Problem 249 - 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 \(x \frac{\partial f}{\partial x}+y \frac{\partial f}{\partial y}=n f(x, y)\). \(f(x, y)=\sqrt{x^{2}+y^{2}}\)

Step-by-Step Solution

Answer

The function is homogeneous of degree 1, and Euler's Theorem holds for it with \(n=1\).

Step 1: Verify Homogeneity

To show that \(f(x, y) = \sqrt{x^2 + y^2}\) is homogeneous of degree \(n\), we need to substitute \((t x, t y)\) into the function and check if it equals \(t^n f(x, y)\). Calculate \(f(tx, ty)\):\[ f(tx, ty) = \sqrt{(tx)^2 + (ty)^2} = \sqrt{t^2 x^2 + t^2 y^2} = \sqrt{t^2 (x^2 + y^2)} = t \sqrt{x^2 + y^2} = t f(x, y) \]Since \(f(tx, ty) = t f(x, y)\), the function is homogeneous of degree 1 (\(n=1\)).

Step 2: Compute Partial Derivatives

Next, we need to find the partial derivatives \(\frac{\partial f}{\partial x}\) and \(\frac{\partial f}{\partial y}\).The function is \(f(x, y) = \sqrt{x^2 + y^2}\).The partial derivative with respect to \(x\) is calculated as follows:\[ \frac{\partial f}{\partial x} = \frac{1}{2\sqrt{x^2 + y^2}} \cdot 2x = \frac{x}{\sqrt{x^2 + y^2}} \]Similarly, the partial derivative with respect to \(y\) is:\[ \frac{\partial f}{\partial y} = \frac{1}{2\sqrt{x^2 + y^2}} \cdot 2y = \frac{y}{\sqrt{x^2 + y^2}} \]

Step 3: Verify Euler's Theorem for Homogeneous Functions

Now verify the equation \(x \frac{\partial f}{\partial x} + y \frac{\partial f}{\partial y} = n f(x, y)\), where we determined \(n=1\).Substitute the partial derivatives:\[ x \frac{\partial f}{\partial x} = x \cdot \frac{x}{\sqrt{x^2 + y^2}} = \frac{x^2}{\sqrt{x^2 + y^2}} \]\[ y \frac{\partial f}{\partial y} = y \cdot \frac{y}{\sqrt{x^2 + y^2}} = \frac{y^2}{\sqrt{x^2 + y^2}} \]Add these results:\[ x \frac{\partial f}{\partial x} + y \frac{\partial f}{\partial y} = \frac{x^2}{\sqrt{x^2 + y^2}} + \frac{y^2}{\sqrt{x^2 + y^2}} = \frac{x^2 + y^2}{\sqrt{x^2 + y^2}} = \sqrt{x^2 + y^2} = f(x, y) \]Since this equals \(f(x, y)\), Euler's Theorem holds for this function with \(n=1\).

Key Concepts

Degree of Homogeneity

When we talk about a function being homogeneous, we refer to its degree of homogeneity. This simply means that if we scale the input variables of the function by a factor \( t \), the output of the function scales by a power of \( t \) called the degree of homogeneity. For example, a function \( f(x, y) \) is homogeneous of degree \( n \) if \( f(tx, ty) = t^n f(x, y) \) for all \( t \).
In our exercise, we checked the function \( f(x, y) = \sqrt{x^2 + y^2} \). By replacing \( x \) with \( tx \) and \( y \) with \( ty \) in the function, we found that \( f(tx, ty) = t f(x, y) \).
- This shows the function scales linearly with \( t \), meaning the function is homogeneous of degree 1. - This property gives us crucial information about how the function behaves under scaling, which is useful in fields like economics, physics, and engineering.

Partial Derivatives

Partial derivatives are crucial in understanding rates of change. For a function with multiple variables like \( f(x, y) \), the partial derivative \( \frac{\partial f}{\partial x} \) tells us how the function changes as \( x \) changes, while holding \( y \) constant. Similarly, \( \frac{\partial f}{\partial y} \) describes the rate of change with respect to \( y \), holding \( x \) constant.
In our step-by-step solution, we calculated these derivatives for \( f(x, y) = \sqrt{x^2 + y^2} \):

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

These calculations were vital for verifying Euler's theorem, a powerful tool in analyzing homogeneous functions.
Understanding partial derivatives helps us grasp how individual changes in variables impact the whole function, which is fundamental in calculus and optimization problems.

Euler's Theorem for Homogeneous Functions

Euler’s theorem provides a significant shortcut when dealing with homogeneous functions. It states that for a function homogeneous of degree \( n \), the relationship \( x \frac{\partial f}{\partial x} + y \frac{\partial f}{\partial y} = n f(x, y) \) must hold.
In our scenario, with the given function \( f(x, y) = \sqrt{x^2 + y^2} \), the theorem was applied by substituting the calculated partial derivatives and confirming the equation:
- We found \( x \frac{\partial f}{\partial x} = \frac{x^2}{\sqrt{x^2+y^2}} \) and \( y \frac{\partial f}{\partial y} = \frac{y^2}{\sqrt{x^2+y^2}} \).- Adding these gave \( \frac{x^2 + y^2}{\sqrt{x^2 + y^2}} = \sqrt{x^2 + y^2} \) which equals \( f(x, y) \).
This step confirms that Euler's theorem holds for this function when \( n=1 \), demonstrating the power of combining homogeneity with partial derivatives. Euler’s theorem is particularly useful in simplifying problems across various scientific disciplines, especially when dealing with scaling properties of physical systems.