{"id":5464,"date":"2022-06-02T18:35:11","date_gmt":"2022-06-02T18:35:11","guid":{"rendered":"https:\/\/courses.lumenlearning.com\/calculus3\/?post_type=chapter&p=5464"},"modified":"2022-10-26T02:58:06","modified_gmt":"2022-10-26T02:58:06","slug":"cylindrical-coordinates","status":"publish","type":"chapter","link":"https:\/\/courses.lumenlearning.com\/calculus3\/chapter\/cylindrical-coordinates\/","title":{"raw":"Cylindrical Coordinates","rendered":"Cylindrical Coordinates"},"content":{"raw":"
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Learning Objectives<\/h3>\r\n
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  • Convert from cylindrical to rectangular coordinates.<\/span><\/li>\r\n \t
  • Convert from rectangular to cylindrical coordinates.<\/span><\/li>\r\n<\/ul>\r\n<\/div>\r\n

    When we expanded the traditional Cartesian coordinate system from two dimensions to three, we simply added a new axis to model the third dimension. Starting with polar coordinates, we can follow this same process to create a new three-dimensional coordinate system, called the cylindrical coordinate system. In this way, cylindrical coordinates provide a natural extension of polar coordinates to three dimensions.<\/p>\r\n\r\n<\/div>\r\n

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    DEFINITION<\/span><\/h3>\r\n\r\n
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    In the\u00a0cylindrical coordinate system<\/span><\/strong>, a point in space (Figure 1) is represented by the ordered triple [latex](r,\\theta,z)[\/latex], where<\/p>\r\n\r\n

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    • [latex](r,\\theta)[\/latex] are the polar coordinates of the point\u2019s projection in the [latex]xy[\/latex]-plane<\/li>\r\n \t
    • [latex]z[\/latex] is the usual [latex]z[\/latex]-coordinate in the Cartesian coordinate system<\/li>\r\n<\/ul>\r\n<\/div>\r\n<\/section><\/div>\r\n[caption id=\"attachment_5265\" align=\"aligncenter\" width=\"427\"]\"This 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]In the [latex]xy[\/latex]-plane, the right triangle shown in\u00a0Figure 1\u00a0provides the key to transformation between cylindrical and Cartesian, or rectangular, coordinates.\r\n
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      THEOREM: conversion between cylindrical and cartesian coordinates<\/span><\/h3>\r\n\r\n
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      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

      [latex]\\begin{aligned}\r\nx&=r\\cos\\theta \\quad & \\text{These equations are used to} \\\\\r\ny&=r\\sin\\theta &\\text{convert from cylindrical coordinates} \\\\\r\nz&=z &\\text{ to rectangular coordinates} \\\\\r\n\\\\\r\n&\\text{and} \\\\\r\n\\\\\r\nr^2&=x^2+y^2 \\quad &\\text{These equations are used to } \\\\\r\n\\tan\\theta&=\\frac{y}{x} &\\text{convert from rectangular coordinates }\\\\\r\nz&=z &\\text{to cylindrical coordinates}\r\n\\end{aligned}[\/latex]<\/p>\r\n\r\n<\/div>\r\n<\/section><\/div>\r\n

      As when we discussed conversion from rectangular coordinates to polar coordinates in two dimensions, it should be noted that the equation [latex]\\tan\\theta=\\frac{y}{x}[\/latex] has an infinite number of solutions. However, if we restrict [latex]\\theta[\/latex] to values between [latex]0[\/latex] and [latex]2\\pi[\/latex], then we can find a unique solution based on the quadrant of the [latex]xy[\/latex]-plane in which original point [latex](x, y, z)[\/latex] is located. Note that if [latex]x=0[\/latex], then the value of [latex]\\theta[\/latex] is either [latex]\\frac{\\pi}2[\/latex], [latex]\\frac{3\\pi}2[\/latex], or [latex]0[\/latex] depending on the value of [latex]y[\/latex].<\/p>\r\n

      Notice that these equations are derived from properties of right triangles. To make this easy to see, consider point [latex]P[\/latex] in the [latex]xy[\/latex]-plane with rectangular coordinates [latex](x, y, 0)[\/latex] and with cylindrical coordinates [latex](r,\\theta,0)[\/latex], as shown in the following figure.<\/p>\r\n\r\n[caption id=\"attachment_5269\" align=\"aligncenter\" width=\"344\"]\"This 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]\r\n

      Let\u2019s consider the differences between rectangular and cylindrical coordinates by looking at the surfaces generated when each of the coordinates is held constant. If [latex]c[\/latex] is a constant, then in rectangular coordinates, surfaces of the form [latex]x=c[\/latex], [latex]y=c[\/latex], or [latex]z=c[\/latex] are all planes. Planes of these forms are parallel to the [latex]yz[\/latex]-plane, the [latex]xz[\/latex]-plane, and the [latex]xy[\/latex]-plane, respectively. When we convert to cylindrical coordinates, the [latex]z[\/latex]-coordinate does not change. Therefore, in cylindrical coordinates, surfaces of the form [latex]z=c[\/latex] are planes parallel to the [latex]xy[\/latex]-plane. Now, let\u2019s think about surfaces of the form [latex]r=c[\/latex]. The points on these surfaces are at a fixed distance from the [latex]z[\/latex]-axis. In other words, these surfaces are vertical circular cylinders. Last, what about [latex]\\theta=c[\/latex]? The points on a surface of the form [latex]\\theta=c[\/latex] are at a fixed angle from the [latex]x[\/latex]-axis, which gives us a half-plane that starts at the [latex]z[\/latex]-axis (Figure 3\u00a0and\u00a0Figure 4).<\/p>\r\n\r\n

      [caption id=\"attachment_5274\" align=\"aligncenter\" width=\"712\"]\"This 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
      [caption id=\"attachment_5275\" align=\"aligncenter\" width=\"712\"]\"This 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
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      Example: converting from cylindrical to rectangular coordinates<\/h3>\r\nPlot the point with cylindrical coordinates [latex]\\left(4,\\frac{2\\pi}3,-2\\right)[\/latex] and express its location in rectangular coordinates.\r\n\r\n[reveal-answer q=\"951616342\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"951616342\"]\r\n

      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

      [latex]\\begin{aligned}\r\nx&=r\\cos\\theta=4\\cos\\frac{2\\pi}3=-2 \\\\\r\ny&=r\\sin\\theta=4\\sin\\frac{2\\pi}3=2\\sqrt3 \\\\\r\nz&=-2\r\n\\end{aligned}[\/latex].<\/p>\r\n

      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\"]\"This 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]\r\n\r\n<\/div>\r\n

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      try it<\/h3>\r\nPoint [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.\r\n\r\n[reveal-answer q=\"572615912\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"572615912\"]\r\n\r\nThe rectangular coordinates of the point are [latex]\\left(\\frac{5\\sqrt3}2,\\frac52,4\\right)[\/latex].\u00a0\r\n<\/span>\r\n\r\n[caption id=\"attachment_5281\" align=\"aligncenter\" width=\"462\"]\"This Figure 6. The rectangular coordinates of the point are [latex]\\left(\\frac{5\\sqrt3}2,\\frac52,4\\right)[\/latex][\/caption][\/hidden-answer]\r\n\r\n<\/div>\r\nWatch the following video to see the worked solution to the above Try IT.\r\n\r\n