If $f(x,y,z) = \frac{x}{y} + \frac{y}{z} + \frac{z}{x}$ then find $xf_x + yf_y + zf_z$

Question

The only thing is, Im not allowed to calculate $f_x, f_y, and f_z$ individually. Which leaves only one way, breaking the function into Euler's homogeneous equation form $x^{n}\phi (\frac{y}{x}, \frac{z}{x})$ . I cannot find out how to formulate the Euler's form from the given function. Any help would be appreciated.

Answer 1

If $f(x,y,z)$ is homogeneous of degree $n$, that is $f(tx,ty,tz)=t^nf(x,y,z)$ for all $x$, $y$, $z$, $t$, then $$xf_x+yf_y+zf_z=nf.$$ This is due to Euler.

Answer 2

Just to verify Lord Shark the Unknown's claim, we get the respective terms in your required sum, as follows $$x{f_x}=\frac{x}{y}-\frac{z}{x}$$ $$y{f_y}=-\frac{x}{y}+\frac{y}{z}$$ $$z{f_z}=-\frac{y}{z}+\frac{z}{x}$$ so that $$x{f_x} + y{f_y} + z{f_z} = 0.$$ This means that $n = 0$ in the other answer. Indeed, $$f(tx,ty,tz)=\frac{tx}{ty}+\frac{ty}{tz}+\frac{tz}{tx}=f(x,y,z)={t^0}\cdot{f(x,y,z)}$$ assuming $t \neq 0$.