lec_week11

lec_week11 - MIT OpenCourseWare http://ocw.mit.edu 18.02...

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Unformatted text preview: MIT OpenCourseWare http://ocw.mit.edu 18.02 Multivariable Calculus Fall 2007 For information about citing these materials or our Terms of Use, visit: http://ocw.mit.edu/terms . 18.02 Lecture 26. Tue, Nov 13, 2007 Spherical coordinates ( ,, ). = rho = distance to origin. = = phi = angle down from z-axis. = same as in cylindrical coordinates. Diagram drawn in space, and picture of 2D slice by vertical plane with z,r coordinates. Formulas to remember: z = cos , r = sin (so x = sin cos , y = sin sin ). 2 2 = x 2 + y 2 + z = r 2 + z . The equation = a defines the sphere of radius a centered at 0. On the surface of the sphere, is similar to latitude , except its at the north pole, / 2 on the equator, at the south pole. is similar to longitude . = / 4 is a cone (asked using ash cards) ( z = r = x 2 + y 2 ). = / 2 is the xy-plane. Volume element: dV = 2 sin ddd . To understand this formula, first study surface area on sphere of radius a : picture shown of a rectangle corresponding to , , with sides = portion of circle of radius a , of length a , and portion of circle of radius r = a sin , of length r = a sin . So S a 2 sin , which gives the surface element dS = a 2 sin dd . The volume element follows: for a small box, V = S , so dV = ddS = 2 sin ddd . Example: recall the complicated example at end of Fridays lecture (region sliced by a plane inside unit sphere). After rotating coordinate system, the question becomes: volume of the portion of unit sphere above the plane z = 1 / 2? (picture drawn). This can be set up in cylindrical (left as exercise) or spherical coordinates. For fixed , we are slicing our region by rays straight out of the origin; ranges from its value on the plane z = 1 / 2 to its value on the sphere = 1. Spherical coordinate equation of the plane: z = cos = 1 / 2, so = sec / 2. The volume is: 2 / 4 1 2 sin ddd....
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lec_week11 - MIT OpenCourseWare http://ocw.mit.edu 18.02...

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