We’ve previously looked at how to derive divergence for cylindrical coordinates. If you’re like me though, knowing the rather lengthy derivation won’t help you understand or memorize the resulting formula. So, let’s take a look at why the result makes sense.

The formula for the gradient of a function in cylindrical coordinates is:

Why is the factor of 1/r in the phi term? Remember what question the divergence is asking. We want to find out the amount the function changes vs. a small change in distance along each coordinate’s direction. For coordinates that actually correspond to distances, like x, y, and z of Cartesian coordinates, or r and z of cylindrical coordinates, this is straightforward. The change in the coordinate corresponds to the change in distance along the coordinate.

For coordinates that correspond to angles in the cylindrical coordinate system, there’s an extra twist. The direction of phi always points tangent to a circle centered on the z axis. The small change in distance along the phi or theta coordinate direction is actually a distance measured along the circumference of a circle. The figure below helps me see the issue more clearly. A small change in phi at a radius of 1 means a change of distance of phi times the radius, or 1 phi. At a radius of 2, the same small change in phi results in a change along the circumference of 2 phi.

The distance used in finding the amount the function changes per small change in distance along the phi or theta coordinate is equal to the change in phi times the radius where the change in phi or theta occurs.

So, to properly account for the distance along the phi direction, the divergence has to be written as:

The formula for the gradient of a function in cylindrical coordinates is:

Why is the factor of 1/r in the phi term? Remember what question the divergence is asking. We want to find out the amount the function changes vs. a small change in distance along each coordinate’s direction. For coordinates that actually correspond to distances, like x, y, and z of Cartesian coordinates, or r and z of cylindrical coordinates, this is straightforward. The change in the coordinate corresponds to the change in distance along the coordinate.

For coordinates that correspond to angles in the cylindrical coordinate system, there’s an extra twist. The direction of phi always points tangent to a circle centered on the z axis. The small change in distance along the phi or theta coordinate direction is actually a distance measured along the circumference of a circle. The figure below helps me see the issue more clearly. A small change in phi at a radius of 1 means a change of distance of phi times the radius, or 1 phi. At a radius of 2, the same small change in phi results in a change along the circumference of 2 phi.

The distance used in finding the amount the function changes per small change in distance along the phi or theta coordinate is equal to the change in phi times the radius where the change in phi or theta occurs.

So, to properly account for the distance along the phi direction, the divergence has to be written as:

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