{"id":533,"date":"2025-10-09T15:18:38","date_gmt":"2025-10-09T15:18:38","guid":{"rendered":"https:\/\/content.one.lumenlearning.com\/calculus3\/?post_type=chapter&#038;p=533"},"modified":"2025-10-22T15:07:32","modified_gmt":"2025-10-22T15:07:32","slug":"alternative-coordinate-systems-and-surfaces-learn-it-1","status":"publish","type":"chapter","link":"https:\/\/content.one.lumenlearning.com\/calculus3\/chapter\/alternative-coordinate-systems-and-surfaces-learn-it-1\/","title":{"raw":"Alternative Coordinate Systems and Surfaces: Learn It 1","rendered":"Alternative Coordinate Systems and Surfaces: Learn It 1"},"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><section id=\"fs-id1163723958933\" data-depth=\"1\">The Cartesian coordinate system works well for many problems, but some surfaces are difficult to model with rectangular equations. Just as polar coordinates provide a useful alternative in 2D (especially for circles), we can extend coordinate systems to 3D space. <strong>Cylindrical coordinates<\/strong> are perfect for problems involving cylinders\u2014like calculating the volume of a water tank or analyzing flow through a pipe. <strong>Spherical coordinates<\/strong> excel with sphere-related problems, such as finding volumes of domed structures.\r\n<h2>Cylindrical Coordinates<\/h2>\r\n<p class=\"whitespace-normal break-words\">Cylindrical coordinates provide a natural extension of polar coordinates to three dimensions. Instead of just adding a third axis like we did going from 2D to 3D Cartesian coordinates, we build on the polar coordinate foundation.<\/p>\r\n\r\n<section class=\"textbox keyTakeaway\" aria-label=\"Key Takeaway\">\r\n<h3>cylindrical coordinate system<\/h3>\r\n<p id=\"fs-id1163723216822\">In the\u00a0<strong><span id=\"293cc618-b0de-4ec9-9029-6efb4eb1b172_term106\" data-type=\"term\">cylindrical coordinate system<\/span><\/strong>, a point in space is represented by the ordered triple [latex](r,\\theta,z)[\/latex], where<\/p>\r\n\r\n<ul id=\"fs-id1163723236536\" data-bullet-style=\"bullet\">\r\n \t<li>[latex](r,\\theta)[\/latex] are the polar coordinates of the point\u2019s projection in the [latex]xy[\/latex]-plane<\/li>\r\n \t<li>[latex]z[\/latex] is the usual [latex]z[\/latex]-coordinate in the Cartesian coordinate system<\/li>\r\n<\/ul>\r\n<\/section>[caption id=\"attachment_5265\" align=\"aligncenter\" width=\"427\"]<img class=\"size-full wp-image-5265\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/5667\/2021\/08\/31225015\/2.89.jpg\" alt=\"This figure is the first octant of the 3-dimensional coordinate system. There is a point labeled \u201c(x, y, z) = (r, theta, z).\u201d In the x y-plane, there is a line segment extending to underneath the point. This line segment is labeled \u201cr.\u201d The angle between the line segment and the x-axis is theta. There is a line segment perpendicular to the x-axis. Along with the line segment labeled r, this line segment and the x-axis form a right triangle.\" width=\"427\" height=\"421\" \/> Figure 1. The right triangle lies in the [latex]xy[\/latex]-plane. The length of the hypotenuse is [latex]r[\/latex] and [latex]\\theta[\/latex] is the measure of the angle formed by the positive [latex]x[\/latex]-axis and the hypotenuse. The [latex]z[\/latex]-coordinate describes the location of the point above or below the [latex]xy[\/latex]-plane.[\/caption]<section class=\"textbox proTip\" aria-label=\"Pro Tip\">Think of it this way: to locate a point in cylindrical coordinates, you first find its \"shadow\" on the [latex]xy[\/latex]-plane using polar coordinates [latex](r,\\theta)[\/latex], then specify how far above or below that plane the point sits using [latex]z[\/latex].<\/section>\r\n<h3 class=\"text-xl font-bold text-text-100 mt-1 -mb-0.5\">Converting Between Coordinate Systems<\/h3>\r\n<p class=\"whitespace-normal break-words\">The conversion formulas come directly from the right triangle relationships in the [latex]xy[\/latex]-plane.<\/p>\r\n\r\n[caption id=\"attachment_5269\" align=\"aligncenter\" width=\"344\"]<img class=\"size-full wp-image-5269\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/5667\/2021\/08\/31225305\/2.90.jpg\" alt=\"This figure is the first quadrant of the rectangular coordinate system. There is a point labeled \u201cP = (x, y, 0) = (r, theta, 0).\u201d There is a line segment from the origin to point P. This line segment is labeled \u201cr.\u201d The angle between the x-axis and the line segment r is labeled \u201ctheta.