\"whitespace-normal break-words\">Rectangular to Spherical:<\/p>\r\n\r\n<ul class=\"[&amp;:not(:last-child)_ul]:pb-1 [&amp;:not(:last-child)_ol]:pb-1 list-disc space-y-2.5 pl-7\">\r\n \t<li class=\"whitespace-normal break-words\">[latex]\\rho = \\sqrt{x^2 + y^2 + z^2}[\/latex]<\/li>\r\n \t<li class=\"whitespace-normal break-words\">[latex]\\tan \\theta = \\frac{y}{x}[\/latex] (choose quadrant)<\/li>\r\n \t<li class=\"whitespace-normal break-words\">[latex]\\varphi = \\arccos\\left(\\frac{z}{\\sqrt{x^2 + y^2 + z^2}}\\right)[\/latex]<\/li>\r\n<\/ul>\r\n<p class=\"whitespace-normal break-words\"><strong>Constant Coordinate Surfaces<\/strong><\/p>\r\n\r\n<ul class=\"[&amp;:not(:last-child)_ul]:pb-1 [&amp;:not(:last-child)_ol]:pb-1 list-disc space-y-2.5 pl-7\">\r\n \t<li class=\"whitespace-normal break-words\">[latex]\\rho = c[\/latex]: <strong>Sphere<\/strong> of radius [latex]c[\/latex] centered at origin<\/li>\r\n \t<li class=\"whitespace-normal break-words\">[latex]\\theta = c[\/latex]: <strong>Half-plane<\/strong> from [latex]z[\/latex]-axis at angle [latex]c[\/latex]<\/li>\r\n \t<li class=\"whitespace-normal break-words\">[latex]\\varphi = c[\/latex]: <strong>Half-cone<\/strong> making angle [latex]c[\/latex] with positive [latex]z[\/latex]-axis<\/li>\r\n<\/ul>\r\n<p class=\"whitespace-normal break-words\"><strong>Why Use Spherical Coordinates?<\/strong><\/p>\r\n<p class=\"whitespace-normal break-words\">They simplify spherical symmetry problems. A sphere [latex]x^2 + y^2 + z^2 = c^2[\/latex] becomes simply [latex]\\rho = c[\/latex].<\/p>\r\n<p class=\"whitespace-normal break-words\"><strong>Real-World Application<\/strong>: Latitude and longitude use spherical-like coordinates to describe Earth locations. The coordinate system matches the natural shape and symmetry of the problem.<\/p>\r\n<p class=\"whitespace-normal break-words\"><strong>Choosing Coordinates<\/strong>: Match the coordinate system to your problem's symmetry\u2014rectangular for general cases, cylindrical for rotation around an axis, spherical for radiation from a point.<\/p>\r\n\r\n<\/div>\r\n<section class=\"textbox example\" aria-label=\"Example\">Plot the point with spherical coordinates [latex]\\left(2,-\\frac{5\\pi}6,\\frac{\\pi}6\\right)[\/latex] and describe its location in both rectangular and cylindrical coordinates.[reveal-answer q=\"387456286\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"387456286\"][caption id=\"attachment_5307\" align=\"aligncenter\" width=\"657\"]<img class=\"size-full wp-image-5307\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/5667\/2021\/08\/31231825\/2.58.jpg\" alt=\"This figure is of the 3-dimensional coordinate system. It has a point. There is a line segment from the origin to the point. The angle between this line segment and the z-axis is phi. There is a line segment in the x y-plane from the origin to the shadow of the point.The angle between the x-axis and rho is theta.\" width=\"657\" height=\"392\" \/> Figure 5. The point with spherical coordinates [latex]\\left(2,-\\frac{5\\pi}6,\\frac{\\pi}6\\right)[\/latex][\/caption]Cartesian: [latex]\\left(-\\frac{\\sqrt{3}}2,-\\frac12,\\sqrt3\\right)[\/latex], cylindrical: [latex]\\left(1,-\\frac{5\\pi}6,\\sqrt3\\right)[\/latex][\/hidden-answer]<\/section><section class=\"textbox watchIt\" aria-label=\"Watch It\">Watch the following video to see the worked solution to the above example.<center><iframe src=\"\/\/plugin.3playmedia.com\/show?mf=7809469&amp;p3sdk_version=1.10.1&amp;p=20361&amp;pt=375&amp;video_id=yrKGmi_JFbM&amp;video_target=tpm-plugin-f16mh34p-yrKGmi_JFbM\" width=\"800px\" height=\"450px\" frameborder=\"0\" marginwidth=\"0px\" marginheight=\"0px\" data-mce-fragment=\"1\"><\/iframe><\/center><center>You can view the transcript for <a href=\"https:\/\/course-building.s3.us-west-2.amazonaws.com\/Calculus+3\/Calc+3+transcripts\/CP2.58_transcript.html\">\u201cCP 2.58\u201d here (opens in new window).<\/a><\/center><\/section><section class=\"textbox example\" aria-label=\"Example\">\r\n<p id=\"fs-id1163723523037\">Describe the surfaces defined by the following equations.<\/p>\r\n\r\n<ol id=\"fs-id1163723523040\" type=\"a\">\r\n \t<li>[latex]\\rho=13[\/latex]<\/li>\r\n \t<li>[latex]\\theta=\\frac{2\\pi}3[\/latex]<\/li>\r\n \t<li>[latex]\\varphi=\\frac{\\pi}4[\/latex]<\/li>\r\n<\/ol>\r\n[reveal-answer q=\"874652711\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"874652711\"]\r\n\r\na. This is the set of all points [latex]13[\/latex] units from the origin. This set forms a sphere with radius [latex]13[\/latex].\r\n\r\nb. This set of points forms a half plane. The angle between the half plane and the positive [latex]x[\/latex]-axis is [latex]\\theta=\\frac{2\\pi}3[\/latex].\r\n\r\nc. Let [latex]P[\/latex] be a point on this surface. The position vector of this point forms an angle of [latex]\\varphi=\\frac{\\pi}4[\/latex] with the positive [latex]z[\/latex]-axis, which means that points closer to the origin are closer to the axis. These points form a half-cone.\r\n\r\n[\/hidden-answer]\r\n\r\n<\/section><section class=\"textbox example\" aria-label=\"Example\">Sydney, Australia is at [latex]34^{\\small\\circ}[\/latex]S and [latex]151^{\\small\\circ}[\/latex]E. Express Sydney\u2019s location in spherical coordinates.[reveal-answer q=\"467253716\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"467253716\"][latex](4,000,151^{\\small\\circ},124^{\\small\\circ})[\/latex][\/hidden-answer]<\/section><section class=\"textbox example\" aria-label=\"Example\">Which coordinate system is most appropriate for creating a star map, as viewed from Earth (see the following figure)?\r\n\r\n[caption id=\"attachment_5322\" align=\"aligncenter\" width=\"731\"]<img class=\"size-full wp-image-5322\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/5667\/2021\/08\/31232545\/2.61.jpg\" alt=\"This figure is a circle with a star chart in the middle.\" width=\"731\" height=\"731\" \/> Figure 11. Star map as viewed from Earth.[\/caption]\r\n\r\nHow should we orient the coordinate axes?