Question

Can someone please write out this what this partial derivative is saying in words?

Answer 1

So what I'm not understanding is
$$ \frac{d^{2}u}{dr\ d\theta} $$
and
$$ \frac{d^{3}u}{d^{2}r\ d\theta} $$

How do you read that?

I understand $$\frac{du}{d\theta}$$

Is take the derivative of u with respect to theta, but how would you read the second and third ones?

Does the $$d\theta$$ just stay there because that was what u was taken with respect to at first, or does it mean you take the derivative of $$\frac{du}{d\theta}$$ *again* with respect to r *and* theta?

Answer 2

in sequence:
we take partial derivative w.r.t. variable theta \( \dfrac {\partial u}{\partial \theta} \)
then we take partial derivative w.r.t. variable r:
\( \dfrac{\partial}{\partial r} \left( \dfrac{\partial u}{\partial \theta} \right) = \dfrac{\partial^2 u}{\partial r \partial \theta} \)
is that what you're asking?

Answer 3

and for third one, we again take partial derivative w.r.t. r:
\( \dfrac{\partial}{\partial r} \left( \dfrac{\partial^2 u}{\partial r \partial \theta} \right) \)
= \( \dfrac{\partial^3 u}{\partial r^2 \partial \theta} \)
it is somewhat like multiplying the du together, but in this sense order should matter (we can't rewrite it as dtheta dr^2 in the denominator freely)

Answer 4

Ah! Thank you. I see it now.

Answer 5

For the third, say we had taken the derivative with respect to theta again, would it then be
$$\frac{\partial^{2}u}{\partial\theta \ \partial r \ \partial \theta}$$
or
$$\frac{\partial^{2}u}{\partial r \ \partial ^{2} \theta}$$
?

Answer 6

i.e. does order matter when combining them?

Answer 7

yes, to make that combination you would effectively be commuting dr and dtheta to accomplish it:
\( \dfrac{\partial ^3 u}{\partial \theta \partial r \partial \theta} \ne \dfrac{\partial^3 u}{\partial r \partial \theta \partial \theta} \)
so you want to keep it written the former way

Answer 8

Awesome! Thank you so much for clearing that up for me!