Help with Partial Derivatives and Implicit Differentiation

[end3r7]

The problem is as follows:

Cartesian and polar coordinates are related by the formulas
x = r\cos\theta
y = r\sin\theta
Determine \frac{\partial r}{\partial x}, \frac{\partial r}{\partial y}, \frac{\partial\theta}{\partial x}, and \frac{\partial\theta}{\partial x}. Differentiate the equations above implcitly adn then solve the resulting system of 4 equations in the four unknowns \frac{\partial r}{\partial x}, \frac{\partial r}{\partial y}, \frac{\partial\theta}{\partial x}, and \frac{\partial\theta}{\partial x}.

Now, this seems a whole lot harder than just differentiating r^2 = x^2 + y^2, but I did anyway and the 4 equations are:

\frac{\partial x}{\partial x} = \frac{\partial r}{\partial x}\cos\theta - \sin\theta\frac{\partial\theta}{\partial x}r

\frac{\partial x}{\partial y} = \frac{\partial r}{\partial y}\cos\theta - \sin\theta\frac{\partial\theta}{\partial y}r

\frac{\partial y}{\partial x} = \frac{\partial r}{\partial x}\sin\theta + \cos\theta\frac{\partial\theta}{\partial x}r

\frac{\partial y}{\partial y} = \frac{\partial r}{\partial y}\sin\theta + \cos\theta\frac{\partial\theta}{\partial y}r

Now, if I try to solve for \frac{\partial r}{\partial x}, I know what the answer should be by differentiating r^2 = x^2 + y^2, and it's \frac xr or \cos\theta, however I can't seem to get that from combining the 4 equations.

What am I doing wrong?

Thanks in advance.

 

[0rthodontist]

Use the fact that dx/dx = 1 and dx/dy = 0, etc.

 

[end3r7]

I did... it still doesn't give me \cos\theta as an answer.

For example, to calculate dr/dx
using eq 1
I get 1 = \frac{\partial r}{\partial x}\cos\theta
using eq 3
0 = \frac{\partial r}{\partial x}\sin\theta

If I use both separatedly or try to combine them, I don't get the same answer as deriving r^2 = x^2 + y^2

 

[0rthodontist]

You seem to have cut off the end parts of those two equations. Try multiplying eq. 1 by cos theta / sin theta and adding it to eq. 3.

 

[end3r7]

Why mutliply times tangent will help?
I cut them off because I multiplied teh other stuff by zero.
I can solve it using the Jacobian, but I just want to know the logic behind it.

 

[0rthodontist]

You multiply eq. 1 by cos theta/sin theta so that when you add it to eq. 3, you cancel the terms involving dtheta/dx.

When you say you cut them off because you "multiplied the other stuff by 0" you can't just multiply one term of an equation by 0 and assume it will still be correct. The stuff you cut off was not in general zero.

 

[end3r7]

Oh ****, you are right. Thanks a lot man, really appreciate it! =D