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Standard Formulae - 1

If u= y z +...

Question

If $u = \frac{y}{z} + \frac{z}{x}$ Show that $x \frac{\partial u}{\partial x} + y \frac{\partial u}{\partial y} + z \frac{\partial u}{\partial z} = 0$

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Solution

$u = \frac{y}{z} + \frac{z}{x}$

$\frac{\partial u}{\partial x} = - \frac{z}{x^{2}}, \frac{\partial u}{\partial y} = \frac{1}{z}, \frac{\partial u}{\partial z} = - \frac{y}{z^{2}} + \frac{1}{x}$

$x \frac{\partial u}{\partial x} + y \frac{\partial u}{\partial y} + z \frac{\partial u}{\partial z} = - \frac{z}{x} + \frac{y}{z} - \frac{y}{z} + \frac{z}{x} = 0$

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