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 […]

A database of number fields, by Jürgen Klüners and Gunter Malle. (Note this is not the same as the one mentioned in this answer.) The site also provides links to similar databases.

As the other answer indicates, the yes answer to your question is known as the De Bruijn-Erdős theorem. This holds regardless of the size of the graph. The De Bruijn–Erdős theorem is a particular instance of what in combinatorics we call a compactness argument or Rado's selection principle, and its truth can be seen as a consequence of the topological c […]

Every $P_c$ has the size of the reals. For instance, suppose $\sum_n a_n=c$ and start by writing $\mathbb N=A\cup B$ where $\sum_{n\in A}a_n$ converges absolutely (to $a$, say). This is possible because $a_n\to 0$: Let $m_0

Consider a subset $\Omega$ of $\mathbb R$ of size $\aleph_1$ and ordered in type $\omega_1$. (This uses the axiom of choice.) Let $\mathcal F$ be the $\sigma$-algebra generated by the initial segments of $\Omega$ under the well-ordering (so all sets in $\mathcal F$ are countable or co-countable), with the measure that assigns $0$ to the countable sets and $1 […]

You assume $\omega_\alpha\subseteq M$ and $X\in M$ so that $X$ belongs to the transitive collapse of $M$ (because if $\pi$ is the collapsing map, $\pi(X)=\pi[X]=X$. You assume $|M|=\aleph_\alpha$ so that the transitive collapse of $M$ has size $\aleph_\alpha$. Since you also have that this transitive collapse is of the form $L_\beta$ for some $\beta$, it fol […]

No, this is not possible. Dave L. Renfro wrote an excellent historical Essay on nowhere analytic $C^\infty$ functions in two parts (with numerous references). See here: 1 (dated May 9, 2002 6:18 PM), and 2 (dated May 19, 2002 8:29 PM). As indicated in part 1, in Zygmunt Zahorski. Sur l'ensemble des points singuliers d'une fonction d'une variab […]

I don't think you need too much in terms of prerequisites. An excellent reference is MR3616119. Tomkowicz, Grzegorz(PL-CEG2); Wagon, Stan(1-MACA-NDM). The Banach-Tarski paradox. Second edition. With a foreword by Jan Mycielski. Encyclopedia of Mathematics and its Applications, 163. Cambridge University Press, New York, 2016. xviii+348 pp. ISBN: 978-1-10 […]

For the second problem, write $x=-3+x'$ and so on. You have $x'+y'+z'=17$ and $x',\dots$ are nonnegative, a case you know how to solve. You can also solve the first problem this way; now you would set $x=1+x'$, etc.

[…] 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 […]