\u201d There is also a vertical line segment labeled \u201cy\u201d from P to the x-axis. It forms a right triangle.\" width=\"344\" height=\"272\" \/> Figure 2. The Pythagorean theorem provides equation [latex]r^{2}=x^{2}+y^{2}[\/latex]. Right-triangle relationships tell us that [latex]x=r\\cos{\\theta}[\/latex], [latex]y=r\\sin{\\theta}[\/latex], and [latex]\\tan{\\theta}=\\frac{y}{x}[\/latex].[\/caption]<\/section><section data-depth=\"1\"><section class=\"textbox keyTakeaway\" aria-label=\"Key Takeaway\">\r\n<h3>conversion between cylindrical and cartesian coordinates<\/h3>\r\n<p id=\"fs-id1163724076636\" style=\"text-align: left;\">The rectangular coordinates [latex](x, y, z)[\/latex] and the cylindrical coordinates [latex](r,\\theta,z)[\/latex] of a point are related as follows:<\/p>\r\n&nbsp;\r\n<p class=\"whitespace-normal break-words\"><strong>Cylindrical to Rectangular<\/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 = 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\"><strong>Rectangular to Cylindrical<\/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^2 = x^2 + y^2[\/latex] (so [latex]r = \\sqrt{x^2 + y^2}[\/latex] for [latex]r \\geq 0[\/latex])<\/li>\r\n \t<li class=\"whitespace-normal break-words\">[latex]\\tan\\theta = \\frac{y}{x}[\/latex] (choose [latex]\\theta[\/latex] based on the quadrant)<\/li>\r\n \t<li class=\"whitespace-normal break-words\">[latex]z = z[\/latex]<\/li>\r\n<\/ul>\r\n<\/section>\r\n<div id=\"fs-id1163724080388\" class=\"ui-has-child-title\" data-type=\"example\">\r\n<div data-type=\"example\"><section class=\"textbox proTip\" aria-label=\"Pro Tip\"><strong>Finding [latex]\\theta[\/latex]<\/strong>: Just like in 2D polar coordinates, [latex]\\tan\\theta = \\frac{y}{x}[\/latex] has infinitely many solutions. Choose [latex]\\theta[\/latex] in [latex][0, 2\\pi)[\/latex] based on the signs of [latex]x[\/latex] and [latex]y[\/latex] to get the correct quadrant.<\/section><section class=\"textbox example\" aria-label=\"Example\">Plot the point with cylindrical coordinates [latex]\\left(4,\\frac{2\\pi}3,-2\\right)[\/latex] and express its location in rectangular coordinates.[reveal-answer q=\"951616342\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"951616342\"]\r\n<p id=\"fs-id1163724079613\">Conversion from cylindrical to rectangular coordinates requires a simple application of the equations listed in\u00a0Theorem: Conversion between Cylindrical and Cartesian Coordinates:<\/p>\r\n<p style=\"text-align: center;\">[latex]\\begin{aligned}\r\nx&amp;=r\\cos\\theta=4\\cos\\frac{2\\pi}3=-2 \\\\\r\ny&amp;=r\\sin\\theta=4\\sin\\frac{2\\pi}3=2\\sqrt3 \\\\\r\nz&amp;=-2\r\n\\end{aligned}[\/latex].<\/p>\r\n<p id=\"fs-id1163723199277\">The point with cylindrical coordinates [latex]\\left(4,\\frac{2\\pi}3,-2\\right)[\/latex] has rectangular coordinates [latex](-2,2\\sqrt3,-2)[\/latex] (see the following figure).<\/p>\r\n\r\n[caption id=\"attachment_5277\" align=\"aligncenter\" width=\"485\"]<img class=\"size-full wp-image-5277\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/5667\/2021\/08\/31225942\/2.93.jpg\" alt=\"This figure is the 3-dimensional coordinate system. It has a point where r = 4, z = -2 and theta = 2 pi \/3.\" width=\"485\" height=\"349\" \/> Figure 5. The projection of the point in the [latex]xy[\/latex]-plane is [latex]4[\/latex] units from the origin. The line from the origin to the point\u2019s projection forms an angle of [latex]\\frac{2\\pi}{3}[\/latex] with the positive [latex]x[\/latex]-axis. The point lies [latex]2[\/latex] units below the [latex]xy[\/latex]-plane.[\/caption][\/hidden-answer]<\/section><section class=\"textbox example\" aria-label=\"Example\">Convert the rectangular coordinates [latex](1, -3, 5)[\/latex] to cylindrical coordinates.[reveal-answer q=\"829547026\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"829547026\"]\r\n<p id=\"fs-id1163723094403\">Use the second set of equations from\u00a0Theorem: Conversion between Cylindrical and Cartesian Coordinates\u00a0to translate from rectangular to cylindrical coordinates:<\/p>\r\n<p style=\"text-align: center;\">[latex]\\begin{aligned}\r\nr^2&amp;=x^2+y^2 \\\\\r\nr&amp;=\\pm\\sqrt{1^2+(-3)^2}=\\pm\\sqrt{10}.\r\n\\end{aligned}[\/latex]<\/p>\r\n<p id=\"fs-id1163723932354\">We choose the positive square root, so [latex]r=\\sqrt{10}[\/latex]. Now, we apply the formula to find [latex]\\theta[\/latex]. In this case, [latex]y[\/latex] is negative and [latex]x[\/latex] is positive, which means we must select the value of [latex]\\theta[\/latex] between [latex]\\frac{3\\pi}2[\/latex] and\u00a0[latex]2\\pi[\/latex]:<\/p>\r\n<p style=\"text-align: center;\">[latex]\\begin{aligned}\r\n\\tan\\theta&amp;=\\frac{y}x=\\frac{-3}1 \\\\\r\n\\theta&amp;=\\arctan(-3)\\approx5.03\\text{ rad}.