Expert Answer to "If u=f(y-z, z-x, x-y), then ux+uy+uz="

pancha3

Answered question

2021-09-16

If u=f(y-z, z-x, x-y), then

u_{x} + u_{y} + u_{z} =

gotovub

Skilled2021-09-17Added 98 answers

Step 1
Let p=y−z, q=z−x and r=x−y
Now determine the partial derivatives.

\frac{\partial p}{\partial x} = 0, \frac{\partial p}{\partial y} = 1

Similarly

\frac{\partial q}{\partial y} = 0, \frac{\partial q}{\partial z} = 1, \frac{\partial r}{\partial z} = 0 and \frac{\partial r}{\partial x} = 1

Now

\frac{\partial u}{\partial x} = \frac{\partial u}{\partial p} \cdot \frac{\partial p}{\partial x} + \frac{\partial u}{\partial r} \cdot \frac{\partial r}{\partial x}

=1
Step 2
Similarly

\frac{\partial u}{\partial y} = \frac{\partial u}{\partial p} \cdot \frac{\partial p}{\partial y} + \frac{\partial u}{\partial q} \cdot \frac{\partial q}{\partial y}

\frac{\partial u}{\partial z} = \frac{\partial u}{\partial q} \cdot \frac{\partial q}{\partial z} + \frac{\partial u}{\partial r} \cdot \frac{\partial r}{\partial z}

=1+0
=1
Therefore

\frac{\partial u}{\partial x} + \frac{\partial u}{\partial y} + \frac{\partial u}{\partial z} = 1 + 1 + 1

=3
Therefore

u_{x} + u_{y} + u_{z} = 3

Jeffrey Jordon

Expert2022-02-01Added 2605 answers

Answer is given below (on video)

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