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Triple Integral in Spherical Coordinates

by cepheid
Tags: coordinates, integral, spherical, triple
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Sep15-04, 04:18 AM
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cepheid's Avatar
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I have a hemispherical surface of radius R with it's base centred on the origin. We are using the convention:

r is the radius i.e. the magnitude of the position vector of a point: its distance from the origin.

theta is the polar angle

phi is the azimuthal angle

I am asked to calculate the integral of the divergence of a given vector field v over this volume (enclosed by the hemisphere).

So far what I have done is to say that a full sphere would be given by the equation:

r = R

What about a hemisphere? It seems to me that the angle theta must be restricted so that points below the x-y plane are not part of the domain. So what are the allowed values of theta? I can't seem to figure out whether it should be -pi/2 < theta < pi/2, or something else? I need to know this to set my bounds of integration for one of the three integrals. Thanks.
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Sep15-04, 06:36 AM
marlon's Avatar
P: 4,006
For a hemisphere : r goes from 0 to R
theta goes from : 0 to 90
phi goes from : 0 to 360

Ofcourse, you know that you need to express the angles in radials


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