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 .

43.614000-116.202000

Advertisements

Like this:

LikeLoading...

Related

This entry was posted on Tuesday, October 7th, 2008 at 5:54 pm and is filed under 275: Calculus III. You can follow any responses to this entry through the RSS 2.0 feed.
You can leave a response, or trackback from your own site.

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?

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 […]

Suppose $M$ is an inner model (of $\mathsf{ZF}$) with the same reals as $V$, and let $A\subseteq \mathbb R$ be a set of reals in $M$. Suppose further that $A$ is determined in $M$. Under these assumptions, $A$ is also determined in $V$. The point is that since winning strategies are coded by reals, and any possible run of the game for $A$ is coded by a real, […]

Yes. This is obvious if there are no such cardinals. (I assume that the natural numbers of the universe of sets are the true natural numbers. Otherwise, the answer is no, and there is not much else to do.) Assume now that there are such cardinals, and that "large cardinal axiom" is something reasonable (so, provably in $\mathsf{ZFC}$, the relevant […]

Please send an email to mathrev@ams.org, explaining the issue. (This is our all-purpose email address; any mistakes you discover, not just regarding references, you can let us know there.) Give us some time, I promise we'll get to it. However, if it seems as if the request somehow fell through the cracks, you can always contact one of your friendly edit […]

The characterization mentioned by Mohammad in his answer really dates back to Lev Bukovský in the early 70s, and, as Ralf and Fabiana recognize in their note, has nothing to do with $L$ or with reals (in their note, they indicate that after proving their result, they realized they had essentially rediscovered Bukovský's theorem). See MR0332477 (48 #1080 […]

No, not even $\mathsf{DC}$ suffices for this. Here, $\mathsf{DC}$ is the axiom of dependent choice, which is strictly stronger than countable choice. For instance, it is a theorem of $\mathsf{ZF}$ that for any set $X$, the set $\mathcal{WO}(X)$ of subsets of $X$ that are well-orderable has size strictly larger than the size of $X$. This is a result of Tarski […]

I give an example (perhaps the best-known example) below, but let me first discuss equiconsistency rather than straight equivalence. Usually an equiconsistency is really the sort of result you are after anyway: You want to establish that certain statements in the universe where choice holds correspond to determinacy, which implies the failure of choice. The […]

The other answers have correctly identified the issue. Let me highlight the difficulty: it is relatively consistent with the axioms of set theory except for the axiom of choice that there are infinite sets which do not contain a copy of the natural numbers (that is, there are infinite sets $X$ such that there is no injection $f\!:\mathbb N\to X$). This means […]

This is $\aleph_\omega^{\aleph_0}$. First of all, this cardinal is an obvious upper bound. Second, if $A\subseteq\omega$ is infinite, $\prod_{i\in A}\aleph_i$ is clearly at least $\aleph_\omega$. The result follows, by splitting $\omega$ into countably many infinite sets. In general, the rules governing infinite products and exponentials are far from being w […]

If $\lambda$ and $\kappa$ are cardinals, $\lambda^\kappa$ represents the cardinality of the set of functions $f\!:A\to B$ where $A,B$ are fixed sets of cardinality $\kappa,\lambda$ respectively. (One needs to check this is independent of which specific sets $A,B$ we pick, of course.) At least for finite numbers, this is something you may have encountered in […]

[…] 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?