Problem 34 - Calculus On Manifolds: A Modern Approach To Classical Theorems Of Advanced Calculus

Question

2-34. A function \(f: \mathbf{R}^{n} \rightarrow \mathbf{R}\) is homogeneous of degree \(m\) if \(f(t x)=\) \(t^{m} f(x)\) for all \(x .\) If \(f\) is also differentiable, show that $$ \sum_{i=1}^{n} x^{i} D_{i} f(x)=m f(x) $$ Hint: If \(g(t)=f(t x)\), find \(g^{\prime}(1)\)

Step-by-Step Solution

Answer

\( \sum_{i=1}^{n} x^i D_i f(x) = m f(x) \)

Step 1: Define the Function

Given a function \( f: abla^{n} \rightarrow abla \), it is homogeneous of degree \( m \) if \( f(t x) = t^{m} f(x) \) for all \( x \).

Step 2: Introduce the Auxiliary Function

Define another function \( g(t) = f(t x) \). By the given property, \( g(t) = t^m f(x) \).

Step 3: Differentiate the Auxiliary Function

Differentiate \( g(t) \) with respect to \( t \). Since \( g(t) = t^m f(x) \), we have:\[ g'(t) = m t^{m-1} f(x) \]

Step 4: Evaluate the Derivative at \( t = 1 \)

Evaluate the derivative \( g'(t) \) at \( t = 1 \). This yields:\[ g'(1) = m f(x) \]

Step 5: Chain Rule Application

Using the chain rule, find \( g'(t) = \sum_{i=1}^{n} x^i D_i f(t x) \). When \( t = 1 \), this becomes:\[ g'(1) = \sum_{i=1}^{n} x^i D_i f(x) \]

Step 6: Combine Results

From Steps 4 and 5, we equate the derivatives:\[ \sum_{i=1}^{n} x^i D_i f(x) = m f(x) \]

Key Concepts

Differentiable Functions

A function is called differentiable if it has a derivative at every point in its domain. This means the function's graph has a tangent line at every point, making it smooth (no sharp corners). To be differentiable, a function must also be continuous; however, not all continuous functions are differentiable. For a function of several variables, differentiability implies the existence of partial derivatives at every point in the domain, along with a certain smoothness condition.
For instance, the function needs to be well-approximated by a linear function around each point. This is essential for applying various calculus tools, like the chain rule and partial derivatives, to solve complex problems.

Partial Derivatives

Partial derivatives are derivatives of functions with multiple variables, taken with respect to one variable while keeping the other variables constant. If you have a function like \( f(x, y) \), the partial derivative with respect to \( x \) is denoted as \( \frac{\partial f}{\partial x} \) and with respect to \( y \) as \( \frac{\partial f}{\partial y} \).
These derivatives describe how the function changes as each variable changes, independently of the others. Partial derivatives are used to understand the slope of the function in different directions, which is crucial for gradient-based optimization and analyzing multivariable functions.
For example, in the given problem, understanding how each component changes allows us to apply the sum of partial derivatives and obtain the desired relation.

Chain Rule

The chain rule is a fundamental concept in calculus used to differentiate composite functions. When dealing with functions of multiple variables, it extends to handle nested functions, allowing us to differentiate more complex expressions.
In the context of the given problem, the chain rule helps us differentiate the auxiliary function \( g(t) = f(t x) \) with respect to \( t \). This is crucial because \( f \) is a function of multiple variables, and we need to understand how each variable affects the composite function.
Specifically, the chain rule states that if you have a composite function \( u(t) = f(g(t)) \), the derivative of \( u \) with respect to \( t \) is found by multiplying the derivative of \( f \) with respect to its argument by the derivative of \( g \) with respect to \( t \). For our problem, considering the partial derivatives of \( f \), we apply:\[ g'(t) = \sum_{i=1}^{n} x^i \frac{\partial f}{\partial x^i} (t x) \]Finally, evaluating at \( t = 1 \) gives us the sum of partial derivatives needed to complete the problem.