\r\n\\end{aligned}[\/latex]<\/p>\r\n<p id=\"fs-id1163724052119\">In this case, the [latex]z[\/latex]-coordinates are the same in both rectangular and cylindrical coordinates:<\/p>\r\n<p style=\"text-align: center;\">[latex]z=5[\/latex].<\/p>\r\n<p id=\"fs-id1163724079567\">The point with rectangular coordinates [latex](1, -3, 5)[\/latex] has cylindrical coordinates approximately equal to [latex](\\sqrt{10},5.03,5)[\/latex].<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/section><section class=\"textbox tryIt\" aria-label=\"Try It\">[ohm_question hide_question_numbers=1]242737[\/ohm_question]\r\n\r\n<\/section>\r\n<h3 class=\"text-xl font-bold text-text-100 mt-1 -mb-0.5\">Surfaces in Cylindrical Coordinates<\/h3>\r\nUnderstanding what surfaces look like when one coordinate is held constant helps you visualize the cylindrical coordinate system.\r\n<p class=\"whitespace-normal break-words\"><strong>Rectangular Coordinates<\/strong> (for comparison):<\/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 = c[\/latex]: plane parallel to [latex]yz[\/latex]-plane<\/li>\r\n \t<li class=\"whitespace-normal break-words\">[latex]y = c[\/latex]: plane parallel to [latex]xz[\/latex]-plane<\/li>\r\n \t<li class=\"whitespace-normal break-words\">[latex]z = c[\/latex]: plane parallel to [latex]xy[\/latex]-plane<\/li>\r\n<\/ul>\r\n<p class=\"whitespace-normal break-words\"><strong>Cylindrical Coordinates<\/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] (points at 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]: plane parallel to [latex]xy[\/latex]-plane (same as rectangular)<\/li>\r\n<\/ul>\r\n<div id=\"CNX_Calc_Figure_12_07_003\" class=\"os-figure\">[caption id=\"attachment_5274\" align=\"aligncenter\" width=\"712\"]<img class=\"size-full wp-image-5274\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/5667\/2021\/08\/31225632\/2.91.jpg\" alt=\"This figure has 3 images. The first image is a plane in the 3-dimensional coordinate system. It is parallel to the y z-plane where x = c. The second image is a plane in the 3-dimensional coordinate system. It is parallel to the x z-plane where y = c. the third image is a plane in the 3-dimensional coordinate system where z = c.\" width=\"712\" height=\"262\" \/> Figure 3. In rectangular coordinates, (a) surfaces of the form [latex]x=c[\/latex] are planes parallel to the [latex]yz[\/latex]-plane, (b) surfaces of the form [latex]y=c[\/latex] are planes parallel to the [latex]xz[\/latex]-plane, and (c) surfaces of the form [latex]z=c[\/latex] are planes parallel to the [latex]xy[\/latex]-plane.[\/caption]<\/div>\r\n<div>[caption id=\"attachment_5275\" align=\"aligncenter\" width=\"712\"]<img class=\"size-full wp-image-5275\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/5667\/2021\/08\/31225639\/2.92.jpg\" alt=\"This figure has 3 images. The first image is a right circular cylinder in the 3-dimensional coordinate system. It has the z-axis in the middle. The second image is a plane in the 3-dimensional coordinate system. It is vertical with the z-axis on one edge. The third image is a plane in the 3-dimensional coordinate system where z = c.\" width=\"712\" height=\"262\" \/> Figure 4. In cylindrical coordinates, (a) surfaces of the form [latex]r=c[\/latex] are vertical cylinders of radius [latex]c[\/latex], (b) surfaces of the form [latex]\\theta=c[\/latex] are half-planes at angle [latex]c[\/latex] from the [latex]x[\/latex]-axis, and (c) surfaces of the form [latex]z=c[\/latex] are planes parallel to the [latex]xy[\/latex]-plane.[\/caption]<\/div>\r\n<\/div>\r\n<\/div>\r\n<div><section class=\"textbox example\" aria-label=\"Example\">\r\n<p id=\"fs-id1163723479630\">Describe the surfaces with the given cylindrical equations.<\/p>\r\n\r\n<ol id=\"fs-id1163723206980\" type=\"a\">\r\n \t<li>[latex]\\theta=\\frac{\\pi}4[\/latex]<\/li>\r\n \t<li>[latex]r^{2}+z^{2}=9[\/latex]<\/li>\r\n \t<li>[latex]z=r[\/latex]<\/li>\r\n<\/ol>\r\n[reveal-answer q=\"124761291\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"124761291\"]\r\n<ol id=\"fs-id1163723569688\" type=\"a\">\r\n \t<li>When the angle [latex]\\theta[\/latex] is held constant while [latex]r[\/latex] and [latex]z[\/latex] are allowed to vary, the result is a half-plane (see the following figure).\r\n[caption id=\"attachment_5284\" align=\"aligncenter\" width=\"447\"]<img class=\"size-full wp-image-5284\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/5667\/2021\/08\/31230343\/2.94.jpg\" alt=\"This figure is the first quadrant of the 3-dimensional coordinate system. There is a plane attached to the z-axis, dividing the x y-plane with a diagonal line. The angle between the x-axis and this plane is pi\/4.\" width=\"447\" height=\"635\" \/> Figure 7. In polar coordinates, the equation [latex]\\theta=\\frac{\\pi}4[\/latex] describes the ray extending diagonally through the first quadrant. In three dimensions, this same equation describes a half-plane.[\/caption]<\/li>\r\n \t<li>Substitute [latex]r^{2}=x^{2}+y^{2}[\/latex] into equation [latex]r^{2}+z^{2}=9[\/latex] to express the rectangular form of the equation: [latex]x^{2}+y^{2}+z^{2}=9[\/latex]. This equation describes a sphere centered at the origin with radius 3 (see the following figure).\r\n[caption id=\"attachment_5286\" align=\"aligncenter\" width=\"473\"]<img class=\"size-full wp-image-5286\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/5667\/2021\/08\/31230418\/2.95.jpg\" alt=\"This figure is a sphere. It has the z-axis through the center vertically. The point of intersection with the z-axis and the sphere is (0, 0, 3). There is also the y-axis through the center of the sphere horizontally. The intersection of the sphere and the y-axis is the point (0, 3, 0).\" width=\"473\" height=\"437\" \/> Figure 8. The sphere centered at the origin with radius 3 can be described by the cylindrical equation [latex]x^{2}+y^{2}+z^{2}=9[\/latex].[\/caption]<\/li>\r\n \t<li>To describe the surface defined by equation [latex]z=r[\/latex], is it useful to examine traces parallel to the [latex]xy[\/latex]-plane. For example, the trace in plane [latex]z=1[\/latex] is circle [latex]r=1[\/latex], the trace in plane [latex]z=3[\/latex] is circle [latex]r=3[\/latex], and so on. Each trace is a circle. As the value of [latex]z[\/latex] increases, the radius of the circle also increases. The resulting surface is a cone (see the following figure).\r\n[caption id=\"attachment_5287\" align=\"aligncenter\" width=\"421\"]<img class=\"size-full wp-image-5287\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/5667\/2021\/08\/31230440\/2.96.jpg\" alt=\"This figure is the 3-dimensional coordinate system. It has an elliptic cone with the z-axis down the center. The two cones, one right side up, the other upside down, meet at the origin.\" width=\"421\" height=\"490\" \/> Figure 9. The traces in planes parallel to the [latex]xy[\/latex]-plane are circles. The radius of the circles increases as [latex]z[\/latex] increases.[\/caption]<\/li>\r\n<\/ol>\r\n[\/hidden-answer]\r\n\r\n<\/section>\r\n<p class=\"whitespace-normal break-words\">Cylindrical coordinates shine when dealing with problems that have cylindrical symmetry. In physics, many systems involving electrical charges, capacitors, and electromagnetic fields have this symmetry.<\/p>\r\n<p class=\"whitespace-normal break-words\">For example, the cylinder [latex]x^2 + y^2 = 25[\/latex] in rectangular coordinates becomes simply [latex]r = 5[\/latex] in cylindrical coordinates\u2014much simpler to work with!<\/p>\r\n\r\n<section class=\"textbox proTip\" aria-label=\"Pro Tip\">This coordinate transformation follows the same pattern as converting from polar to rectangular coordinates in 2D. We're just adding the unchanged [latex]z[\/latex]-coordinate to extend into three dimensions.<\/section><\/div>\r\n<\/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<section id=\"fs-id1163723958933\" data-depth=\"1\">The Cartesian coordinate system works well for many problems, but some surfaces are difficult to model with rectangular equations. Just as polar coordinates provide a useful alternative in 2D (especially for circles), we can extend coordinate systems to 3D space. <strong>Cylindrical coordinates<\/strong> are perfect for problems involving cylinders\u2014like calculating the volume of a water tank or analyzing flow through a pipe. <strong>Spherical coordinates<\/strong> excel with sphere-related problems, such as finding volumes of domed structures.<\/p>\n<h2>Cylindrical Coordinates<\/h2>\n<p class=\"whitespace-normal break-words\">Cylindrical coordinates provide a natural extension of polar coordinates to three dimensions. Instead of just adding a third axis like we did going from 2D to 3D Cartesian coordinates, we build on the polar coordinate foundation.