Partial Differentiation

**Aryth** · October 15th 2009, 04:33 PM

A function $f(x,y,z)$ is called homogeneous of degree n if $f(tx,ty,tz) = t^nf(x,y,z)$ . For example, $z^2\ln{\left(\frac{x}{y}\right)}$ is homogeneous of degree 2 since

$(tz)^2\ln{\left(\frac{tx}{ty}\right)} = t^2\left(x^2\ln{\frac{x}{y}}\right)$

Euler's theorem on homogeneous functions says that if $f$ is homogeneous of degree n, then

$x\frac{\partial{f}}{\partial{x}} + y\frac{\partial{f}}{\partial{y}} + z\frac{\partial{f}}{\partial{z}} = nf$

Prove this theorem.

Can someone just get me started?

**Opalg** · October 15th 2009, 11:40 PM

Originally Posted by Aryth

A function $f(x,y,z)$ is called homogeneous of degree n if $f(tx,ty,tz) = t^nf(x,y,z)$ . For example, $z^2\ln{\left(\frac{x}{y}\right)}$ is homogeneous of degree 2 since

$(tz)^2\ln{\left(\frac{tx}{ty}\right)} = t^2\left(x^2\ln{\frac{x}{y}}\right)$

Euler's theorem on homogeneous functions says that if $f$ is homogeneous of degree n, then

$x\frac{\partial{f}}{\partial{x}} + y\frac{\partial{f}}{\partial{y}} + z\frac{\partial{f}}{\partial{z}} = nf$

Prove this theorem.

Can someone just get me started?

Differentiate both sides of the equation $f(tx,ty,tz) = t^nf(x,y,z)$ with respect to t, using the chain rule. Then put t=1.

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