- #1
rhdinah
- 17
- 1
Homework Statement
If ## z=x^2+2y^2 ##, find the following partial derivative:
[tex]\Big(\frac{∂z}{∂\theta}\Big)_x[/tex]
Homework Equations
## x=r cos(\theta), ~y=r sin(\theta),~r^2=x^2+y^2,~\theta=tan^{-1}\frac{y}{x} ##
The Attempt at a Solution
I've been using Boas for self-study and been working on this problem for almost a week now and cannot get the expected answer. I'm persistent though and have decided to look outside my study area for some help. So if you can add some direction or pointers I'd appreciate it.
The first thing we need to do is to simplify the original equation since keeping ##x## constant means the ##x^2## term is a constant and will differentiate to 0 ... so ## z=2y^2 ##.
[tex]\frac{∂z}{∂\theta}=\frac{∂z}{∂y}\frac{∂y}{∂\theta}=4 r sin(\theta)*r cos(\theta)=4 r^2sin(\theta)cos(\theta)[/tex]
Now the expected answer is ##4 r^2 tan(\theta)## and the apparent only way to get to the expected answer from here is to divide by ##cos^2(\theta)## ... which is working backwards of course. I've done and redone this work from a number of approaches and have gotten the same answer. I just don't see where this ##tan## function is coming from.
One of my problems is that I don't know how to do the special case of partial derivatives on Wolfram|Alpha ... while writing standard partials is easy, setting the constraint on what stays constant is difficult to conjure. Plus the engine doesn't convert easily from polar to cartesian well. So if there are some hints there ... so far, I've not found them on my googling of the net.
I've had no problems with any other of the partials, plus all the exercises I've found on the Internet have been straight-forward. So I wonder what I'm doing wrong here. Her next problem 14, ##\Big(\frac{∂z}{∂\theta}\Big)_y##, goes down a similar path so knowing how to approach this problem will help with the next. Thanks!