\r\n\r\n[reveal-answer q=\"883476235\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"883476235\"]\r\n\r\nSpherical coordinates with the origin located at the center of the earth, the [latex]z[\/latex]-axis aligned with the North Pole, and the [latex]x[\/latex]-axis aligned with the prime meridian.\r\n\r\n[\/hidden-answer]\r\n\r\n<\/section>\r\n<h2 class=\"text-2xl font-bold mt-1 text-text-100\">Cylinders and Surfaces in 3D<\/h2>\r\n<div class=\"textbox shaded\">\r\n\r\n<strong>The Main Idea\u00a0<\/strong>\r\n<p class=\"whitespace-normal break-words\">The mathematical definition of a <strong>cylinder<\/strong> is broader than the everyday \"hollow tube\" image\u2014any curve extended by parallel lines creates a cylinder.<\/p>\r\n<p class=\"whitespace-normal break-words\">A <strong>cylinder<\/strong> (or cylindrical surface) is formed by a set of parallel lines (called <strong>rulings<\/strong>) passing through a given curve. The curve can be any shape, and the rulings can be parallel to any direction.<\/p>\r\n<p class=\"whitespace-normal break-words\"><strong>Identifying Cylinders from Equations<\/strong><\/p>\r\n<p class=\"whitespace-normal break-words\">An equation describes a cylinder if <strong>one variable is missing<\/strong>. The missing variable tells you the direction of the rulings:<\/p>\r\n\r\n<ul class=\"[&amp;:not(:last-child)_ul]:pb-1 [&amp;:not(:last-child)_ol]:pb-1 list-disc space-y-2.5 pl-7\">\r\n \t<li class=\"whitespace-normal break-words\">Missing [latex]z[\/latex]: rulings parallel to [latex]z[\/latex]-axis<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Missing [latex]y[\/latex]: rulings parallel to [latex]y[\/latex]-axis<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Missing [latex]x[\/latex]: rulings parallel to [latex]x[\/latex]-axis<\/li>\r\n<\/ul>\r\n<p class=\"whitespace-normal break-words\"><strong>Examples<\/strong>:<\/p>\r\n\r\n<ul class=\"[&amp;:not(:last-child)_ul]:pb-1 [&amp;:not(:last-child)_ol]:pb-1 list-disc space-y-2.5 pl-7\">\r\n \t<li class=\"whitespace-normal break-words\">[latex]x^2 + y^2 = 9[\/latex]: circular cylinder (rulings parallel to [latex]z[\/latex]-axis)<\/li>\r\n \t<li class=\"whitespace-normal break-words\">[latex]x^2 + z^2 = 25[\/latex]: circular cylinder (rulings parallel to [latex]y[\/latex]-axis)<\/li>\r\n \t<li class=\"whitespace-normal break-words\">[latex]y = \\sin x[\/latex]: sinusoidal cylinder (rulings parallel to [latex]z[\/latex]-axis)<\/li>\r\n<\/ul>\r\n<p class=\"whitespace-normal break-words\"><strong>Traces<\/strong><\/p>\r\n<p class=\"whitespace-normal break-words\">Traces are the curves formed when a surface intersects planes parallel to coordinate planes ([latex]xy[\/latex], [latex]xz[\/latex], or [latex]yz[\/latex]).<\/p>\r\n<p class=\"whitespace-normal break-words\"><strong>For cylinders<\/strong>: Only one trace is informative\u2014the one perpendicular to the rulings shows the generating curve. Traces parallel to the rulings just show parallel lines.<\/p>\r\n<p class=\"whitespace-normal break-words\"><strong>Example<\/strong>: For [latex]y = \\sin x[\/latex]:<\/p>\r\n\r\n<ul class=\"[&amp;:not(:last-child)_ul]:pb-1 [&amp;:not(:last-child)_ol]:pb-1 list-disc space-y-2.5 pl-7\">\r\n \t<li class=\"whitespace-normal break-words\">[latex]xz[\/latex]-plane trace ([latex]y = 0[\/latex]): Shows the sine wave<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Other traces: Just parallel lines (not helpful)<\/li>\r\n<\/ul>\r\n<p class=\"whitespace-normal break-words\"><strong>Key Insight<\/strong>: Cylinders are the simplest 3D surfaces\u2014they're just 2D curves extended in a straight direction. More complex surfaces require examining multiple traces to understand their full 3D shape.<\/p>\r\n\r\n<\/div>\r\n<section class=\"textbox example\" aria-label=\"Example\">Sketch or use a graphing tool to view the graph of the cylindrical surface defined by equation [latex]z=y^{2}[\/latex].[reveal-answer q=\"378465244\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"378465244\"][caption id=\"attachment_5210\" align=\"aligncenter\" width=\"288\"]<img class=\"size-full wp-image-5210\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/5667\/2021\/08\/31203348\/2.52.jpg\" alt=\"This figure is a surface above the x y plane. A cross section of this surface parallel to the y z plane would be a parabola. The surface sits on top of the x y plane.\" width=\"288\" height=\"324\" \/> Figure 6. The cylindrical surface defined by equation [latex]z=y^{2}[\/latex].[\/caption][\/hidden-answer]<\/section><section class=\"textbox watchIt\" aria-label=\"Watch It\">Watch the following video to see the worked solution to the above example.<center><iframe src=\"\/\/plugin.3playmedia.com\/show?mf=7809453&amp;p3sdk_version=1.10.1&amp;p=20361&amp;pt=375&amp;video_id=ygBugCYrufQ&amp;video_target=tpm-plugin-mxtub258-ygBugCYrufQ\" width=\"800px\" height=\"450px\" frameborder=\"0\" marginwidth=\"0px\" marginheight=\"0px\" data-mce-fragment=\"1\"><\/iframe><\/center><center><span style=\"font-size: 1rem; orphans: 1; text-align: initial;\">You can view the transcript for <a href=\"https:\/\/course-building.s3.us-west-2.amazonaws.com\/Calculus+3\/Calc+3+transcripts\/CP2.52_transcript.html\">\u201cCP 2.52\u201d here (opens in new window).<\/a><\/span><\/center><\/section>\r\n<h2 data-type=\"title\">Quadric Surfaces<\/h2>\r\n<div class=\"textbox shaded\">\r\n\r\n<strong>The Main Idea\u00a0<\/strong>\r\n<p class=\"whitespace-normal break-words\"><strong>Quadric surfaces<\/strong> are 3D extensions of conic sections (ellipses, parabolas, hyperbolas), described by second-degree equations. Understanding their traces reveals their shape.<\/p>\r\n<p class=\"whitespace-normal break-words\"><strong>General Form<\/strong><\/p>\r\n<p class=\"whitespace-normal break-words\" style=\"text-align: center;\">[latex]Ax^2 + By^2 + Cz^2 + Dxy + Exz + Fyz + Gx + Hy + Jz + K = 0[\/latex]<\/p>\r\n<p class=\"whitespace-normal break-words\"><strong>Key Surfaces<\/strong><\/p>\r\n<p class=\"whitespace-normal break-words\"><strong>Ellipsoid<\/strong>: [latex]\\frac{x^2}{a^2} + \\frac{y^2}{b^2} + \\frac{z^2}{c^2} = 1[\/latex]<\/p>\r\n\r\n<ul class=\"[&amp;:not(:last-child)_ul]:pb-1 [&amp;:not(:last-child)_ol]:pb-1 list-disc space-y-2.5 pl-7\">\r\n \t<li class=\"whitespace-normal break-words\">All traces are ellipses<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Special case: [latex]a = b = c[\/latex] gives a sphere<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Extends [latex]a[\/latex], [latex]b[\/latex], [latex]c[\/latex] units along each axis<\/li>\r\n<\/ul>\r\n<p class=\"whitespace-normal break-words\"><strong>Elliptic Paraboloid<\/strong>: [latex]\\frac{x^2}{a^2} + \\frac{y^2}{b^2} = \\frac{z}{c}[\/latex]<\/p>\r\n\r\n<ul class=\"[&amp;:not(:last-child)_ul]:pb-1 [&amp;:not(:last-child)_ol]:pb-1 list-disc space-y-2.5 pl-7\">\r\n \t<li class=\"whitespace-normal break-words\">Bowl-shaped surface<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Traces perpendicular to axis: ellipses (or circles if [latex]a = b[\/latex])<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Traces parallel to axis: parabolas<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Opens up if [latex]c &gt; 0[\/latex], down if [latex]c &lt; 0[\/latex]<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Variable to first power determines axis direction<\/li>\r\n<\/ul>\r\n<p class=\"whitespace-normal break-words\"><strong>Using Traces to Identify Surfaces<\/strong><\/p>\r\n<p class=\"whitespace-normal break-words\"><strong>Traces<\/strong> are cross-sections where the surface meets coordinate planes. To identify a quadric surface:<\/p>\r\n\r\n<ol class=\"[&amp;:not(:last-child)_ul]:pb-1 [&amp;:not(:last-child)_ol]:pb-1 list-decimal space-y-2.5 pl-7\">\r\n \t<li class=\"whitespace-normal break-words\">Check which variables are squared vs. first power<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Look at signs (positive vs. negative coefficients)<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Find traces by setting [latex]x = 0[\/latex], [latex]y = 0[\/latex], and [latex]z = 0[\/latex]<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Put equation in standard form if needed<\/li>\r\n<\/ol>\r\n<p class=\"whitespace-normal break-words\"><strong>Identification Strategy<\/strong>:<\/p>\r\n\r\n<ul class=\"[&amp;:not(:last-child)_ul]:pb-1 [&amp;:not(:last-child)_ol]:pb-1 list-disc space-y-2.5 pl-7\">\r\n \t<li class=\"whitespace-normal break-words\">All squared terms with same sign \u2192 ellipsoid or sphere<\/li>\r\n \t<li class=\"whitespace-normal break-words\">One first-power term, two squared (same sign) \u2192 elliptic paraboloid<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Mixed signs in squared terms \u2192 hyperboloid or hyperbolic paraboloid<\/li>\r\n<\/ul>\r\n<p class=\"whitespace-normal break-words\"><strong>Real-World Applications<\/strong><\/p>\r\n\r\n<ul class=\"[&amp;:not(:last-child)_ul]:pb-1 [&amp;:not(:last-child)_ol]:pb-1 list-disc space-y-2.5 pl-7\">\r\n \t<li class=\"whitespace-normal break-words\"><strong>Parabolic reflectors<\/strong>: Satellite dishes and solar collectors use elliptic paraboloids because energy hitting the surface concentrates at the focal point<\/li>\r\n \t<li class=\"whitespace-normal break-words\"><strong>Hyperboloids<\/strong>: Nuclear cooling towers use hyperboloids of one sheet\u2014they can be built with straight beams, making them strong yet economical<\/li>\r\n<\/ul>\r\n<\/div>\r\n<section class=\"textbox example\" aria-label=\"Example\">A hyperboloid of one sheet is any surface that can be described with an equation of the form [latex]\\frac{x^2}{a^2}+\\frac{y^2}{b^2}-\\frac{z^2}{c^2}=1[\/latex]. Describe the traces of the hyperboloid of one sheet given by equation [latex]\\frac{x^2}{3^2}+\\frac{y^2}{2^2}-\\frac{z^2}{5^2}=1[\/latex].[reveal-answer q=\"785629873\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"785629873\"]\r\n<p id=\"fs-id1163724104708\">The traces parallel to the [latex]xy[\/latex]-plane are ellipses and the traces parallel to the [latex]xz[\/latex]- and [latex]yz[\/latex]-planes are hyperbolas. Specifically, the trace in the [latex]xy[\/latex]-plane is ellipse [latex]\\frac{x^2}{3^2}+\\frac{y^2}{2^2}=1[\/latex], the trace in the [latex]xz[\/latex]-plane is hyperbola [latex]\\frac{x^2}{3^2}-\\frac{z^2}{5^2}=1[\/latex], and the trace in the [latex]yz[\/latex]-plane is hyperbola [latex]\\frac{y^2}{2^2}-\\frac{z^2}{5^2}=1[\/latex] (see the following figure).<span data-type=\"newline\">\r\n<\/span><\/p>\r\n\r\n[caption id=\"attachment_5224\" align=\"aligncenter\" width=\"876\"]<img class=\"size-full wp-image-5224\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/5667\/2021\/08\/31204619\/2.53.jpg\" alt=\"A hyperboloid of one sheet is any surface that can be described with an equation of the form\" width=\"876\" height=\"914\" \/> Figure 5. (a) The trace in the [latex]xy[\/latex]-plane. (b) The trace in the [latex]xz[\/latex]-plane. (c) The trace in the [latex]yz[\/latex]-plane. (d) The hyperboloid [latex]\\frac{x^2}{3^2}+\\frac{y^2}{2^2}-\\frac{z^2}{5^2}=1[\/latex].[\/caption][\/hidden-answer]<\/section><section class=\"textbox watchIt\" aria-label=\"Watch It\">Watch the following video to see the worked solution to the above example.<center><iframe src=\"\/\/plugin.3playmedia.com\/show?mf=7809454&amp;p3sdk_version=1.10.1&amp;p=20361&amp;pt=375&amp;video_id=qYftkr5uwzo&amp;video_target=tpm-plugin-znpykjkm-qYftkr5uwzo\" width=\"800px\" height=\"450px\" frameborder=\"0\" marginwidth=\"0px\" marginheight=\"0px\" data-mce-fragment=\"1\"><\/iframe><\/center><center><span style=\"font-size: 1rem; orphans: 1; text-align: initial;\">You can view the transcript for <a href=\"https:\/\/course-building.s3.us-west-2.amazonaws.com\/Calculus+3\/Calc+3+transcripts\/CP2.53_transcript.html\">\u201cCP 2.53\u201d here (opens in new window).<\/a><\/span><\/center><\/section><section class=\"textbox example\" aria-label=\"Example\">Identify the surface represented by equation [latex]9x^{2}+y^{2}-z^{2}+2z-10=0[\/latex].[reveal-answer q=\"752345122\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"752345122\"]Hyperboloid of one sheet, centered at [latex](0, 0, 1)[\/latex].