Tough calculus...

**andreas** · December 12th 2008, 08:34 AM

I have the following from Vector Calculus:

$\frac{\delta\phi}{\delta x}=\frac{yz(y+z)}{(x+y+z)^2}$

$\frac{\delta\phi}{\delta y}=\frac{xz(x+z)}{(x+y+z)^2}$

$\frac{\delta\phi}{\delta x}=\frac{xy(x+y)}{(x+y+z)^2}$

I have to determine whether this system has solutions or not. If there is a solution I have to find it. I also have to determine the domain on which solution is defined. Also some words about uniqueness must be said...

Can anyone give me some help?

**running-gag** · December 12th 2008, 08:56 AM

Hi

Integrating

$\frac{\delta\phi}{\delta x}=\frac{yz(y+z)}{(x+y+z)^2}$

gives

$\phi(x,y,z)=-\frac{yz(y+z)}{x+y+z} + f(y,z)$

$\phi(x,y,z)=-yz + \frac{xyz}{x+y+z} + f(y,z)$

Derivation with respect to y gives

$\frac{\delta\phi}{\delta y} = -z + \frac{xz(x+z)}{(x+y+z)^2} + \frac{\delta f}{\delta y}$

But $\frac{\delta\phi}{\delta y} = \frac{xz(x+z)}{(x+y+z)^2}$

Therefore
$\frac{\delta f}{\delta y} = z$ and f(y,z) = yz + g(z)

The same approach for z leads to

$\phi(x,y,z)=\frac{xyz}{x+y+z} + constant$

**moolimanj** · December 12th 2008, 08:57 AM

I think this is where you might use the formulas for gradient, curl, divergence and laplacian equations:

You may need to wrte out teh Jacobian matrix first to help you.

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