<\/p>\n<section class=\"textbox keyTakeaway\" aria-label=\"Key Takeaway\">\n<h3>cylindrical coordinate system<\/h3>\n<p id=\"fs-id1163723216822\">In the\u00a0<strong><span id=\"293cc618-b0de-4ec9-9029-6efb4eb1b172_term106\" data-type=\"term\">cylindrical coordinate system<\/span><\/strong>, a point in space is represented by the ordered triple [latex](r,\\theta,z)[\/latex], where<\/p>\n<ul id=\"fs-id1163723236536\" data-bullet-style=\"bullet\">\n<li>[latex](r,\\theta)[\/latex] are the polar coordinates of the point\u2019s projection in the [latex]xy[\/latex]-plane<\/li>\n<li>[latex]z[\/latex] is the usual [latex]z[\/latex]-coordinate in the Cartesian coordinate system<\/li>\n<\/ul>\n<\/section>\n<figure id=\"attachment_5265\" aria-describedby=\"caption-attachment-5265\" style=\"width: 427px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" class=\"size-full wp-image-5265\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/5667\/2021\/08\/31225015\/2.89.jpg\" alt=\"This figure is the first octant of the 3-dimensional coordinate system. There is a point labeled \u201c(x, y, z) = (r, theta, z).\u201d In the x y-plane, there is a line segment extending to underneath the point. This line segment is labeled \u201cr.\u201d The angle between the line segment and the x-axis is theta. There is a line segment perpendicular to the x-axis. Along with the line segment labeled r, this line segment and the x-axis form a right triangle.\" width=\"427\" height=\"421\" \/><figcaption id=\"caption-attachment-5265\" class=\"wp-caption-text\">Figure 1. The right triangle lies in the [latex]xy[\/latex]-plane. The length of the hypotenuse is [latex]r[\/latex] and [latex]\\theta[\/latex] is the measure of the angle formed by the positive [latex]x[\/latex]-axis and the hypotenuse. The [latex]z[\/latex]-coordinate describes the location of the point above or below the [latex]xy[\/latex]-plane.<\/figcaption><\/figure>\n<section class=\"textbox proTip\" aria-label=\"Pro Tip\">Think of it this way: to locate a point in cylindrical coordinates, you first find its &#8220;shadow&#8221; on the [latex]xy[\/latex]-plane using polar coordinates [latex](r,\\theta)[\/latex], then specify how far above or below that plane the point sits using [latex]z[\/latex].<\/section>\n<h3 class=\"text-xl font-bold text-text-100 mt-1 -mb-0.5\">Converting Between Coordinate Systems<\/h3>\n<p class=\"whitespace-normal break-words\">The conversion formulas come directly from the right triangle relationships in the [latex]xy[\/latex]-plane.<\/p>\n<figure id=\"attachment_5269\" aria-describedby=\"caption-attachment-5269\" style=\"width: 344px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" class=\"size-full wp-image-5269\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/5667\/2021\/08\/31225305\/2.90.jpg\" alt=\"This figure is the first quadrant of the rectangular coordinate system. There is a point labeled \u201cP = (x, y, 0) = (r, theta, 0).\u201d There is a line segment from the origin to point P. This line segment is labeled \u201cr.\u201d The angle between the x-axis and the line segment r is labeled \u201ctheta.\u201d There is also a vertical line segment labeled \u201cy\u201d from P to the x-axis. It forms a right triangle.\" width=\"344\" height=\"272\" \/><figcaption id=\"caption-attachment-5269\" class=\"wp-caption-text\">Figure 2. The Pythagorean theorem provides equation [latex]r^{2}=x^{2}+y^{2}[\/latex]. Right-triangle relationships tell us that [latex]x=r\\cos{\\theta}[\/latex], [latex]y=r\\sin{\\theta}[\/latex], and [latex]\\tan{\\theta}=\\frac{y}{x}[\/latex].<\/figcaption><\/figure>\n<\/section>\n<section data-depth=\"1\">\n<section class=\"textbox keyTakeaway\" aria-label=\"Key Takeaway\">\n<h3>conversion between cylindrical and cartesian coordinates<\/h3>\n<p id=\"fs-id1163724076636\" style=\"text-align: left;\">The rectangular coordinates [latex](x, y, z)[\/latex] and the cylindrical coordinates [latex](r,\\theta,z)[\/latex] of a point are related as follows:<\/p>\n<p>&nbsp;<\/p>\n<p class=\"whitespace-normal break-words\"><strong>Cylindrical to Rectangular<\/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 = 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\"><strong>Rectangular to Cylindrical<\/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^2 = x^2 + y^2[\/latex] (so [latex]r = \\sqrt{x^2 + y^2}[\/latex] for [latex]r \\geq 0[\/latex])<\/li>\n<li class=\"whitespace-normal break-words\">[latex]\\tan\\theta = \\frac{y}{x}[\/latex] (choose [latex]\\theta[\/latex] based on the quadrant)<\/li>\n<li class=\"whitespace-normal break-words\">[latex]z = z[\/latex]<\/li>\n<\/ul>\n<\/section>\n<div id=\"fs-id1163724080388\" class=\"ui-has-child-title\" data-type=\"example\">\n<div data-type=\"example\">\n<section class=\"textbox proTip\" aria-label=\"Pro Tip\"><strong>Finding [latex]\\theta[\/latex]<\/strong>: Just like in 2D polar coordinates, [latex]\\tan\\theta = \\frac{y}{x}[\/latex] has infinitely many solutions. Choose [latex]\\theta[\/latex] in [latex][0, 2\\pi)[\/latex] based on the signs of [latex]x[\/latex] and [latex]y[\/latex] to get the correct quadrant.