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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5 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?

[…] 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 […]

Georgii: Let me start with some brief remarks. In a series of three papers: a. Wacław Sierpiński, "Contribution à la théorie des séries divergentes", Comp. Rend. Soc. Sci. Varsovie 3 (1910) 89–93 (in Polish). b. Wacław Sierpiński, "Remarque sur la théorème de Riemann relatif aux séries semi-convergentes", Prac. Mat. Fiz. XXI (1910) 17–20 […]

It is not possible to provide an explicit expression for a non-linear solution. The reason is that (it is a folklore result that) an additive $f:{\mathbb R}\to{\mathbb R}$ is linear iff it is measurable. (This result can be found in a variety of places, it is a standard exercise in measure theory books. As of this writing, there is a short proof here (Intern […]

Stefan, "low" cardinalities do not change by passing from $L({\mathbb R})$ to $L({\mathbb R})[{\mathcal U}]$, so the answer to the second question is that the existence of a nonprincipal ultrafilter does not imply the existence of a Vitali set. More precisely: Assume determinacy in $L({\mathbb R})$. Then $2^\omega/E_0$ is a successor cardinal to ${ […]

Marginalia to a theorem of Silver (see also this link) by Keith I. Devlin and R. B. Jensen, 1975. A humble title and yet, undoubtedly, one of the most important papers of all time in set theory.

Given a positive integer $a$, the Ramsey number $R(a)$ is the least $n$ such that whenever the edges of the complete graph $K_n$ are colored using only two colors, we necessarily have a copy of $K_a$ with all its edges of the same color. For example, $R(3)= 6$, which is usually stated by saying that in a party of 6 people, necessarily there are 3 that know e […]

Equality is part of the background (first-order) logic, so it is included, but there is no need to mention it. The situation is the same in many other theories. If you want to work in a language without equality, on the other hand, then this is mentioned explicitly. It is true that from extensionality (and logical axioms), one can prove that two sets are equ […]

$L$ has such a nice canonical structure that one can use it to define a global well-ordering. That is, there is a formula $\phi(u,v)$ that (provably in $\mathsf{ZF}$) well-orders all of $L$, so that its restriction to any specific set $A$ in $L$ is a set well-ordering of $A$. The well-ordering $\varphi$ you are asking about can be obtained as the restriction […]

Gödel sentences are by construction $\Pi^0_1$ statements, that is, they have the form "for all $n$ ...", where ... is a recursive statement (think "a statement that a computer can decide"). For instance, the typical Gödel sentence for a system $T$ coming from the second incompleteness theorem says that "for all $n$ that code a proof […]

When I first saw the question, I remembered there was a proof on MO using Ramsey theory, but couldn't remember how the argument went, so I came up with the following, that I first posted as a comment: A cute proof using Schur's theorem: Fix $a$ in your semigroup $S$, and color $n$ and $m$ with the same color whenever $a^n=a^m$. By Schur's theo […]

It depends on what you are doing. I assume by lower level you really mean high level, or general, or 2-digit class. In that case, 54 is general topology, 26 is real functions, 03 is mathematical logic and foundations. "Point-set topology" most likely refers to the stuff in 54, or to the theory of Baire functions, as in 26A21, or to descriptive set […]

[…] 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?

[…] 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 […]