Differentiation problem

**champrock** · March 12th 2009, 03:26 AM

2006

**sitho** · March 14th 2009, 01:50 PM

$u = \phi(x - y, y - z, z - x) = \phi(a_1, a_2, a_3)$

$a_1 = x - y$
$a_2 = y - z$
$a_3 = z - x$

$\frac{du}{dx} = \sum_{i = 1}^{3}\frac{d\phi}{da_i}\frac{da_i}{dx} = \frac{d\phi}{da_1} - \frac{d\phi}{da_3}$
$\frac{du}{dy} = -\frac{d\phi}{da_1} + \frac{d\phi}{da_2}$
$\frac{du}{dz} = -\frac{d\phi}{da_2} + \frac{d\phi}{da_3}$

$\Rightarrow \frac{du}{dx} + \frac{du}{dy} + \frac{du}{dz} = \frac{d\phi}{da_1} - \frac{d\phi}{da_3} - \frac{d\phi}{da_1} + \frac{d\phi}{da_2} - \frac{d\phi}{da_2} + \frac{d\phi}{da_3} = 0$
yey!! ^^

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