<\/section>\n<section class=\"textbox example\" aria-label=\"Example\">Plot the point with cylindrical coordinates [latex]\\left(4,\\frac{2\\pi}3,-2\\right)[\/latex] and express its location in rectangular coordinates.<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"q951616342\">Show Solution<\/button><\/p>\n<div id=\"q951616342\" class=\"hidden-answer\" style=\"display: none\">\n<p id=\"fs-id1163724079613\">Conversion from cylindrical to rectangular coordinates requires a simple application of the equations listed in\u00a0Theorem: Conversion between Cylindrical and Cartesian Coordinates:<\/p>\n<p style=\"text-align: center;\">[latex]\\begin{aligned}  x&=r\\cos\\theta=4\\cos\\frac{2\\pi}3=-2 \\\\  y&=r\\sin\\theta=4\\sin\\frac{2\\pi}3=2\\sqrt3 \\\\  z&=-2  \\end{aligned}[\/latex].<\/p>\n<p id=\"fs-id1163723199277\">The point with cylindrical coordinates [latex]\\left(4,\\frac{2\\pi}3,-2\\right)[\/latex] has rectangular coordinates [latex](-2,2\\sqrt3,-2)[\/latex] (see the following figure).<\/p>\n<figure id=\"attachment_5277\" aria-describedby=\"caption-attachment-5277\" style=\"width: 485px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" class=\"size-full wp-image-5277\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/5667\/2021\/08\/31225942\/2.93.jpg\" alt=\"This figure is the 3-dimensional coordinate system. It has a point where r = 4, z = -2 and theta = 2 pi \/3.\" width=\"485\" height=\"349\" \/><figcaption id=\"caption-attachment-5277\" class=\"wp-caption-text\">Figure 5. The projection of the point in the [latex]xy[\/latex]-plane is [latex]4[\/latex] units from the origin. The line from the origin to the point\u2019s projection forms an angle of [latex]\\frac{2\\pi}{3}[\/latex] with the positive [latex]x[\/latex]-axis. The point lies [latex]2[\/latex] units below the [latex]xy[\/latex]-plane.<\/figcaption><\/figure>\n<\/div>\n<\/div>\n<\/section>\n<section class=\"textbox example\" aria-label=\"Example\">Convert the rectangular coordinates [latex](1, -3, 5)[\/latex] to cylindrical coordinates.<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"q829547026\">Show Solution<\/button><\/p>\n<div id=\"q829547026\" class=\"hidden-answer\" style=\"display: none\">\n<p id=\"fs-id1163723094403\">Use the second set of equations from\u00a0Theorem: Conversion between Cylindrical and Cartesian Coordinates\u00a0to translate from rectangular to cylindrical coordinates:<\/p>\n<p style=\"text-align: center;\">[latex]\\begin{aligned}  r^2&=x^2+y^2 \\\\  r&=\\pm\\sqrt{1^2+(-3)^2}=\\pm\\sqrt{10}.  \\end{aligned}[\/latex]<\/p>\n<p id=\"fs-id1163723932354\">We choose the positive square root, so [latex]r=\\sqrt{10}[\/latex]. Now, we apply the formula to find [latex]\\theta[\/latex]. In this case, [latex]y[\/latex] is negative and [latex]x[\/latex] is positive, which means we must select the value of [latex]\\theta[\/latex] between [latex]\\frac{3\\pi}2[\/latex] and\u00a0[latex]2\\pi[\/latex]:<\/p>\n<p style=\"text-align: center;\">[latex]\\begin{aligned}  \\tan\\theta&=\\frac{y}x=\\frac{-3}1 \\\\  \\theta&=\\arctan(-3)\\approx5.03\\text{ rad}.  \\end{aligned}[\/latex]<\/p>\n<p id=\"fs-id1163724052119\">In this case, the [latex]z[\/latex]-coordinates are the same in both rectangular and cylindrical coordinates:<\/p>\n<p style=\"text-align: center;\">[latex]z=5[\/latex].<\/p>\n<p id=\"fs-id1163724079567\">The point with rectangular coordinates [latex](1, -3, 5)[\/latex] has cylindrical coordinates approximately equal to [latex](\\sqrt{10},5.03,5)[\/latex].<\/p>\n<\/div>\n<\/div>\n<\/section>\n<section class=\"textbox tryIt\" aria-label=\"Try It\"><iframe loading=\"lazy\" id=\"ohm242737\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=242737&theme=lumen&iframe_resize_id=ohm242737&source=tnh\" width=\"100%\" height=\"150\"><\/iframe><\/p>\n<\/section>\n<h3 class=\"text-xl font-bold text-text-100 mt-1 -mb-0.5\">Surfaces in Cylindrical Coordinates<\/h3>\n<p>Understanding what surfaces look like when one coordinate is held constant helps you visualize the cylindrical coordinate system.