implicit Partial differentiation

**gotFermi** · January 3rd 2012, 04:19 PM

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:
$f(u,v) = 0$
$u = lx + my + nz$
$v = x^{2} + y^{2} + z^{2}$
Prove:
$(ly - mx) + (ny - mx) \frac{\partial z}{\partial x} + (lz - nx)\frac{\partial z}{\partial y} = 0$

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

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

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

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

Any suggestions? Thanks!

**Prove It** · January 3rd 2012, 04:21 PM

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:
$f(u,v) = 0$
$u = lx + my + nz$
$v = x^{2} + y^{2} + z^{2}$
Prove:
$(ly - mx) + (ny - mx) \frac{\partial z}{\partial x} + (lz - nx)\frac{\partial z}{\partial y} = 0$

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

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

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

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

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.

**gotFermi** · January 3rd 2012, 04:28 PM

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.

Thread: implicit Partial differentiation

LinkBack

Thread Tools

Display

implicit Partial differentiation

Re: implicit Partial differentiation

Re: implicit Partial differentiation

Similar Math Help Forum Discussions

[SOLVED] Implicit partial differentiation. Need my answer checked please

Implicit differentiation/ partial differentiation

Implicit Differentiation of Partial Derivatives

Calculating partial derivatives by implicit differentiation of the Law of Cosines

Implicit vs. partial

Search Tags