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

Similarly, or

Thus,

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 .

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4 Responses to 275 -The Laplacian in polar coordinates

[…] proof that the method of partial fractions decomposition works. For 275, you may want to review the polar expression of the Laplacian and how to derive it. For both courses, a safe assumption is that if something was […]

thanks i am really impressed how you explained all this i am undergraduate student i am having problem with spherical polar coordinate conversion how we find derivative of Theta w.r.t x, y,z kindly help me

I’m still just a little confused in the second application of the chain rule, when we differentiate to get the second derivative of f in respect to x. Can anyone help me here?

The description below comes from József Beck. Combinatorial games. Tic-tac-toe theory, Encyclopedia of Mathematics and its Applications, 114. Cambridge University Press, Cambridge, 2008, MR2402857 (2009g:91038). Given a finite set $S$ of points in the plane $\mathbb R^2$, consider the following game between two players Maker and Breaker. The players alternat […]

Yes. This is a consequence of the Davis-Matiyasevich-Putnam-Robinson work on Hilbert's 10th problem, and some standard number theory. A number of papers have details of the $\Pi^0_1$ sentence. To begin with, take a look at the relevant paper in Mathematical developments arising from Hilbert's problems (Proc. Sympos. Pure Math., Northern Illinois Un […]

I am looking for references discussing two inequalities that come up in the study of the dynamics of Newton's method on real-valued polynomials (in one variable). The inequalities are fairly different, but it seems to make sense to ask about both of them in the same post. Most of the details below are fairly elementary, they are mostly included for comp […]

Let $C$ be the standard Cantor middle-third set. As a consequence of the Baire category theorem, there are numbers $r$ such that $C+r$ consists solely of irrational numbers, see here. What would be an explicit example of a number $r$ with this property? Short of an explicit example, are there any references addressing this question? A natural approach would […]

Not necessarily. That $\mathfrak m$ is consistently singular is proved in MR0947850 (89m:03045) Kunen, Kenneth. Where $\mathsf{MA}$ first fails. J. Symbolic Logic 53(2), (1988), 429–433. There, Ken shows that $\mathfrak{m}$ can be singular of cofinality $\omega_1$. (Both links above are behind paywalls.)

No, the rank of a set $x$ is the least $\alpha$ such that $x\in V_{\alpha+1}$. Note that if $\alpha$ is limit, any $x\in V_\alpha$ belongs to some $V_\beta$ with $\beta

The real numbers are the usual thing. Surreal numbers are not real numbers, so no, they are not an example of non-constructible reals. Any real $r$ can be written as an infinite sequence $(n;d_1,d_2,\dots)$ where $n$ in an integer and the $d_i$ are digits. Whether the real is rational, constructible or not, is irrelevant. Any rational number, in fact, any al […]

Following Tomas's suggestion, I am posting this as an answer: I encountered this problem while directing a Master's thesis two years ago, and again (in a different setting) with another thesis last year. I seem to recall that I somehow got to this while reading slides of a talk by Paul Pollack. Anyway, I like to deduce the results asked in the prob […]

This is a beautiful and truly fundamental result, and so there are several good quality presentations. Try MR1321144. Kanamori, Akihiro. The higher infinite. Large cardinals in set theory from their beginnings. Perspectives in Mathematical Logic. Springer-Verlag, Berlin, 1994. xxiv+536 pp. ISBN: 3-540-57071-3, or any of the newer editions (the 2003 second ed […]

[…] proof that the method of partial fractions decomposition works. For 275, you may want to review the polar expression of the Laplacian and how to derive it. For both courses, a safe assumption is that if something was […]

Thanks for that post!

I was making some mistakes in these computations,

and this has helped!

thanks i am really impressed how you explained all this i am undergraduate student i am having problem with spherical polar coordinate conversion how we find derivative of Theta w.r.t x, y,z kindly help me

I’m still just a little confused in the second application of the chain rule, when we differentiate to get the second derivative of f in respect to x. Can anyone help me here?