# Positively Homogenous Functions of Several Variables

Definition: Let $z = f(x_1, x_2, ..., x_n)$ be an $n$ variable real-valued function. $f$ is said to be Positively Homogenous of Degree $k$ if for all $(x_1, x_2, ..., x_n) \in D(f)$ and for every real number $t > 0$ we have that $f(tx_1, tx_2, ..., tx_n) = t^k f(x_1, x_2, ..., x_n)$. |

Let's look at some examples of homogenous functions. Let $f(x, y) = \frac{3x^2y^2}{\sqrt{x^4 + y^4}}$, and let $t > 0$. Then we have that:

(1)Therefore $f$ is positively homogenous of degree $2$. The graph of $f$ is shown below.

Let's look at another example. Let $g(x, y, z) = xyz + x^3 + y^3 + z^3$ and let $t > 0$. Then we have that:

(2)Therefore $g$ is positively homogenous of degree $3$. Of course, a positively homogenous function need not be of positive degree, i.e., the degree of a positively homogenous function can be nonpositive. For example, let $h(x, y) = \frac{x + y}{x^3y^3}$, and let $t > 0$. Then:

(3)Therefore $h$ is positively homogenous of degree $-5$. The graph of $h$ is shown below.

One important property of positively homogenous functions of degree $k$ of two variables is that they have characteristics of their degree. For the function $f$ above, observe parabola shapes in the graph of $f$. Similarly, for the function $h$ above, observe the hyperbola shapes present. If the degree of a positively homogenous function is known, then this can considerably help us in graphing these two variable functions.

## Euler's Positive Homogenous Function Theorem

Theorem 1 (Euler's Positive Homogenous Function Theorem): Let $z = f(x_1, x_2, ..., x_n)$ be an $n$ variable real-valued function that is positively homogenous of degree $k$, and suppose that $f$ has continuous first partial derivatives. Then $\sum_{i=1}^n x_i \frac{\partial z}{\partial x_i} = k f(x_1, x_2, ..., x_n)$. |

**Proof:**Since $z = f(x_1, x_2, ..., x_n)$ is a positively homogenous function of degree $k$ then we have that:

- If we differentiate both sides with respect to $t$ using The Chain Rule Type 2 for Functions of Several Variables we get that:

- Since $f$ is positively homogenous for all $t > 0$, then by letting $t = 1$ we get that: