If u = f (x - y, y - z, z - x), prove that partial u / partial x + partial u / partial y +...


If {eq}\displaystyle u = f (x - y,\ y - z,\ z - x) {/eq}, prove that {eq}\displaystyle \frac{\partial u}{\partial x} + \frac{\partial u}{\partial y} + \frac{\partial u}{\partial z} = 0 {/eq}.

Partial Derivative by using Chain Rule :

If {eq}z=g(x,y) {/eq} and {eq}x=f(r,\theta),y=h(r,\theta) {/eq}

Then we can say by using chain rule:

{eq}\displaystyle \frac{{\partial g}}{{\partial \theta }} = \displaystyle \frac{{\partial g}}{{\partial x}}\displaystyle \frac{{\partial x}}{{\partial \theta }} +\displaystyle \frac{{\partial g}}{{\partial y}}\displaystyle \frac{{\partial y}}{{\partial \theta }} \\ \displaystyle \frac{{\partial g}}{{\partial r}} = \displaystyle \frac{{\partial g}}{{\partial x}}\displaystyle \frac{{\partial x}}{{\partial r}} + \displaystyle \frac{{\partial g}}{{\partial y}}\displaystyle \frac{{\partial y}}{{\partial r}} \\ {/eq}

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Here in this case the given function is: {eq}\displaystyle u = f (x - y,\ y - z,\ z - x) {/eq}

Now let us choose:

{eq}\eqalign{ & l = x - y ...

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Learn more about this topic:

The Chain Rule for Partial Derivatives


Chapter 14 / Lesson 4

This lesson defines the chain rule. It goes on to explore the chain rule with partial derivatives and integrals of partial derivatives.

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