[\/hidden-answer]<\/section>","rendered":"<section class=\"textbox learningGoals\" aria-label=\"Learning Goals\">\n<ul>\n<li>Switch between cylindrical and rectangular coordinates<\/li>\n<li>Switch between spherical and rectangular coordinates<\/li>\n<li>Recognize and describe different types of quadric surfaces<\/li>\n<li>Analyze 3D surfaces by looking at their cross-sections and traces<\/li>\n<\/ul>\n<\/section>\n<h2>Cylindrical Coordinates<\/h2>\n<div class=\"textbox shaded\">\n<p><strong>The Main Idea\u00a0<\/strong><\/p>\n<p class=\"whitespace-normal break-words\">Just as polar coordinates simplify circular problems in 2D, <strong>cylindrical coordinates<\/strong> extend this advantage to 3D, making problems with cylindrical symmetry much easier to handle.<\/p>\n<p class=\"whitespace-normal break-words\"><strong>Cylindrical Coordinate System<\/strong><\/p>\n<p class=\"whitespace-normal break-words\">A point in space is represented by [latex](r, \\theta, z)[\/latex]:<\/p>\n<ul class=\"[&amp;:not(:last-child)_ul]:pb-1 [&amp;:not(:last-child)_ol]:pb-1 list-disc space-y-2.5 pl-7\">\n<li class=\"whitespace-normal break-words\">[latex](r, \\theta)[\/latex]: polar coordinates of the point&#8217;s projection onto the [latex]xy[\/latex]-plane<\/li>\n<li class=\"whitespace-normal break-words\">[latex]z[\/latex]: height above or below the [latex]xy[\/latex]-plane<\/li>\n<\/ul>\n<p class=\"whitespace-normal break-words\"><strong>Tip<\/strong>: Find the point&#8217;s &#8220;shadow&#8221; on the floor using polar coordinates, then specify how high\/low it sits.<\/p>\n<p class=\"whitespace-normal break-words\"><strong>Conversion Formulas<\/strong><\/p>\n<p class=\"whitespace-normal break-words\">Cylindrical to Rectangular:<\/p>\n<ul class=\"[&amp;:not(:last-child)_ul]:pb-1 [&amp;:not(:last-child)_ol]:pb-1 list-disc space-y-2.5 pl-7\">\n<li class=\"whitespace-normal break-words\">[latex]x = r\\cos\\theta[\/latex]<\/li>\n<li class=\"whitespace-normal break-words\">[latex]y = r\\sin\\theta[\/latex]<\/li>\n<li class=\"whitespace-normal break-words\">[latex]z = z[\/latex]<\/li>\n<\/ul>\n<p class=\"whitespace-normal break-words\">Rectangular to Cylindrical:<\/p>\n<ul class=\"[&amp;:not(:last-child)_ul]:pb-1 [&amp;:not(:last-child)_ol]:pb-1 list-disc space-y-2.5 pl-7\">\n<li class=\"whitespace-normal break-words\">[latex]r = \\sqrt{x^2 + y^2}[\/latex]<\/li>\n<li class=\"whitespace-normal break-words\">[latex]\\tan\\theta = \\frac{y}{x}[\/latex] (choose [latex]\\theta[\/latex] based on quadrant)<\/li>\n<li class=\"whitespace-normal break-words\">[latex]z = z[\/latex]<\/li>\n<\/ul>\n<p class=\"whitespace-normal break-words\"><strong>Constant Coordinate Surfaces<\/strong><\/p>\n<ul class=\"[&amp;:not(:last-child)_ul]:pb-1 [&amp;:not(:last-child)_ol]:pb-1 list-disc space-y-2.5 pl-7\">\n<li class=\"whitespace-normal break-words\">[latex]r = c[\/latex]: Vertical circular cylinder of radius [latex]c[\/latex] (fixed distance from [latex]z[\/latex]-axis)<\/li>\n<li class=\"whitespace-normal break-words\">[latex]\\theta = c[\/latex]: Half-plane starting at [latex]z[\/latex]-axis, making angle [latex]c[\/latex] with positive [latex]x[\/latex]-axis<\/li>\n<li class=\"whitespace-normal break-words\">[latex]z = c[\/latex]: Horizontal plane (same as rectangular coordinates)<\/li>\n<\/ul>\n<p class=\"whitespace-normal break-words\"><strong>Why Use Cylindrical Coordinates?<\/strong><\/p>\n<p class=\"whitespace-normal break-words\">They simplify equations with cylindrical symmetry. For example:<\/p>\n<ul class=\"[&amp;:not(:last-child)_ul]:pb-1 [&amp;:not(:last-child)_ol]:pb-1 list-disc space-y-2.5 pl-7\">\n<li class=\"whitespace-normal break-words\">Rectangular: [latex]x^2 + y^2 = 25[\/latex] (a cylinder)<\/li>\n<li class=\"whitespace-normal break-words\">Cylindrical: [latex]r = 5[\/latex] (much simpler!)<\/li>\n<\/ul>\n<\/div>\n<section class=\"textbox example\" aria-label=\"Example\">Point [latex]R[\/latex] has cylindrical coordinates [latex]\\left(5,\\frac{\\pi}6,4\\right)[\/latex]. Plot [latex]R[\/latex] and describe its location in space using rectangular, or Cartesian, coordinates.<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"q572615912\">Show Solution<\/button><\/p>\n<div id=\"q572615912\" class=\"hidden-answer\" style=\"display: none\">The rectangular coordinates of the point are [latex]\\left(\\frac{5\\sqrt3}2,\\frac52,4\\right)[\/latex].\u00a0<span data-type=\"newline\"><br \/>\n<\/span><\/p>\n<figure id=\"attachment_5281\" aria-describedby=\"caption-attachment-5281\" style=\"width: 462px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" class=\"size-full wp-image-5281\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/5667\/2021\/08\/31230153\/2.55.jpg\" alt=\"This figure is the 3-dimensional coordinate system. There is a point labeled \u201c(5, pi\/6, 4).\u201d The point is located above a line segment in the x y-plane labeled r = 5 that is pi\/6 degrees from the x-axis. The distance from the x y-plane to the point is labeled \u201cz = 4.\u201d\" width=\"462\" height=\"370\" \/><figcaption id=\"caption-attachment-5281\" class=\"wp-caption-text\">Figure 6. The rectangular coordinates of the point are [latex]\\left(\\frac{5\\sqrt3}2,\\frac52,4\\right)[\/latex]<\/figcaption><\/figure>\n<\/div>\n<\/div>\n<\/section>\n<section class=\"textbox watchIt\" aria-label=\"Watch It\">Watch the following video to see the worked solution to the above example.<\/p>\n<div style=\"text-align: center;\"><iframe loading=\"lazy\" src=\"\/\/plugin.3playmedia.com\/show?mf=7809468&amp;p3sdk_version=1.10.1&amp;p=20361&amp;pt=375&amp;video_id=glvfBsNTgRc&amp;video_target=tpm-plugin-tf4lbpbl-glvfBsNTgRc\" width=\"800px\" height=\"450px\" frameborder=\"0\" marginwidth=\"0px\" marginheight=\"0px\" data-mce-fragment=\"1\"><\/iframe><\/div>\n<div style=\"text-align: center;\">You can view the transcript for <a href=\"https:\/\/course-building.s3.us-west-2.amazonaws.com\/Calculus+3\/Calc+3+transcripts\/CP2.55_transcript.html\">\u201cCP 2.55\u201d here (opens in new window).<\/a><\/div>\n<\/section>\n<section class=\"textbox example\" aria-label=\"Example\">Convert point [latex](-8, 8, -7)[\/latex] from Cartesian coordinates to cylindrical coordinates.