10.2 Homogeneous functions

A function $f:{ℝ}^n\mapsto ℝ,\vec{x}\mapsto f\left(\vec{x}\right)$ is called homogeneous of degree $h\in ℝ$, if for all $\vec{x}\in {ℝ}^n$ applies:

$$f\left(k\vec{x}\right)=k^{h}f\left(\vec{x}\right)\text{for all}k\in {ℝ}_0^{+}$$

If the input variables $\vec{x}{\sum <!--\; FUNCTION\; APPLICATION\; -->}_i^b$ are multiplied by a positive number $k\ge 0$, the function value is multiplied by the factor $k^h$.

Homogeneous functions have special properties, which are briefly listed below and illustrated on the following pages for functions of two variables. For homogeneous functions, the following two theorems apply in particular:

$$f\left(\vec{x}\right)\text{is homogeneous of the degree}h\iff x_1\frac{\partial }{\partial x_1}f\left(\vec{x}\right)+...+x_n\frac{\partial }{\partial x_n}f\left(\vec{x}\right)=\mathit{hf}\left(\vec{x}\right).$$

$$f\left(\vec{x}\right)\text{is homogeneous of the degree}h\Rightarrow \frac{\partial }{\partial x_i}f\left(\vec{x}\right)\text{is homogeneous of the degree}h-1\text{ for all }i.$$