Question

We say that \(f=f\left(x_{1}, x_{2}, \ldots, x_{n}\right)\) is homogeneous of degree \(r\) if \(D_{f}\) is open and there is a constant \(r\) such that $$ f\left(t x_{1}, t x_{2}, \ldots, t x_{n}\right)=t^{r} f\left(x_{1}, x_{2}, \ldots, x_{n}\right) $$ whenever \(t>0\) and \(\left(x_{1}, x_{2}, \ldots, x_{n}\right)\) and \(\left(t x_{1}, t x_{2}, \ldots, t x_{n}\right)\) are in \(D_{f} .\) Prove: If \(f\) is differentiable and homogeneous of degree \(r,\) then $$ \sum_{i=1}^{n} x_{i} f_{x_{i}}\left(x_{1}, x_{2}, \ldots, x_{n}\right)=r f\left(x_{1}, x_{2}, \ldots, x_{n}\right) $$ (This is Euler's theorem for homogeneous functions.)

Short answer

Using the definition of a homogeneous function and the chain rule for differentiable functions, show that if the function \(f=f(x_1, x_2, ..., x_n)\) is homogeneous of degree \(r\), then it satisfies Euler's theorem for homogeneous functions.

Step-by-step solution

Differentiate the Homogeneous Function Equation

First, we need to differentiate both sides of the equation defining a homogeneous function of degree \(r\): $$ f(tx_1, tx_2, \ldots, tx_n)=t^r f(x_1, x_2, \ldots, x_n) $$ To do this, we use the chain rule. For the left side of the equation, we differentiate \(f(tx_1, tx_2, \ldots, tx_n)\) with respect to \(t\). Using the chain rule, this differentiation involves: $$ \frac{d}{dt} f(tx_1, tx_2, \ldots, tx_n) = \sum_{i=1}^n \frac{\partial f}{\partial (tx_i)} \frac{d(tx_i)}{dt} $$ For the right side of the equation, we differentiate \(t^r f(x_1, x_2, \ldots, x_n)\) with respect to \(t\): $$ \frac{d}{dt}[t^r f(x_1, x_2, \ldots, x_n)] = r t^{r-1}f(x_1, x_2, \ldots, x_n) $$

Evaluate the Differentiated Equations

Next, we will evaluate the differentiated equations from Step 1: $$ \sum_{i=1}^n \frac{\partial f}{\partial (tx_i)} \frac{d(tx_i)}{dt} = r t^{r-1} f(x_1, x_2, \ldots, x_n) $$ We notice that \(\frac{d(tx_i)}{dt} = x_i\). Thus, we can simplify the left side of the equation: $$ \sum_{i=1}^n \frac{\partial f}{\partial (tx_i)} x_i = r t^{r-1} f(x_1, x_2, \ldots, x_n) $$ Now, using the chain rule again, and notice that \(\frac{\partial f}{\partial (tx_i)} = f_{x_i}(tx_1, tx_2, \ldots, tx_n)\). So, substituting this into our equation, we have: $$ \sum_{i=1}^n f_{x_i}(tx_1, tx_2, \ldots, tx_n) x_i = r t^{r-1} f(x_1, x_2, \ldots, x_n) $$

Replace the Homogeneous Function

Finally, we will use the definition of the homogeneous function again to replace the function \(f(tx_1, tx_2, \ldots, tx_n)\): $$ f(tx_1, tx_2, \ldots, tx_n) = t^r f(x_1, x_2, \ldots, x_n) $$ Applying this substitution to our equation from Step 2, we get: $$ \sum_{i=1}^n x_i f_{x_i}(x_1, x_2, \ldots, x_n) = r f(x_1, x_2, \ldots, x_n) $$ This is Euler's theorem for homogeneous functions, and it's proven for differentiable functions \(f\) of degree \(r\).

Key concepts

Homogeneous Functions

A function is called homogeneous if it exhibits a particular symmetry: its output scales in a specific way when the inputs are scaled. Given a function \( f(x_1, x_2, \ldots, x_n) \), it is considered homogeneous of degree \( r \) if scaling all input variables by a positive constant \( t \) results in the function being scaled by \( t^r \). Specifically, this means:

• If you multiply each \( x_i \) by \( t \), the output of the function becomes \( t^r \) times the original function.
• The expression summarizing this property is \( f(tx_1, tx_2, \ldots, tx_n) = t^r f(x_1, x_2, \ldots, x_n) \).

This property is particularly useful in various mathematical and physical contexts where scalings are considered, such as in economics to model production functions or physics for symmetry laws.

Degree of a Function

The degree of a homogeneous function refers to the exponent \( r \) in the scaling law: a crucial characteristic that tells how the function changes with scaling of inputs. You can determine this by observing how the function behaves as you multiply every variable by a scalar \( t \).

• A function is of degree zero, the outputs remain constant regardless of input scaling.
• If the degree of a function is positive, the function increases with increasing scale, indicating a proportional relationship.
• A negative degree implies that the function decreases as inputs scale up, pointing to an inverse relationship.

Understanding the degree is vital for applying Euler's theorem, as it helps in further deducing relationships between differentials.

Partial Differentiation

Partial differentiation is a method to take the derivative of multivariable functions with respect to one variable while keeping others constant. This approach breaks down changes in the function relative to changes in each input.

• If \( f(x_1, x_2, \ldots, x_n) \) is a multivariable function, \( \frac{\partial f}{\partial x_i} \) is its partial derivative with respect to \( x_i \).
• In practical terms, it assesses how \( f \) changes as \( x_i \) changes, holding \( x_1, x_2, \ldots, x_{i-1}, x_{i+1}, \ldots, x_n \) constant.
• This is key when working with homogeneous functions to explore their derivatives and apply Euler's theorem.

Partial derivatives provide foundational tools to understand the behavior of the function across different dimensions.

Chain Rule

The chain rule is a crucial concept in calculus, allowing you to differentiate a composite function. When you have a function composed of multiple nested functions, the chain rule helps you find the derivative.

• It's often applied when dealing with homogeneous functions, because scaling variables necessitates composite functions.
• In simpler terms, it tells us that the derivative of a composite function \( g(f(x)) \) is the derivative of \( g \) evaluated at \( f(x) \), multiplied by the derivative of \( f \).
• For example, if we have \( f(tx_1, tx_2, \ldots, tx_n) \), it requires differentiating with respect to \( t \), involving partial derivatives of \( f \) with respect to each scaled term \( tx_i \).

Using the chain rule in combination with partial differentiation allows us to tackle complex functions, especially in proving theorems like Euler's for homogeneous functions.

Differentiability

Differentiability refers to the property of a function that makes it possible to take derivatives. For a function of several variables to be differentiable at a point, it must be locally linearizable at that point.

• In the context of homogeneous functions and Euler's theorem, differentiability is crucial. It guarantees we can perform derivative calculations reliably.
• A differentiable function is smooth, meaning no corners or discontinuities at the point in question.
• This property is essential for proving statements, such as establishing relationships between function value scaling and the sum of weighted partial derivatives.

Understanding differentiability gives assurance that the function behaves predictably in a small neighborhood around any point, a prerequisite for using advanced calculus concepts.