Problem 32 - Calculus Early Transcendentals: Pearson New International Edition

Question

Call a function \(f(x, y)\) homogeneous of degree 1 if \(f(t x, t y)=t f(x, y)\) for all \(t>0 .\) For example, \(f(x, y)=\) \(x+y e^{y / x}\) satisfies this criterion. Prove Euler's Theorem that such a function satisfies $$ f(x, y)=x \frac{\partial f}{\partial x}+y \frac{\partial f}{\partial y} $$

Step-by-Step Solution

Answer

The function satisfies Euler's Theorem as proven for homogeneous functions of degree 1.

Step 1: Understand the Given Condition

Given that the function \( f(x, y) \) is homogeneous of degree 1, it satisfies the property \( f(tx, ty) = t f(x, y) \) for all \( t > 0 \). This means when you scale the inputs \( x \) and \( y \) by a factor \( t \), the output of the function is scaled by the same factor \( t \).

Step 2: Differentiate with Respect to t

Apply the chain rule to differentiate both sides of the equation \( f(tx, ty) = t f(x, y) \) with respect to \( t \). Begin by differentiating the left side:\[ \frac{d}{dt} [f(tx, ty)] = \frac{\partial f}{\partial x}(tx, ty) \cdot x + \frac{\partial f}{\partial y}(tx, ty) \cdot y \]This results from the chain rule, as \( x \) and \( y \) are held constant while \( t \) varies.

Step 3: Differentiate the Right Side

Differentiate the right side of the equation \( t f(x, y) \):\[ \frac{d}{dt} [t f(x, y)] = f(x, y) \]The differentiation is straightforward since this is a product of \( t \) and \( f(x, y) \).

Step 4: Set the Right and Left Differentiated Sides Equal

Equate the derivatives obtained from Steps 2 and 3:\[ \frac{\partial f}{\partial x}(tx, ty) \cdot x + \frac{\partial f}{\partial y}(tx, ty) \cdot y = f(x, y) \]

Step 5: Evaluate at t = 1

Since the relationship holds for all \( t > 0 \), it must specifically hold for \( t = 1 \). Substituting \( t = 1 \) into both sides yields:\[ \frac{\partial f}{\partial x}(x, y) \cdot x + \frac{\partial f}{\partial y}(x, y) \cdot y = f(x, y) \]This concludes the proof that if a function \( f \) is homogeneous of degree 1, then Euler's Theorem holds precisely as stated.

Key Concepts

Homogeneous Functions

In mathematics, a function is termed "homogeneous" if its behavior remains consistent under a particular type of transformation. Specifically, for a function like \( f(x, y) \), it is homogeneous of degree 1 if the equation \( f(tx, ty) = t f(x, y) \) holds true for any positive scalar \( t \). This property implies:

• If we scale the inputs \( x \) and \( y \) by a factor \( t \), the output scales by the same factor \( t \).
• It reflects a direct proportionality in behavior when inputs are scaled.
By understanding this intrinsic scaling property, we see how such functions embody symmetry and uniformity in transformations—integral aspects of analyzing differential and integral problems.

Partial Differentiation

Partial differentiation is a fundamental concept in multivariable calculus where we differentiate a function with respect to one of its variables while keeping other variables constant. In the context of Euler's Theorem, you encounter partial derivatives:

• \( \frac{\partial f}{\partial x} \): This represents the rate at which \( f \) changes with respect to \( x \), holding \( y \) constant.
• \( \frac{\partial f}{\partial y} \): Similarly, this reveals how \( f \) changes concerning \( y \), keeping \( x \) constant.

These derivatives are crucial when proving Euler’s Theorem for homogeneous functions, as they help articulate how the scaling properties unfold through differentiation. Understanding partial derivatives allows you to explore intricate changes which multivariable functions undergo.

Chain Rule

The chain rule is a principle used in calculus to differentiate composite functions. When you differentiate \( f(tx, ty) \), as needed in Euler's Theorem, you apply the chain rule since you are differentiating with respect to \( t \), implying:

• First, differentiate the function \( f \) inside, as if \( x \) and \( y \) are constants.
• Second, multiply these derivatives by the derivatives of \( tx \) and \( ty \) with respect to \( t \), which are simply \( x \) and \( y \) respectively.

This method ensures the links between nested functions are handled correctly and allows the solitary examination of each component's rate of change. Using the chain rule effectively demonstrates how small changes in each part affect the overall behavior of composite functions.

Calculus Proof

The logical framework provided by calculus proof is what validates mathematical theorems. Euler’s Theorem relies on showing that if \( f(x, y) \) is homogeneous of degree 1, then it follows that \( f(x, y) = x \frac{\partial f}{\partial x} + y \frac{\partial f}{\partial y} \). Here's how the proof unfolds:

• