Saturday, March 5, 2011

Understanding Spherical Gradients

Previously we looked at where the 1/r term comes from in the gradient in cylindrical coordinates. This time, we're looking at the gradient for spherical coordinates.

The spherical coordinate values are shown in the figure below. The new theta coordinate is another angular coordinate similar to the phi coordinate introduced in the cylindrical system. It's angle sweeps down from the positive z axis of the Cartesian coordinate system to the negative z axis. Theres a another change. Instead of being anchored on the z axis and moving up and down, the r coordinate is anchored permanently at the coordinate origin.

In this coordinate system, the the angle theta and the radial direction sweep out circles in vertical planes similar to the horizontal plane circles discussed in the cylindrical case. Because of this, the theta coordinate has the same 1/r multiplier discussed in the cylindrical case.

phi and r sweep out circles in horizontal planes exactly as in the cylindrical system. But now, the radius of the circles are not just dependent on the value of r anymore. They also depend on the value of theta. If theta is zero radians, then the circle swept out by r and phi is a point, a circle with 0 radius. If theta is pi/2 radians, then the circle swept out by r and phi actually has a radius of r. The figure below shows a view of the relation ship between r and theta. The phi circles are swept out by the radial line with lenght r and angle theta from the z axis.

Because phi no longer sweeps out circles with radius r, but with radius r sin theta, it's element of length change must be modified in the same manner and we get the length factor assciate with the phi coordinate shown in the gradient above.

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