<\/p>\n<p class=\"whitespace-normal break-words\"><strong>Rectangular Coordinates<\/strong> (for comparison):<\/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 = c[\/latex]: plane parallel to [latex]yz[\/latex]-plane<\/li>\n<li class=\"whitespace-normal break-words\">[latex]y = c[\/latex]: plane parallel to [latex]xz[\/latex]-plane<\/li>\n<li class=\"whitespace-normal break-words\">[latex]z = c[\/latex]: plane parallel to [latex]xy[\/latex]-plane<\/li>\n<\/ul>\n<p class=\"whitespace-normal break-words\"><strong>Cylindrical Coordinates<\/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] (points at 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]: plane parallel to [latex]xy[\/latex]-plane (same as rectangular)<\/li>\n<\/ul>\n<div id=\"CNX_Calc_Figure_12_07_003\" class=\"os-figure\">\n<figure id=\"attachment_5274\" aria-describedby=\"caption-attachment-5274\" style=\"width: 712px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" class=\"size-full wp-image-5274\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/5667\/2021\/08\/31225632\/2.91.jpg\" alt=\"This figure has 3 images. The first image is a plane in the 3-dimensional coordinate system. It is parallel to the y z-plane where x = c. The second image is a plane in the 3-dimensional coordinate system. It is parallel to the x z-plane where y = c. the third image is a plane in the 3-dimensional coordinate system where z = c.\" width=\"712\" height=\"262\" \/><figcaption id=\"caption-attachment-5274\" class=\"wp-caption-text\">Figure 3. In rectangular coordinates, (a) surfaces of the form [latex]x=c[\/latex] are planes parallel to the [latex]yz[\/latex]-plane, (b) surfaces of the form [latex]y=c[\/latex] are planes parallel to the [latex]xz[\/latex]-plane, and (c) surfaces of the form [latex]z=c[\/latex] are planes parallel to the [latex]xy[\/latex]-plane.<\/figcaption><\/figure>\n<\/div>\n<div>\n<figure id=\"attachment_5275\" aria-describedby=\"caption-attachment-5275\" style=\"width: 712px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" class=\"size-full wp-image-5275\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/5667\/2021\/08\/31225639\/2.92.jpg\" alt=\"This figure has 3 images. The first image is a right circular cylinder in the 3-dimensional coordinate system. It has the z-axis in the middle. The second image is a plane in the 3-dimensional coordinate system. It is vertical with the z-axis on one edge. The third image is a plane in the 3-dimensional coordinate system where z = c.\" width=\"712\" height=\"262\" \/><figcaption id=\"caption-attachment-5275\" class=\"wp-caption-text\">Figure 4. In cylindrical coordinates, (a) surfaces of the form [latex]r=c[\/latex] are vertical cylinders of radius [latex]c[\/latex], (b) surfaces of the form [latex]\\theta=c[\/latex] are half-planes at angle [latex]c[\/latex] from the [latex]x[\/latex]-axis, and (c) surfaces of the form [latex]z=c[\/latex] are planes parallel to the [latex]xy[\/latex]-plane.<\/figcaption><\/figure>\n<\/div>\n<\/div>\n<\/div>\n<div>\n<section class=\"textbox example\" aria-label=\"Example\">\n<p id=\"fs-id1163723479630\">Describe the surfaces with the given cylindrical equations.<\/p>\n<ol id=\"fs-id1163723206980\" type=\"a\">\n<li>[latex]\\theta=\\frac{\\pi}4[\/latex]<\/li>\n<li>[latex]r^{2}+z^{2}=9[\/latex]<\/li>\n<li>[latex]z=r[\/latex]<\/li>\n<\/ol>\n<div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"q124761291\">Show Solution<\/button><\/p>\n<div id=\"q124761291\" class=\"hidden-answer\" style=\"display: none\">\n<ol id=\"fs-id1163723569688\" type=\"a\">\n<li>When the angle [latex]\\theta[\/latex] is held constant while [latex]r[\/latex] and [latex]z[\/latex] are allowed to vary, the result is a half-plane (see the following figure).<br \/>\n<figure id=\"attachment_5284\" aria-describedby=\"caption-attachment-5284\" style=\"width: 447px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" class=\"size-full wp-image-5284\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/5667\/2021\/08\/31230343\/2.94.jpg\" alt=\"This figure is the first quadrant of the 3-dimensional coordinate system. There is a plane attached to the z-axis, dividing the x y-plane with a diagonal line. The angle between the x-axis and this plane is pi\/4.