<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"q004357132\">Show Solution<\/button><\/p>\n<div id=\"q004357132\" class=\"hidden-answer\" style=\"display: none\">[latex](8\\sqrt2,\\frac{3\\pi}4,-7)[\/latex]<\/div>\n<\/div>\n<\/section>\n<section class=\"textbox example\" aria-label=\"Example\">Describe the surface with cylindrical equation [latex]r=6[\/latex].<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"q765298743\">Show Solution<\/button><\/p>\n<div id=\"q765298743\" class=\"hidden-answer\" style=\"display: none\">Describe the surface with cylindrical equation [latex]r=6[\/latex].<\/p>\n<figure id=\"attachment_5291\" aria-describedby=\"caption-attachment-5291\" style=\"width: 276px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" class=\"size-full wp-image-5291\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/5667\/2021\/08\/31230631\/2.57.jpg\" alt=\"This figure is a right circular cylinder. It is upright with the z-axis through the center. It is on top of the x y plane.\" width=\"276\" height=\"380\" \/><figcaption id=\"caption-attachment-5291\" class=\"wp-caption-text\">Figure 10. The surface with cylindrical equation [latex]r=6[\/latex].<\/figcaption><\/figure>\n<\/div>\n<\/div>\n<\/section>\n<h2>Spherical Coordinates<\/h2>\n<div class=\"textbox shaded\">\n<p><strong>The Main Idea\u00a0<\/strong><\/p>\n<p class=\"whitespace-normal break-words\"><strong>Spherical coordinates<\/strong> use one distance and two angles to locate points in 3D space. They excel at problems with spherical symmetry\u2014like planetary fields, domed structures, or radiation from a point source.<\/p>\n<p class=\"whitespace-normal break-words\"><strong>Spherical Coordinate System<\/strong><\/p>\n<p class=\"whitespace-normal break-words\">A point is represented by [latex](\\rho, \\theta, \\varphi)[\/latex]:<\/p>\n<ul class=\"[&amp;:not(:last-child)_ul]:pb-1 [&amp;:not(:last-child)_ol]:pb-1 list-disc space-y-2.5 pl-7\">\n<li class=\"whitespace-normal break-words\">[latex]\\rho[\/latex] (rho): distance from the origin to the point<\/li>\n<li class=\"whitespace-normal break-words\">[latex]\\theta[\/latex]: angle in the [latex]xy[\/latex]-plane (same as cylindrical)<\/li>\n<li class=\"whitespace-normal break-words\">[latex]\\varphi[\/latex] (phi): angle down from the positive [latex]z[\/latex]-axis ([latex]0 \\leq \\varphi \\leq \\pi[\/latex])<\/li>\n<\/ul>\n<p class=\"whitespace-normal break-words\"><strong>Key point<\/strong>: [latex]\\varphi[\/latex] measures from the [latex]z[\/latex]-axis, not from the [latex]xy[\/latex]-plane. So [latex]\\varphi = 0[\/latex] points straight up, [latex]\\varphi = \\frac{\\pi}{2}[\/latex] lies in the [latex]xy[\/latex]-plane, and [latex]\\varphi = \\pi[\/latex] points straight down.<\/p>\n<p class=\"whitespace-normal break-words\"><strong>Conversion Formulas<\/strong><\/p>\n<p class=\"whitespace-normal break-words\">Spherical to Rectangular:<\/p>\n<ul class=\"[&amp;:not(:last-child)_ul]:pb-1 [&amp;:not(:last-child)_ol]:pb-1 list-disc space-y-2.5 pl-7\">\n<li class=\"whitespace-normal break-words\">[latex]x = \\rho \\sin \\varphi \\cos \\theta[\/latex]<\/li>\n<li class=\"whitespace-normal break-words\">[latex]y = \\rho \\sin \\varphi \\sin \\theta[\/latex]<\/li>\n<li class=\"whitespace-normal break-words\">[latex]z = \\rho \\cos \\varphi[\/latex]<\/li>\n<\/ul>\n<p class=\"whitespace-normal break-words\">Rectangular to Spherical:<\/p>\n<ul class=\"[&amp;:not(:last-child)_ul]:pb-1 [&amp;:not(:last-child)_ol]:pb-1 list-disc space-y-2.5 pl-7\">\n<li class=\"whitespace-normal break-words\">[latex]\\rho = \\sqrt{x^2 + y^2 + z^2}[\/latex]<\/li>\n<li class=\"whitespace-normal break-words\">[latex]\\tan \\theta = \\frac{y}{x}[\/latex] (choose quadrant)<\/li>\n<li class=\"whitespace-normal break-words\">[latex]\\varphi = \\arccos\\left(\\frac{z}{\\sqrt{x^2 + y^2 + z^2}}\\right)[\/latex]<\/li>\n<\/ul>\n<p class=\"whitespace-normal break-words\"><strong>Constant Coordinate Surfaces<\/strong><\/p>\n<ul class=\"[&amp;:not(:last-child)_ul]:pb-1 [&amp;:not(:last-child)_ol]:pb-1 list-disc space-y-2.5 pl-7\">\n<li class=\"whitespace-normal break-words\">[latex]\\rho = c[\/latex]: <strong>Sphere<\/strong> of radius [latex]c[\/latex] centered at origin<\/li>\n<li class=\"whitespace-normal break-words\">[latex]\\theta = c[\/latex]: <strong>Half-plane<\/strong> from [latex]z[\/latex]-axis at angle [latex]c[\/latex]<\/li>\n<li class=\"whitespace-normal break-words\">[latex]\\varphi = c[\/latex]: <strong>Half-cone<\/strong> making angle [latex]c[\/latex] with positive [latex]z[\/latex]-axis<\/li>\n<\/ul>\n<p class=\"whitespace-normal break-words\"><strong>Why Use Spherical Coordinates?<\/strong><\/p>\n<p class=\"whitespace-normal break-words\">They simplify spherical symmetry problems. A sphere [latex]x^2 + y^2 + z^2 = c^2[\/latex] becomes simply [latex]\\rho = c[\/latex].<\/p>\n<p class=\"whitespace-normal break-words\"><strong>Real-World Application<\/strong>: Latitude and longitude use spherical-like coordinates to describe Earth locations. The coordinate system matches the natural shape and symmetry of the problem.<\/p>\n<p class=\"whitespace-normal break-words\"><strong>Choosing Coordinates<\/strong>: Match the coordinate system to your problem&#8217;s symmetry\u2014rectangular for general cases, cylindrical for rotation around an axis, spherical for radiation from a point.<\/p>\n<\/div>\n<section class=\"textbox example\" aria-label=\"Example\">Plot the point with spherical coordinates [latex]\\left(2,-\\frac{5\\pi}6,\\frac{\\pi}6\\right)[\/latex] and describe its location in both rectangular and cylindrical coordinates.<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"q387456286\">Show Solution<\/button><\/p>\n<div id=\"q387456286\" class=\"hidden-answer\" style=\"display: none\">\n<figure id=\"attachment_5307\" aria-describedby=\"caption-attachment-5307\" style=\"width: 657px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" class=\"size-full wp-image-5307\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/5667\/2021\/08\/31231825\/2.58.jpg\" alt=\"This figure is of the 3-dimensional coordinate system. It has a point. There is a line segment from the origin to the point. The angle between this line segment and the z-axis is phi. There is a line segment in the x y-plane from the origin to the shadow of the point.The angle between the x-axis and rho is theta.