- #1

KungPeng Zhou

- 22

- 7

- Homework Statement
- Supposed f is a function of several variables that satisfies the equation f(tx, ty, tz) =t^{n}f(x, y, z),(t as any number).

Prove:

##xf_{x}+yf_{y}+zf_{z}=nf(x, y, z) ##

- Relevant Equations
- Partial derivative related formulas

For the first equation:

##f(tx, ty, tz)=f(u, v, w) ##, ##u=tx, v=ty, w=tz##,##k=f(u, v, w) ####

t^{n}f_{x}=\frac{\partial f}{\partial u} \cdot \frac{\partial u}{\partial x}##

As the same calculation

##xf_{x}+yf_{y}+zf_{z}=[\frac{\partial f}{\partial x} + \frac{\partial f}{\partial y} +\frac{\partial f}{\partial z}] t^{1-n}##

But I can't continue with it.

##f(tx, ty, tz)=f(u, v, w) ##, ##u=tx, v=ty, w=tz##,##k=f(u, v, w) ####

t^{n}f_{x}=\frac{\partial f}{\partial u} \cdot \frac{\partial u}{\partial x}##

As the same calculation

##xf_{x}+yf_{y}+zf_{z}=[\frac{\partial f}{\partial x} + \frac{\partial f}{\partial y} +\frac{\partial f}{\partial z}] t^{1-n}##

But I can't continue with it.