\" width=\"447\" height=\"635\" \/><figcaption id=\"caption-attachment-5284\" class=\"wp-caption-text\">Figure 7. In polar coordinates, the equation [latex]\\theta=\\frac{\\pi}4[\/latex] describes the ray extending diagonally through the first quadrant. In three dimensions, this same equation describes a half-plane.<\/figcaption><\/figure>\n<\/li>\n<li>Substitute [latex]r^{2}=x^{2}+y^{2}[\/latex] into equation [latex]r^{2}+z^{2}=9[\/latex] to express the rectangular form of the equation: [latex]x^{2}+y^{2}+z^{2}=9[\/latex]. This equation describes a sphere centered at the origin with radius 3 (see the following figure).<br \/>\n<figure id=\"attachment_5286\" aria-describedby=\"caption-attachment-5286\" style=\"width: 473px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" class=\"size-full wp-image-5286\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/5667\/2021\/08\/31230418\/2.95.jpg\" alt=\"This figure is a sphere. It has the z-axis through the center vertically. The point of intersection with the z-axis and the sphere is (0, 0, 3). There is also the y-axis through the center of the sphere horizontally. The intersection of the sphere and the y-axis is the point (0, 3, 0).\" width=\"473\" height=\"437\" \/><figcaption id=\"caption-attachment-5286\" class=\"wp-caption-text\">Figure 8. The sphere centered at the origin with radius 3 can be described by the cylindrical equation [latex]x^{2}+y^{2}+z^{2}=9[\/latex].<\/figcaption><\/figure>\n<\/li>\n<li>To describe the surface defined by equation [latex]z=r[\/latex], is it useful to examine traces parallel to the [latex]xy[\/latex]-plane. For example, the trace in plane [latex]z=1[\/latex] is circle [latex]r=1[\/latex], the trace in plane [latex]z=3[\/latex] is circle [latex]r=3[\/latex], and so on. Each trace is a circle. As the value of [latex]z[\/latex] increases, the radius of the circle also increases. The resulting surface is a cone (see the following figure).<br \/>\n<figure id=\"attachment_5287\" aria-describedby=\"caption-attachment-5287\" style=\"width: 421px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" class=\"size-full wp-image-5287\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/5667\/2021\/08\/31230440\/2.96.jpg\" alt=\"This figure is the 3-dimensional coordinate system. It has an elliptic cone with the z-axis down the center. The two cones, one right side up, the other upside down, meet at the origin.\" width=\"421\" height=\"490\" \/><figcaption id=\"caption-attachment-5287\" class=\"wp-caption-text\">Figure 9. The traces in planes parallel to the [latex]xy[\/latex]-plane are circles. The radius of the circles increases as [latex]z[\/latex] increases.<\/figcaption><\/figure>\n<\/li>\n<\/ol>\n<\/div>\n<\/div>\n<\/section>\n<p class=\"whitespace-normal break-words\">Cylindrical coordinates shine when dealing with problems that have cylindrical symmetry. In physics, many systems involving electrical charges, capacitors, and electromagnetic fields have this symmetry.<\/p>\n<p class=\"whitespace-normal break-words\">For example, the cylinder [latex]x^2 + y^2 = 25[\/latex] in rectangular coordinates becomes simply [latex]r = 5[\/latex] in cylindrical coordinates\u2014much simpler to work with!<\/p>\n<section class=\"textbox proTip\" aria-label=\"Pro Tip\">This coordinate transformation follows the same pattern as converting from polar to rectangular coordinates in 2D. We&#8217;re just adding the unchanged [latex]z[\/latex]-coordinate to extend into three dimensions.<\/section>\n<\/div>\n<\/section>\n","protected":false},"author":15,"menu_order":18,"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-533","chapter","type-chapter","status-publish","hentry"],"part":422,"module-header":"learn_it","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\/533","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":6,"href":"https:\/\/content.one.lumenlearning.com\/calculus3\/wp-json\/pressbooks\/v2\/chapters\/533\/revisions"}],"predecessor-version":[{"id":1682,"href":"https:\/\/content.one.lumenlearning.com\/calculus3\/wp-json\/pressbooks\/v2\/chapters\/533\/revisions\/1682"}],"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\/533\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/content.one.lumenlearning.com\/calculus3\/wp-json\/wp\/v2\/media?parent=533"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/calculus3\/wp-json\/pressbooks\/v2\/chapter-type?post=533"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/calculus3\/wp-json\/wp\/v2\/contributor?post=533"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/calculus3\/wp-json\/wp\/v2\/license?post=533"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}