\" width=\"657\" height=\"392\" \/><figcaption id=\"caption-attachment-5307\" class=\"wp-caption-text\">Figure 5. The point with spherical coordinates [latex]\\left(2,-\\frac{5\\pi}6,\\frac{\\pi}6\\right)[\/latex]<\/figcaption><\/figure>\n<p>Cartesian: [latex]\\left(-\\frac{\\sqrt{3}}2,-\\frac12,\\sqrt3\\right)[\/latex], cylindrical: [latex]\\left(1,-\\frac{5\\pi}6,\\sqrt3\\right)[\/latex]<\/p><\/div>\n<\/div>\n<\/section>\n<section class=\"textbox watchIt\" aria-label=\"Watch It\">Watch the following video to see the worked solution to the above example.<\/p>\n<div style=\"text-align: center;\"><iframe loading=\"lazy\" src=\"\/\/plugin.3playmedia.com\/show?mf=7809469&amp;p3sdk_version=1.10.1&amp;p=20361&amp;pt=375&amp;video_id=yrKGmi_JFbM&amp;video_target=tpm-plugin-f16mh34p-yrKGmi_JFbM\" width=\"800px\" height=\"450px\" frameborder=\"0\" marginwidth=\"0px\" marginheight=\"0px\" data-mce-fragment=\"1\"><\/iframe><\/div>\n<div style=\"text-align: center;\">You can view the transcript for <a href=\"https:\/\/course-building.s3.us-west-2.amazonaws.com\/Calculus+3\/Calc+3+transcripts\/CP2.58_transcript.html\">\u201cCP 2.58\u201d here (opens in new window).<\/a><\/div>\n<\/section>\n<section class=\"textbox example\" aria-label=\"Example\">\n<p id=\"fs-id1163723523037\">Describe the surfaces defined by the following equations.<\/p>\n<ol id=\"fs-id1163723523040\" type=\"a\">\n<li>[latex]\\rho=13[\/latex]<\/li>\n<li>[latex]\\theta=\\frac{2\\pi}3[\/latex]<\/li>\n<li>[latex]\\varphi=\\frac{\\pi}4[\/latex]<\/li>\n<\/ol>\n<div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"q874652711\">Show Solution<\/button><\/p>\n<div id=\"q874652711\" class=\"hidden-answer\" style=\"display: none\">\n<p>a. This is the set of all points [latex]13[\/latex] units from the origin. This set forms a sphere with radius [latex]13[\/latex].<\/p>\n<p>b. This set of points forms a half plane. The angle between the half plane and the positive [latex]x[\/latex]-axis is [latex]\\theta=\\frac{2\\pi}3[\/latex].<\/p>\n<p>c. Let [latex]P[\/latex] be a point on this surface. The position vector of this point forms an angle of [latex]\\varphi=\\frac{\\pi}4[\/latex] with the positive [latex]z[\/latex]-axis, which means that points closer to the origin are closer to the axis. These points form a half-cone.<\/p>\n<\/div>\n<\/div>\n<\/section>\n<section class=\"textbox example\" aria-label=\"Example\">Sydney, Australia is at [latex]34^{\\small\\circ}[\/latex]S and [latex]151^{\\small\\circ}[\/latex]E. Express Sydney\u2019s location in spherical coordinates.<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"q467253716\">Show Solution<\/button><\/p>\n<div id=\"q467253716\" class=\"hidden-answer\" style=\"display: none\">[latex](4,000,151^{\\small\\circ},124^{\\small\\circ})[\/latex]<\/div>\n<\/div>\n<\/section>\n<section class=\"textbox example\" aria-label=\"Example\">Which coordinate system is most appropriate for creating a star map, as viewed from Earth (see the following figure)?<\/p>\n<figure id=\"attachment_5322\" aria-describedby=\"caption-attachment-5322\" style=\"width: 731px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" class=\"size-full wp-image-5322\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/5667\/2021\/08\/31232545\/2.61.jpg\" alt=\"This figure is a circle with a star chart in the middle.\" width=\"731\" height=\"731\" \/><figcaption id=\"caption-attachment-5322\" class=\"wp-caption-text\">Figure 11. Star map as viewed from Earth.<\/figcaption><\/figure>\n<p>How should we orient the coordinate axes?<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"q883476235\">Show Solution<\/button><\/p>\n<div id=\"q883476235\" class=\"hidden-answer\" style=\"display: none\">\n<p>Spherical coordinates with the origin located at the center of the earth, the [latex]z[\/latex]-axis aligned with the North Pole, and the [latex]x[\/latex]-axis aligned with the prime meridian.<\/p>\n<\/div>\n<\/div>\n<\/section>\n<h2 class=\"text-2xl font-bold mt-1 text-text-100\">Cylinders and Surfaces in 3D<\/h2>\n<div class=\"textbox shaded\">\n<p><strong>The Main Idea\u00a0<\/strong><\/p>\n<p class=\"whitespace-normal break-words\">The mathematical definition of a <strong>cylinder<\/strong> is broader than the everyday &#8220;hollow tube&#8221; image\u2014any curve extended by parallel lines creates a cylinder.<\/p>\n<p class=\"whitespace-normal break-words\">A <strong>cylinder<\/strong> (or cylindrical surface) is formed by a set of parallel lines (called <strong>rulings<\/strong>) passing through a given curve. The curve can be any shape, and the rulings can be parallel to any direction.<\/p>\n<p class=\"whitespace-normal break-words\"><strong>Identifying Cylinders from Equations<\/strong><\/p>\n<p class=\"whitespace-normal break-words\">An equation describes a cylinder if <strong>one variable is missing<\/strong>. The missing variable tells you the direction of the rulings:<\/p>\n<ul class=\"[&amp;:not(:last-child)_ul]:pb-1 [&amp;:not(:last-child)_ol]:pb-1 list-disc space-y-2.5 pl-7\">\n<li class=\"whitespace-normal break-words\">Missing [latex]z[\/latex]: rulings parallel to [latex]z[\/latex]-axis<\/li>\n<li class=\"whitespace-normal break-words\">Missing [latex]y[\/latex]: rulings parallel to [latex]y[\/latex]-axis<\/li>\n<li class=\"whitespace-normal break-words\">Missing [latex]x[\/latex]: rulings parallel to [latex]x[\/latex]-axis<\/li>\n<\/ul>\n<p class=\"whitespace-normal break-words\"><strong>Examples<\/strong>:<\/p>\n<ul class=\"[&amp;:not(:last-child)_ul]:pb-1 [&amp;:not(:last-child)_ol]:pb-1 list-disc space-y-2.5 pl-7\">\n<li class=\"whitespace-normal break-words\">[latex]x^2 + y^2 = 9[\/latex]: circular cylinder (rulings parallel to [latex]z[\/latex]-axis)<\/li>\n<li class=\"whitespace-normal break-words\">[latex]x^2 + z^2 = 25[\/latex]: circular cylinder (rulings parallel to [latex]y[\/latex]-axis)<\/li>\n<li class=\"whitespace-normal break-words\">[latex]y = \\sin x[\/latex]: sinusoidal cylinder (rulings parallel to [latex]z[\/latex]-axis)<\/li>\n<\/ul>\n<p class=\"whitespace-normal break-words\"><strong>Traces<\/strong><\/p>\n<p class=\"whitespace-normal break-words\">Traces are the curves formed when a surface intersects planes parallel to coordinate planes ([latex]xy[\/latex], [latex]xz[\/latex], or [latex]yz[\/latex]).