Problem 55 - Calculus Early Transcendentals

Question

A function \(f\) is called homogeneous of degree \(n\) if it satisfies the equation \(f(t X, t y)=t^{n} f(x, y)\) for all \(t\) , where \(n\) is a positive integer and \(f\) has continuous second-order partial derivatives. (a) Verify that \(f(x, y)=x^{2} y+2 x y^{2}+5 y^{3}\) is homogeneous of degree \(3 .\) (b) Show that if \(f\) is homogeneous of degree \(n,\) then $$x \frac{\partial f}{\partial x}+y \frac{\partial f}{\partial y}=n f(x, y)$$ I Hint: Use the Chain Rule to differentiate \(f(t x, t y)\) with respect to \(t ]\)

Step-by-Step Solution

Answer

(a) Yes, it is homogeneous of degree 3. (b) The given identity holds.

Step 1: Verify Homogeneity

To verify if the function \(f(x, y) = x^2y + 2xy^2 + 5y^3\) is homogeneous of degree 3, consider the expression \(f(tx, ty)\). Substitute \(tx\) for \(x\) and \(ty\) for \(y\) in the function:\[f(tx, ty) = (tx)^2(ty) + 2(tx)(ty)^2 + 5(ty)^3\] This simplifies to:\[=t^3(x^2y + 2xy^2 + 5y^3) = t^3 f(x, y)\]Since we end up with \(t^3 f(x, y)\), \(f(x, y)\) is indeed homogeneous of degree 3.

Step 2: Setup the Chain Rule

We need to show that if \(f\) is homogeneous of degree \(n\), then \(x \frac{\partial f}{\partial x} + y \frac{\partial f}{\partial y} = nf(x, y)\). Assume \(f(tx, ty) = t^n f(x, y)\) and differentiate both sides with respect to \(t\) using the Chain Rule, noting that \(x\) and \(y\) are constants with respect to \(t\).

Key Concepts

Partial Derivatives

Partial derivatives are a fundamental concept in multivariable calculus, allowing us to understand how a function changes with respect to each of its variables independently.
Unlike regular derivatives that look at the rate of change of a function with one variable, partial derivatives focus on change while keeping all other variables constant. For instance, for a function \( f(x, y) \), the partial derivative \( \frac{\partial f}{\partial x} \) is found by treating \( y \) as a constant and differentiating with respect to \( x \), and similarly, \( \frac{\partial f}{\partial y} \) is found by treating \( x \) as a constant.

To apply this to our function \( f(x, y) = x^2y + 2xy^2 + 5y^3 \):

• First, we calculate \( \frac{\partial f}{\partial x} = 2xy + 2y^2 \) to find the rate of change along the \( x \)-axis while keeping \( y \) constant.
• Next, calculate \( \frac{\partial f}{\partial y} = x^2 + 4xy + 15y^2 \) to determine the change along the \( y \)-axis while \( x \) is constant.

These derivatives are essential for the next steps in verifying the homogeneity of functions.

Chain Rule

The Chain Rule is an essential technique in calculus for handling the differentiation of composite functions. It provides a way to break down the differentiation process when the function is a composition of two or more functions.

When you have a function \( f \) that depends indirectly on a variable \( t \) through other variables \( x \) and \( y \), the Chain Rule helps us find the derivative concerning \( t \). In our problem, to prove homogeneity using the chain rule, we consider the function \( f(tx, ty) = t^n f(x, y) \) and differentiate with respect to \( t \).

Here's how the application goes:

• Compute \( \frac{\partial}{\partial t}[f(tx, ty)] = \frac{\partial f}{\partial (tx)} \cdot \frac{d(tx)}{dt} + \frac{\partial f}{\partial (ty)} \cdot \frac{d(ty)}{dt} \).
• Evaluate these expressions to ultimately show that \( x \frac{\partial f}{\partial x} + y \frac{\partial f}{\partial y} = n f(x, y) \).

Each step uses the Chain Rule to handle respective partial derivatives, linking back to the degree of homogeneity.

Degree of Homogeneity

The degree of homogeneity of a function is a measure of how scaling the inputs affects the scaling of outputs.
A function is homogeneous of degree \( n \) if, when all the function arguments are scaled by a factor \( t \), the entire function is scaled by \( t^n \). In mathematical terms, this means \( f(tx, ty) = t^n f(x, y) \).

For our function \( f(x, y) = x^2y + 2xy^2 + 5y^3 \), substituting \( tx \) and \( ty \) leads to the equation \( t^3(x^2y + 2xy^2 + 5y^3) = t^3f(x, y) \). Hence, it confirms that the function is homogeneous of degree 3.

Understanding the degree of homogeneity helps us determine how changes in inputs affect the function's output, and it is crucial for analyzing system behaviors and invariances. It's like knowing the exact factor by which an output grows when we stretch inputs, making it a powerful tool in many applied math scenarios.

Differentiation

Differentiation is a fundamental concept that helps us understand and compute the rate of change of functions.
In single-variable calculus, it involves finding derivatives with respect to a single variable. However, for multivariable functions like \( f(x, y) \), differentiation involves finding partial derivatives.

Here’s a quick recap:

• The differentiation process helps verify the homogeneity by working with the equation \( x \frac{\partial f}{\partial x} + y \frac{\partial f}{\partial y} = n f(x, y) \).
• We take the partial derivatives of \( f \) concerning \( x \) and \( y \) and then analyze these to see how they relate back to the original function through the homogeneity condition.
• The differentiation verifies if the sum of scaled partial derivatives equals the scaled value of the function itself, confirming properties like degree of homogeneity.

Differentiation, coupled with other rules, is indispensable for analyzing and transforming functions under various mathematical operations.