Problem 249 - Calculus 3

Question

For the following exercises, 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 \(x \frac{\partial f}{\partial x}+y \frac{\partial f}{\partial y}=n f(x, y)\) $$ f(x, y)=\sqrt{x^{2}+y^{2}} $$

Step-by-Step Solution

Answer

The function is homogeneous of degree 1 and satisfies the condition for the given equation.

Step 1: Check Homogeneity

To verify that the function \( f(x, y) = \sqrt{x^2 + y^2} \) is homogeneous of some degree \( n \), check if it satisfies the definition. Substitute \( t x \) and \( t y \) into the function: \[ f(t x, t y) = \sqrt{(t x)^2 + (t y)^2} = \sqrt{t^2 x^2 + t^2 y^2} = \sqrt{t^2(x^2 + y^2)} = t \sqrt{x^2 + y^2} = t f(x, y) \]. This shows the function is homogeneous of degree \( n = 1 \).

Step 2: Calculate Partial Derivatives

Find the partial derivatives of the function. First, calculate \( \frac{\partial f}{\partial x} \):\[ \frac{\partial f}{\partial x} = \frac{x}{\sqrt{x^2 + y^2}} \].Then calculate \( \frac{\partial f}{\partial y} \):\[ \frac{\partial f}{\partial y} = \frac{y}{\sqrt{x^2 + y^2}} \].

Step 3: Verify the Homogeneous Function Equation

Use the calculated partial derivatives to check if the equation \( x \frac{\partial f}{\partial x} + y \frac{\partial f}{\partial y} = n f(x, y) \) holds. Substitute \( n = 1 \), and substitute the partial derivatives:\[ x \frac{\partial f}{\partial x} + y \frac{\partial f}{\partial y} = x \left( \frac{x}{\sqrt{x^2 + y^2}} \right) + y \left( \frac{y}{\sqrt{x^2 + y^2}} \right) \]\[ = \frac{x^2}{\sqrt{x^2 + y^2}} + \frac{y^2}{\sqrt{x^2 + y^2}} = \frac{x^2 + y^2}{\sqrt{x^2 + y^2}} = \sqrt{x^2 + y^2} = f(x, y) \]. This confirms the equation holds.

Key Concepts

Partial Derivatives

In multivariable calculus, a function can depend on more than one variable. To understand how the function changes with respect to one of these variables, while keeping the others constant, we use partial derivatives.
For a function like \( f(x, y) = \sqrt{x^2 + y^2} \), the partial derivatives tell us how \( f \) changes as \( x \) changes, keeping \( y \) fixed, and vice versa.

• The partial derivative with respect to \( x \) is \( \frac{\partial f}{\partial x} = \frac{x}{\sqrt{x^2 + y^2}} \).
• The partial derivative with respect to \( y \) is \( \frac{\partial f}{\partial y} = \frac{y}{\sqrt{x^2 + y^2}} \).

These derivatives are crucial for examining how the function behaves locally and are instrumental in verifying the properties of homogeneous functions.

Degree of Homogeneity

The degree of homogeneity of a function indicates how the function scales when its variables are multiplied by a common factor. A function \( f(x, y) \) is said to be homogeneous of degree \( n \) if plugging in \( tx \) and \( ty \) results in:

• \( f(tx, ty) = t^n f(x, y) \)

For the function \( f(x, y) = \sqrt{x^2 + y^2} \), substituting \( tx \) and \( ty \) gives \( t \sqrt{x^2 + y^2} \) which shows that our function is homogeneous of degree \( n = 1 \).
This property helps in establishing the identity \( x \frac{\partial f}{\partial x} + y \frac{\partial f}{\partial y} = n f(x, y) \), which verifies the degree of homogeneity with respect to partial derivatives.

Multivariable Calculus

Multivariable calculus is an extension of single-variable calculus to functions of several variables. It is essential for modeling and solving problems where multiple factors impact the outcome. Here, functions do not just have one influence but often several independent shifts.
A core concept is understanding how each variable influences the function, analyzing their interactions through tools like gradients and partial derivatives.

• Gradients combine all partial derivatives into a vector, pointing in the direction of the greatest rate of increase.
• Homogeneous functions and their degrees provide insights into how scaling inputs affects the output.
• We analyze functions in terms of level curves, where each curve represents points with the same function value.

Functions such as \( f(x, y) = \sqrt{x^2 + y^2} \), widely used in physics and engineering, showcase how multivariable calculus enables complex scenario analysis with interactions of variables.