<\/p>\n<p class=\"whitespace-normal break-words\"><strong>For cylinders<\/strong>: Only one trace is informative\u2014the one perpendicular to the rulings shows the generating curve. Traces parallel to the rulings just show parallel lines.<\/p>\n<p class=\"whitespace-normal break-words\"><strong>Example<\/strong>: For [latex]y = \\sin x[\/latex]:<\/p>\n<ul class=\"[&amp;:not(:last-child)_ul]:pb-1 [&amp;:not(:last-child)_ol]:pb-1 list-disc space-y-2.5 pl-7\">\n<li class=\"whitespace-normal break-words\">[latex]xz[\/latex]-plane trace ([latex]y = 0[\/latex]): Shows the sine wave<\/li>\n<li class=\"whitespace-normal break-words\">Other traces: Just parallel lines (not helpful)<\/li>\n<\/ul>\n<p class=\"whitespace-normal break-words\"><strong>Key Insight<\/strong>: Cylinders are the simplest 3D surfaces\u2014they&#8217;re just 2D curves extended in a straight direction. More complex surfaces require examining multiple traces to understand their full 3D shape.<\/p>\n<\/div>\n<section class=\"textbox example\" aria-label=\"Example\">Sketch or use a graphing tool to view the graph of the cylindrical surface defined by equation [latex]z=y^{2}[\/latex].<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"q378465244\">Show Solution<\/button><\/p>\n<div id=\"q378465244\" class=\"hidden-answer\" style=\"display: none\">\n<figure id=\"attachment_5210\" aria-describedby=\"caption-attachment-5210\" style=\"width: 288px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" class=\"size-full wp-image-5210\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/5667\/2021\/08\/31203348\/2.52.jpg\" alt=\"This figure is a surface above the x y plane. A cross section of this surface parallel to the y z plane would be a parabola. The surface sits on top of the x y plane.\" width=\"288\" height=\"324\" \/><figcaption id=\"caption-attachment-5210\" class=\"wp-caption-text\">Figure 6. The cylindrical surface defined by equation [latex]z=y^{2}[\/latex].<\/figcaption><\/figure>\n<\/div>\n<\/div>\n<\/section>\n<section class=\"textbox watchIt\" aria-label=\"Watch It\">Watch the following video to see the worked solution to the above example.<\/p>\n<div style=\"text-align: center;\"><iframe loading=\"lazy\" src=\"\/\/plugin.3playmedia.com\/show?mf=7809453&amp;p3sdk_version=1.10.1&amp;p=20361&amp;pt=375&amp;video_id=ygBugCYrufQ&amp;video_target=tpm-plugin-mxtub258-ygBugCYrufQ\" width=\"800px\" height=\"450px\" frameborder=\"0\" marginwidth=\"0px\" marginheight=\"0px\" data-mce-fragment=\"1\"><\/iframe><\/div>\n<div style=\"text-align: center;\"><span style=\"font-size: 1rem; orphans: 1; text-align: initial;\">You can view the transcript for <a href=\"https:\/\/course-building.s3.us-west-2.amazonaws.com\/Calculus+3\/Calc+3+transcripts\/CP2.52_transcript.html\">\u201cCP 2.52\u201d here (opens in new window).<\/a><\/span><\/div>\n<\/section>\n<h2 data-type=\"title\">Quadric Surfaces<\/h2>\n<div class=\"textbox shaded\">\n<p><strong>The Main Idea\u00a0<\/strong><\/p>\n<p class=\"whitespace-normal break-words\"><strong>Quadric surfaces<\/strong> are 3D extensions of conic sections (ellipses, parabolas, hyperbolas), described by second-degree equations. Understanding their traces reveals their shape.<\/p>\n<p class=\"whitespace-normal break-words\"><strong>General Form<\/strong><\/p>\n<p class=\"whitespace-normal break-words\" style=\"text-align: center;\">[latex]Ax^2 + By^2 + Cz^2 + Dxy + Exz + Fyz + Gx + Hy + Jz + K = 0[\/latex]<\/p>\n<p class=\"whitespace-normal break-words\"><strong>Key Surfaces<\/strong><\/p>\n<p class=\"whitespace-normal break-words\"><strong>Ellipsoid<\/strong>: [latex]\\frac{x^2}{a^2} + \\frac{y^2}{b^2} + \\frac{z^2}{c^2} = 1[\/latex]<\/p>\n<ul class=\"[&amp;:not(:last-child)_ul]:pb-1 [&amp;:not(:last-child)_ol]:pb-1 list-disc space-y-2.5 pl-7\">\n<li class=\"whitespace-normal break-words\">All traces are ellipses<\/li>\n<li class=\"whitespace-normal break-words\">Special case: [latex]a = b = c[\/latex] gives a sphere<\/li>\n<li class=\"whitespace-normal break-words\">Extends [latex]a[\/latex], [latex]b[\/latex], [latex]c[\/latex] units along each axis<\/li>\n<\/ul>\n<p class=\"whitespace-normal break-words\"><strong>Elliptic Paraboloid<\/strong>: [latex]\\frac{x^2}{a^2} + \\frac{y^2}{b^2} = \\frac{z}{c}[\/latex]<\/p>\n<ul class=\"[&amp;:not(:last-child)_ul]:pb-1 [&amp;:not(:last-child)_ol]:pb-1 list-disc space-y-2.5 pl-7\">\n<li class=\"whitespace-normal break-words\">Bowl-shaped surface<\/li>\n<li class=\"whitespace-normal break-words\">Traces perpendicular to axis: ellipses (or circles if [latex]a = b[\/latex])<\/li>\n<li class=\"whitespace-normal break-words\">Traces parallel to axis: parabolas<\/li>\n<li class=\"whitespace-normal break-words\">Opens up if [latex]c > 0[\/latex], down if [latex]c < 0[\/latex]<\/li>\n<li class=\"whitespace-normal break-words\">Variable to first power determines axis direction<\/li>\n<\/ul>\n<p class=\"whitespace-normal break-words\"><strong>Using Traces to Identify Surfaces<\/strong><\/p>\n<p class=\"whitespace-normal break-words\"><strong>Traces<\/strong> are cross-sections where the surface meets coordinate planes. To identify a quadric surface:<\/p>\n<ol class=\"[&amp;:not(:last-child)_ul]:pb-1 [&amp;:not(:last-child)_ol]:pb-1 list-decimal space-y-2.5 pl-7\">\n<li class=\"whitespace-normal break-words\">Check which variables are squared vs. first power<\/li>\n<li class=\"whitespace-normal break-words\">Look at signs (positive vs. negative coefficients)<\/li>\n<li class=\"whitespace-normal break-words\">Find traces by setting [latex]x = 0[\/latex], [latex]y = 0[\/latex], and [latex]z = 0[\/latex]<\/li>\n<li class=\"whitespace-normal break-words\">Put equation in standard form if needed<\/li>\n<\/ol>\n<p class=\"whitespace-normal break-words\"><strong>Identification Strategy<\/strong>:<\/p>\n<ul class=\"[&amp;:not(:last-child)_ul]:pb-1 [&amp;:not(:last-child)_ol]:pb-1 list-disc space-y-2.5 pl-7\">\n<li