In many situations in which symmetries are involved, it is useful to write the equations describing the physical objects under study with respect to frames related to said symmetries. For situations involving rotational symmetry, polar (or cylindrical) coordinates are particularly useful.
For this reason, it is a good exercise to see how to transform something like Laplace equation into polar coordinates.
Remember that the Laplacian of is . If we write in polar coordinates as , we then need to write the Laplacian directly in terms of derivatives with respect to and .
Following a common abuse of notation, I will write , etc, for what one should more precisely write , etc, with as above.
Let’s begin by finding , , etc.
Since , then
We have . Similarly, .
Since , then , so .
Similarly, , so .
Before we continue, a word of caution is in order: Notice that so, for example, . This means that is not , as it is the case with functions of one variable.
Now we use the chain rule: Suppose is a differentiable function of . Then we obtain .
Assume now that is sufficiently differentiable so all the derivatives to follow exist and are continuous, mixed derivatives commute, etc. Apply the above twice, first with and then with to obtain or
We have obtained the Laplacian in polar coordinates as
Example: Let . We know from lecture that is harmonic. Writing in polar coordinates, we obtain . Then , , , and , so as expected.
Exercise: To test your understanding of the algebra involved in the operations above, you may want to find an expression for the wave equation in polar coordinates.
Cylindrical coordinates: Finally, if is a function of three variables and so , then we see immediately from the work above that we can write the Laplacian in cylindrical coordinates as .