CAL C: POLAR, CYLINDRICAL, & SPHERICAL COORDINATES

Polar and Cylindrical Coordinates: Jacobian of the transformation

∬ 푓(푥, 푦) 푑푥 푑푦 = ∬ 푓(푟푐표푠휃, 푟푠푖푛휃) ⋅ 푟 푑푟 푑휃 푅 퐺

 For limits of integration for a polar double integral, draw a ray extending radially outward from the and determine the functions at which it enters and at which it exits. For the 휃 limits, determine the initial and final of this ray. Note: 휃 will usually be the last of integration.

 The Cylindrical is built on the polar coordinate system with the addition of a variable to describe the from the 푟-휃 (thought of as height”), notated as 푧. Note that this 푧 is equivalent to that of the rectangular Jacobian of the coordinate 푧. transformation

∭ 푓 푑푉 = ∭ 푓 푑푧 푟 푑푟 푑휃 퐷 퐷

 The limits of integration are similar to polar for 푟 and 휃 and to rectangular for 푧. Note: 휃 will usually be the last variable of integration.

Spherical Coordinates:

 Spherical coordinates are defined by three parameters: 1) 휌, the radial distance from a to the origin. 2) 휙, the polar angle between a point and the positive z-axis. Source: Wikipedia 3) 휃, the angle between the shadow of 휌 on the x-y plane and positive x-axis. Jacobian of the transformation ∭ 푓(푥, 푦, 푧)푑푉 = ∭ 푓(휌푠푖푛휙푐표푠휃, 휌푠푖푛휙푠푖푛휃, 휌푐표푠휃)푑푉 = ∭ 푓(휌, 휙, 휃) 휌2푠푖푛휙 푑휌 푑휙 푑휃 퐷 퐷 퐷

 The limits of integration are found by similar fashion as cylindrical. Typically, first find 휌 limits by drawing a ray from the origin through the region at angle 휙 and observe the functions which it enters and exits. Next, determine 휙 limits by observing the minimum (≤ 0°) and maximum (≥ 180°) angle made with the positive z-axis. Finally, observe the “shadow” of ray 휌 on the x-y plane and determine the minimum and maximum it makes with the positive x-axis as it sweeps through the entire region.

Cylindrical to Rectangular Spherical to Rectangular Spherical to Cylindrical 푥 = 푟 푐표푠휃 푥 = 휌 푠푖푛휙 푐표푠휃 푟 = 휌 푠푖푛휙 푦 = 푟 푠푖푛휃 푦 = 휌 푠푖푛휙 푠푖푛휃 푧 = 휌푐표푠휙 푧 = 푧 푧 = 휌 푐표푠휙 휃 = 휃

푑푉 = 푑푥 푑푦 푑푧 Rectangular

푑푧 푟 푑푟 푑휃 Cylindrical 휌2푠푖푛휙 푑휌 푑휙 푑휃 Spherical

Change of Variables:

 A or coordinate systems is often useful in solving complex integrals. When doing so, a Jacobian is a necessary accompaniment:

휕푥 휕푥 휕푥 휕푦 휕푦 휕푥 휕(푥, 푦) 퐽(푢, 푣) = |휕푢 휕푣| = ⋅ − ⋅ = 휕푦 휕푦 휕푢 휕푣 휕푢 휕푣 휕(푢, 푣) 휕푢 휕푣

 For example, the following Jacobian is required to transform cartesian to polar coordinates:

휕푥 휕푥 휕(푟푐표푠휃) 휕(푟푐표푠휃) 푐표푠휃 −푟푠푖푛휃 퐽(푟, 휃) = | 휕푟 휕휃| = | 휕푟 휕휃 | = | | = 푟(cos2 휃 + sin2 휃) = 푟 휕푦 휕푦 휕(푟푠푖푛휃) 휕(푟푠푖푛휃) 푠푖푛휃 푟푐표푠휃

휕푟 휕휃 휕휃 휕휃

휕푧  Likewise, it can be shown that 퐽(휌, 휙, 휃) = 휌2 푠푖푛휙 by adding a third row ( ) and third 휕( ) ( ) column (휕 ). 휕휙

For more information, visit a tutor. All appointments are available in-person at the Student Success Center, located in the Library, or online. Adapted from Hass, J., Weir, M.D., & Thomas, G.B. (2012). University : Early Transcendentals (2nd ed.). Boston: Pearson Education.