implicit Partial differentiation

Hi,

I have a problem that has to do with implicit partial differentiation. I was wondering how I could solve it using Jacobian determinants, if not possible, then a straight forward approach is okay. Here we go!

Given:

$\displaystyle f(u,v) = 0$

$\displaystyle u = lx + my + nz$

$\displaystyle v = x^{2} + y^{2} + z^{2}$

Prove:

$\displaystyle (ly - mx) + (ny - mx) \frac{\partial z}{\partial x} + (lz - nx)\frac{\partial z}{\partial y} = 0$

I have a problem finding the two partial terms. This is what I have done:

$\displaystyle \frac{\partial z}{\partial x} = -\frac{\frac{\partial (u,v)}{\partial (x,y)}}{\frac{\partial (u,v)}{\partial (z,y)}}$

$\displaystyle \frac{\partial z}{\partial y} = -\frac{\frac{\partial (u,v)}{\partial (y,x)}}{\frac{\partial (u,v)}{\partial (z,x)}}$

but I end up with $\displaystyle ly - mx = 0$, not $\displaystyle 0 = 0$

Any suggestions? Thanks!

Re: implicit Partial differentiation

Quote:

Originally Posted by

**gotFermi** Hi,

I have a problem that has to do with implicit partial differentiation. I was wondering how I could solve it using Jacobian determinants, if not possible, then a straight forward approach is okay. Here we go!

Given:

$\displaystyle f(u,v) = 0$

$\displaystyle u = lx + my + nz$

$\displaystyle v = x^{2} + y^{2} + z^{2}$

Prove:

$\displaystyle (ly - mx) + (ny - mx) \frac{\partial z}{\partial x} + (lz - nx)\frac{\partial z}{\partial y} = 0$

I have a problem finding the two partial terms. This is what I have done:

$\displaystyle \frac{\partial z}{\partial x} = -\frac{\frac{\partial (u,v)}{\partial (x,y)}}{\frac{\partial (u,v)}{\partial (z,y)}}$

$\displaystyle \frac{\partial z}{\partial y} = -\frac{\frac{\partial (u,v)}{\partial (y,x)}}{\frac{\partial (u,v)}{\partial (z,x)}}$

but I end up with $\displaystyle ly - mx = 0$, not $\displaystyle 0 = 0$

Any suggestions? Thanks!

Your notation doesn't make any sense. I would advise writing each function as z in terms of the other variables, then doing the differentiation.

Re: implicit Partial differentiation

What terms don't make sense? I can change that, I am just copying how my book sells the ideas.

I'll give the rewriting a try, thanks.