Problem 38 - Calculus with Concepts in Calculus

Question

A function \(f\) of two variables is homogeneous of degree \(\boldsymbol{n}\) if for any real number \(t\) we have $$ f(t x, t y)=t^{n} f(x, y) $$ Show that in this case $$ x f_{x}(x, y)+y f_{y}(x, y)=n f(x, y) $$ (Hint: Differentiate both sides of \((8)\) with respect to \(t\), and then set \(t=1 .\) )

Step-by-Step Solution

Answer

The identity holds: \( x f_x(x, y) + y f_y(x, y) = n f(x, y) \).

Step 1: Start with the Given Homogeneity Condition

The function \(f\) is given as homogeneous of degree \(n\). This means:\[ f(tx, ty) = t^n f(x, y) \]This equation should hold true for any real number \(t\).

Step 2: Differentiate Both Sides with Respect to \(t\)

Now, differentiate both sides of the equation \( f(tx, ty) = t^n f(x, y) \) with respect to \(t\):- The left side differentiation results in: \[ \frac{d}{dt}[f(tx, ty)] = f_x(tx, ty) \cdot x + f_y(tx, ty) \cdot y \] using the chain rule.- The right side differentiation yields: \[ \frac{d}{dt}[t^n f(x, y)] = n t^{n-1} f(x, y) \]Equating both differentiated sides:\[ f_x(tx, ty) \cdot x + f_y(tx, ty) \cdot y = n t^{n-1} f(x, y) \]

Step 3: Set \(t = 1\) in the Equation

Now, set \( t = 1 \) in the equation:\[ f_x(1 \cdot x, 1 \cdot y) \cdot x + f_y(1 \cdot x, 1 \cdot y) \cdot y = n \cdot 1^{n-1} f(x, y) \]This simplifies to:\[ x f_x(x, y) + y f_y(x, y) = n f(x, y) \]Therefore, this shows the desired identity holds when the function \( f \) is homogeneous of degree \( n \).

Key Concepts

Partial Derivatives

Understanding partial derivatives is crucial when dealing with functions of multiple variables. Partial derivatives represent the rate at which a function changes with respect to one of its variables while keeping the others constant. For a function \( f(x, y) \), the partial derivative with respect to \( x \), denoted as \( f_x(x, y) \), measures how \( f \) changes as \( x \) changes, assuming that \( y \) remains unchanged.
Similarly, the partial derivative with respect to \( y \), denoted as \( f_y(x, y) \), considers the change in \( f \) as \( y \) varies, with \( x \) held constant.

These derivatives are essential tools in understanding the behavior of multi-variable functions, and become particularly useful when applying the chain rule in differentiation.

Chain Rule

The chain rule is a fundamental technique in calculus, used to differentiate composite functions. When we have a function, such as \( f(tx, ty) \), where the inputs themselves depend on another variable \( t \), the chain rule helps us find the necessary derivatives.
By applying the chain rule, we determine the partial derivatives \( f_x(tx, ty) \) and \( f_y(tx, ty) \). Each derivative is then multiplied by the rate of change of the inner functions, which in the case of \( f(tx, ty) \) would be \( x \) and \( y \) respectively.

Thus, differentiating \( f(tx, ty) \) with respect to \( t \) gives us:

• \( f_x(tx, ty) \cdot x \)
• \( f_y(tx, ty) \cdot y \)

This result showcases the power of the chain rule in analyzing changes across multiple dependencies.

Differentiation

Differentiation is the core process of calculating derivatives, which are essentially measures of how a function changes as its variables change. In the context of homogeneous functions, differentiation helps us explore how the function's output alters with respect to changes in each input variable.
Through differentiation, especially by applying the chain rule, we can understand complex relationships within the function.

By differentiating both sides of the homogeneity condition with respect to \( t \), we establish critical relationships:

• Left side: \( f_x(tx, ty) \cdot x + f_y(tx, ty) \cdot y \)
• Right side: \( n t^{n-1} f(x, y) \)

Once differentiated, setting \( t = 1 \) simplifies the expression, bridging the derivative findings with the original function. Differentiation thus provides insights into the function's behavior under scaling transformations.

Degree of Homogeneity

The degree of homogeneity of a function indicates how the function scales when its input variables are multiplied by a scalar. A function \( f(x, y) \) is homogeneous of degree \( n \) if multiplying both variables by a scalar \( t \) results in the function being scaled by \( t^n \). Mathematically, this is expressed as \( f(tx, ty) = t^n f(x, y) \).
Understanding the degree of homogeneity is pivotal for solving problems involving scaling, such as in the given exercise.

When the function is homogeneous of degree \( n \), we can derive the relation:

• \( x f_x(x, y) + y f_y(x, y) = n f(x, y) \)

This equation verifies that the sum of the contributions from each variable, weighted by their respective partial derivatives, equates to the function scaled by its degree of homogeneity. This identity plays a key role in fields such as economics, physics, and engineering, where understanding scaling properties is essential.