- #1

- 53

- 0

## Main Question or Discussion Point

Hi all. I've recently started working a lot on my background in math and physics, since this year I began a new masters program which is quite math/physics heavy and I don't have a formal background in either field. I will try to get active on this forum, since I've been luring for some time and I find it really useful.

Right now I'm going through partial derivatives and some of the questions in the book I'm working with are quite confusing and I'm stuck. Here is one few examples:

z=x[itex]^{2}[/itex]+2y[itex]^{2}[/itex]

x=rcos(θ)

y=rsin(θ)

Question: [itex]\frac{∂^{2}z}{∂y∂θ}[/itex]

Here's what I've done so far:

I took the partial derivative with respect to θ = 2r[itex]^{2}[/itex]cosθsinθ

Now, from here I'm supposed to take the partial of the above with respect to y. I could either express it in terms of x and y:

[itex]\frac{∂2xy}{∂y}[/itex]

or in terms of r and θ:

[itex]\frac{∂2r^{2}sinθcosθ}{∂rsinθ}[/itex]

The thing is that I have no idea how to solve any of the above. The first I don't know how to solve because x and y aren't independent, so I can't just treat x as a constant. I got stuck when I attempted using the chain rule as well. The second I can't solve, since I don't know how to take a partial derivative with respect to a function, rather than a variable.

I would appreciate it if somebody explained how I can solve the problem using both methods (if both are possible, that is).

Right now I'm going through partial derivatives and some of the questions in the book I'm working with are quite confusing and I'm stuck. Here is one few examples:

z=x[itex]^{2}[/itex]+2y[itex]^{2}[/itex]

x=rcos(θ)

y=rsin(θ)

Question: [itex]\frac{∂^{2}z}{∂y∂θ}[/itex]

Here's what I've done so far:

I took the partial derivative with respect to θ = 2r[itex]^{2}[/itex]cosθsinθ

Now, from here I'm supposed to take the partial of the above with respect to y. I could either express it in terms of x and y:

[itex]\frac{∂2xy}{∂y}[/itex]

or in terms of r and θ:

[itex]\frac{∂2r^{2}sinθcosθ}{∂rsinθ}[/itex]

The thing is that I have no idea how to solve any of the above. The first I don't know how to solve because x and y aren't independent, so I can't just treat x as a constant. I got stuck when I attempted using the chain rule as well. The second I can't solve, since I don't know how to take a partial derivative with respect to a function, rather than a variable.

I would appreciate it if somebody explained how I can solve the problem using both methods (if both are possible, that is).