Let f be defined on an open set Φof R^p into R^q and satisfy the relation f(t x)=t^k f(x) for

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Let f be defined on an open set Φof R^p into R^q and satisfy...

Let $f$ be defined on an open set $\Phi$ of $R^p$ into $R^q$ and satisfy the relation $$ f(t x)=t^k f(x) \quad \text { for } t \in \mathbf{R}, x \in \mathrm{D} \text {. } $$ In this case we say that $f$ is homogeneous of degree $k$. If this function $f$ is differentiable at $x$, show that $$ D f(x)(x)=k f(x) . $$ Conclude that Euler's $\dagger$ Relation (20.34) holds even when $f$ is positively homogeneous in the sense that (20.33) holds only for $t \geq 0$. if $q=1$ and $x=\left(\xi_1, \ldots, \xi_p\right)$, then Euler's Relation becomes $$ k f(x)=\xi_1 \frac{\partial f}{\partial \xi_1}(x)+\cdots+\xi_p \frac{\partial f}{\partial \xi_p}(x) . $$

Let $f$ be defined on an open set $\Phi$ of $R^p$ into $R^q$ and satisfy the relation
$$
f(t x)=t^k f(x) \quad \text { for } t \in \mathbf{R}, x \in \mathrm{D} \text {. }
$$

In this case we say that $f$ is homogeneous of degree $k$. If this function $f$ is differentiable at $x$, show that
$$
D f(x)(x)=k f(x) .
$$
Conclude that Euler's $\dagger$ Relation (20.34) holds even when $f$ is positively homogeneous in the sense that (20.33) holds only for $t \geq 0$. if $q=1$ and $x=\left(\xi_1, \ldots, \xi_p\right)$, then Euler's Relation becomes
$$
k f(x)=\xi_1 \frac{\partial f}{\partial \xi_1}(x)+\cdots+\xi_p \frac{\partial f}{\partial \xi_p}(x) .
$$

Key Concepts

Euler's Theorem on Homogeneous Functions

Euler's Theorem states that if a function is homogeneous of degree k and is differentiable, then the derivative of the function at any point, when applied to that point, equals k times the function value at that point. In scalar form for a function f: R^p -> R, this is expressed as k f(x) = x_1 (?f/?x_1)(x) + ... + x_p (?f/?x_p)(x). This theorem connects the scaling behavior of homogeneous functions with their differential properties.

Chain Rule

The chain rule is a fundamental concept in calculus that permits the differentiation of composite functions. When dealing with a function defined on R^p that is evaluated along a path, such as the scaled input tx, the chain rule helps differentiate f(tx) with respect to the scaling parameter t, thereby linking the derivative of f to its homogeneity property.

Homogeneous Functions

A function f is called homogeneous of degree k if it satisfies the property f(tx) = t^k f(x) for all scalars t (or for t in a specified subset, such as t ? 0) and for all x in its domain. This property indicates that scaling the input by a factor of t results in scaling the output by a factor of t^k, reflecting a consistent scaling behavior inherent to the function.

Differentiability

Differentiability at a point means that a function can be well approximated by a linear map near that point. In the context of homogeneous functions, differentiability allows us to use the derivative (or Jacobian in higher dimensions) to approximate changes in the function when the input is scaled or perturbed, which is crucial for establishing identities like Euler's relation.

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