class=\"whitespace-normal break-words\">All squared terms with same sign \u2192 ellipsoid or sphere<\/li>\n<li class=\"whitespace-normal break-words\">One first-power term, two squared (same sign) \u2192 elliptic paraboloid<\/li>\n<li class=\"whitespace-normal break-words\">Mixed signs in squared terms \u2192 hyperboloid or hyperbolic paraboloid<\/li>\n<\/ul>\n<p class=\"whitespace-normal break-words\"><strong>Real-World Applications<\/strong><\/p>\n<ul class=\"[&amp;:not(:last-child)_ul]:pb-1 [&amp;:not(:last-child)_ol]:pb-1 list-disc space-y-2.5 pl-7\">\n<li class=\"whitespace-normal break-words\"><strong>Parabolic reflectors<\/strong>: Satellite dishes and solar collectors use elliptic paraboloids because energy hitting the surface concentrates at the focal point<\/li>\n<li class=\"whitespace-normal break-words\"><strong>Hyperboloids<\/strong>: Nuclear cooling towers use hyperboloids of one sheet\u2014they can be built with straight beams, making them strong yet economical<\/li>\n<\/ul>\n<\/div>\n<section class=\"textbox example\" aria-label=\"Example\">A hyperboloid of one sheet is any surface that can be described with an equation of the form [latex]\\frac{x^2}{a^2}+\\frac{y^2}{b^2}-\\frac{z^2}{c^2}=1[\/latex]. Describe the traces of the hyperboloid of one sheet given by equation [latex]\\frac{x^2}{3^2}+\\frac{y^2}{2^2}-\\frac{z^2}{5^2}=1[\/latex].<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"q785629873\">Show Solution<\/button><\/p>\n<div id=\"q785629873\" class=\"hidden-answer\" style=\"display: none\">\n<p id=\"fs-id1163724104708\">The traces parallel to the [latex]xy[\/latex]-plane are ellipses and the traces parallel to the [latex]xz[\/latex]&#8211; and [latex]yz[\/latex]-planes are hyperbolas. Specifically, the trace in the [latex]xy[\/latex]-plane is ellipse [latex]\\frac{x^2}{3^2}+\\frac{y^2}{2^2}=1[\/latex], the trace in the [latex]xz[\/latex]-plane is hyperbola [latex]\\frac{x^2}{3^2}-\\frac{z^2}{5^2}=1[\/latex], and the trace in the [latex]yz[\/latex]-plane is hyperbola [latex]\\frac{y^2}{2^2}-\\frac{z^2}{5^2}=1[\/latex] (see the following figure).<span data-type=\"newline\"><br \/>\n<\/span><\/p>\n<figure id=\"attachment_5224\" aria-describedby=\"caption-attachment-5224\" style=\"width: 876px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" class=\"size-full wp-image-5224\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/5667\/2021\/08\/31204619\/2.53.jpg\" alt=\"A hyperboloid of one sheet is any surface that can be described with an equation of the form\" width=\"876\" height=\"914\" \/><figcaption id=\"caption-attachment-5224\" class=\"wp-caption-text\">Figure 5. (a) The trace in the [latex]xy[\/latex]-plane. (b) The trace in the [latex]xz[\/latex]-plane. (c) The trace in the [latex]yz[\/latex]-plane. (d) The hyperboloid [latex]\\frac{x^2}{3^2}+\\frac{y^2}{2^2}-\\frac{z^2}{5^2}=1[\/latex].<\/figcaption><\/figure>\n<\/div>\n<\/div>\n<\/section>\n<section class=\"textbox watchIt\" aria-label=\"Watch It\">Watch the following video to see the worked solution to the above example.<\/p>\n<div style=\"text-align: center;\"><iframe loading=\"lazy\" src=\"\/\/plugin.3playmedia.com\/show?mf=7809454&amp;p3sdk_version=1.10.1&amp;p=20361&amp;pt=375&amp;video_id=qYftkr5uwzo&amp;video_target=tpm-plugin-znpykjkm-qYftkr5uwzo\" width=\"800px\" height=\"450px\" frameborder=\"0\" marginwidth=\"0px\" marginheight=\"0px\" data-mce-fragment=\"1\"><\/iframe><\/div>\n<div style=\"text-align: center;\"><span style=\"font-size: 1rem; orphans: 1; text-align: initial;\">You can view the transcript for <a href=\"https:\/\/course-building.s3.us-west-2.amazonaws.com\/Calculus+3\/Calc+3+transcripts\/CP2.53_transcript.html\">\u201cCP 2.53\u201d here (opens in new window).<\/a><\/span><\/div>\n<\/section>\n<section class=\"textbox example\" aria-label=\"Example\">Identify the surface represented by equation [latex]9x^{2}+y^{2}-z^{2}+2z-10=0[\/latex].<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"q752345122\">Show Solution<\/button><\/p>\n<div id=\"q752345122\" class=\"hidden-answer\" style=\"display: none\">Hyperboloid of one sheet, centered at [latex](0, 0, 1)[\/latex].<\/div>\n<\/div>\n<\/section>\n","protected":false},"author":15,"menu_order":23,"template":"","meta":{"_candela_citation":"[]","pb_show_title":"on","pb_short_title":"","pb_subtitle":"","pb_authors":[],"pb_section_license":""},"chapter-type":[],"contributor":[],"license":[],"class_list":["post-535","chapter","type-chapter","status-publish","hentry"],"part":422,"module-header":"fresh_take","content_attributions":[],"internal_book_links":[],"video_content":null,"cc_video_embed_content":{"cc_scripts":"","media_targets":[]},"try_it_collection":null,"_links":{"self":[{"href":"https:\/\/content.one.lumenlearning.com\/calculus3\/wp-json\/pressbooks\/v2\/chapters\/535","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/content.one.lumenlearning.com\/calculus3\/wp-json\/pressbooks\/v2\/chapters"}],"about":[{"href":"https:\/\/content.one.lumenlearning.com\/calculus3\/wp-json\/wp\/v2\/types\/chapter"}],"author":[{"embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/calculus3\/wp-json\/wp\/v2\/users\/15"}],"version-history":[{"count":7,"href":"https:\/\/content.one.lumenlearning.com\/calculus3\/wp-json\/pressbooks\/v2\/chapters\/535\/revisions"}],"predecessor-version":[{"id":2193,"href":"https:\/\/content.one.lumenlearning.com\/calculus3\/wp-json\/pressbooks\/v2\/chapters\/535\/revisions\/2193"}],"part":[{"href":"https:\/\/content.one.lumenlearning.com\/calculus3\/wp-json\/pressbooks\/v2\/parts\/422"}],"metadata":[{"href":"https:\/\/content.one.lumenlearning.com\/calculus3\/wp-json\/pressbooks\/v2\/chapters\/535\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/content.one.lumenlearning.com\/calculus3\/wp-json\/wp\/v2\/media?parent=535"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/calculus3\/wp-json\/pressbooks\/v2\/chapter-type?post=535"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/calculus3\/wp-json\/wp\/v2\/contributor?post=535"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/calculus3\/wp-json\/wp\/v2\/license?post=535"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}