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Working in Different Coordinate Systems

In this class, you will probably always solve problems in one of the following coordinate systems:
☞ Cartesian coordinates
☞ Cylindrical coordinates
☞ Spherical coordinates

I will not waste your time be defining all of these systems for you, nor will I rehash techniquies of changing from one system to the other. What I will do is try to explain why the different systems are used and why you need to pay attention to which system you are in.

The idea behind using a different coordinate system is to make the problem you are solving as simple as possible. If you use a system that is not helpful, you will wind up doing more work than necessary.

Common sense will normally dictate which system to use. Finite distributions of points are usually amenable to Cartesian coordinates. Fields surrounding wires are often easiest to express in cylindrical coordinates. Fields surrounding spheres, clouds, etc. will probably be best handled in spherical coordinates.

But remember as you go from system to system that you must be careful about how you define different quantities. For example, volume elements are usually expressed as follows:

☞ Cartesian coordinates : dV = dx dy dz
☞ Cylindrical coordinates : dV = r dr d(phi) dz
☞ Spherical coordinates : dV = R^2 sin(theta) dR d(theta) d(phi)

Our definition of the del operator also changes from coordinate system to coordinate system. I would not recommend memorizing every definition for every system, because :

☞ It will drive you nuts
☞ You will probably forget it just when you need it.

A better idea is to understand why we have to redefine these quantities. This will allow you to remember that you do need to change the definition, so you won't forget to look them up and use the correct ones.

The reason that quantities such as volume appear to have different definitions in each system is that not all coordinates of all systems represent length. In cartesian coordinates, we know that all three coordinates represent length. Since a volume is determined by a length cubed, we get all of our ordinary definitions. However, cylindrical and Spherical coordinates contain terms that represent angles. In order to use such a direction to represent a length, we must change the angle into a length.

In cylindrical coordinates, for example, to change the phi component d(phi) to a length, you must multiply by the radius r. That is why a factor of r appears in the definition of dV for cylindrical coordinates. Similar arguments exist for spherical coordinates.

The bottom line is that if you are dealing with any formular that relies on length (and you will, because many formulas involve an integral over dL, dS, or dV), check your coordinates system. If you are in a system where not all of the coordinates are lengths, remember to add in the proper factors so that they may be treated as lengths.

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