Problem 250 - Calculus Volume 3

Question

Use this information: A function \(f(x, y)\) is said to be homogeneous of degree \(n\) if \(f(t x, t y)=t^{n} f(x, y) .\) For all homogeneous functions of degree \(n, \quad\) the following equation is true: \(x \frac{\partial f}{\partial x}+y \frac{\partial f}{\partial y}=n f(x, y) .\) Show that the given function is homogeneous and verify that \(x \frac{\partial f}{\partial x}+y \frac{\partial f}{\partial y}=n f(x, y)\). \(f(x, y)=x^{2} y-2 y^{3}\)

Step-by-Step Solution

Answer

The function is homogeneous of degree 3 and satisfies the homogeneous function equation.

Step 1: Determine Homogeneity

To check if the function \( f(x, y) = x^2y - 2y^3 \) is homogeneous, substitute \( t \/ x \) and \( t \/ y \) into the function to get \( f(tx, ty) = (tx)^2(ty) - 2(ty)^3 = t^2x^2ty - 2t^3y^3 = t^3(x^2y - 2y^3) = t^3f(x, y) \). This shows that \( f(x, y) \) is homogeneous of degree 3.

Step 2: Calculate Partial Derivatives

Calculate the partial derivatives of \( f \): - \( \frac{\partial f}{\partial x} = \frac{\partial}{\partial x}(x^2y - 2y^3) = 2xy \).- \( \frac{\partial f}{\partial y} = \frac{\partial}{\partial y}(x^2y - 2y^3) = x^2 - 6y^2 \).

Step 3: Verify Homogeneous Function Equation

Substitute the partial derivatives and \( f(x, y) \) into the given equation \( x \frac{\partial f}{\partial x} + y \frac{\partial f}{\partial y} = n f(x, y) \):- \( x \cdot 2xy + y \cdot (x^2 - 6y^2) = 2x^2y + yx^2 - 6y^3 = 3x^2y - 6y^3 \).- Comparing with \( 3f(x, y) = 3(x^2y - 2y^3) = 3x^2y - 6y^3 \), we see that both sides are equal, confirming the equation.

Key Concepts

Partial Derivatives

Partial derivatives are essential tools in multivariable calculus. They measure the rate of change of a function with respect to one variable while keeping other variables constant. Let's consider the function given in the exercise, \( f(x, y) = x^2 y - 2 y^3 \). It has two variables: \( x \) and \( y \).

• To find the partial derivative with respect to \( x \), denoted \( \frac{\partial f}{\partial x} \), treat \( y \) as a constant and differentiate \( f \) with respect to \( x \). This yields: \( 2xy \).
• Similarly, the partial derivative with respect to \( y \), denoted \( \frac{\partial f}{\partial y} \), involves treating \( x \) as constant when differentiating. This results in: \( x^2 - 6y^2 \).

Partial derivatives are vital not only in verifying the degree of homogeneity but also in analyzing the behavior of multivariable functions.

Degree of Homogeneity

The degree of homogeneity of a function indicates how the function behaves when all its inputs are multiplied by the same factor. Specifically, if \( f(tx, ty) = t^n f(x, y) \), the function is homogeneous of degree \( n \).
For the function \( f(x, y) = x^2 y - 2 y^3 \), substituting \( tx \) and \( ty \) gives us \( t^3 f(x, y) \). This indicates it is homogeneous of degree 3 because multiplying \( x \) and \( y \) by \( t \) results in the entire function being multiplied by \( t^3 \).
Determining the degree of homogeneity helps in various applications, such as in physics and economics, where scaling properties are essential.

Multivariable Calculus

Multivariable calculus extends the concepts of calculus to functions with more than one variable. It plays a crucial role in various fields of science and engineering where systems are influenced by multiple factors.
In multivariable calculus, a function's behavior can be examined through concepts such as partial derivatives and homogeneity. These help us understand how changes in one variable affect the overall function. For example, in the exercise, we looked at \( f(x, y) = x^2 y - 2 y^3 \) and explored its homogeneous properties and derivatives.

• Understanding multivariable functions involves visualizing them in higher-dimensional spaces, which can be complex but fascinating.
• Techniques such as finding gradients, solving systems of equations, and utilizing partial derivatives are vital to navigating these functions effectively.

Learning multivariable calculus equips students to tackle real-world problems where multiple variables interact dynamically.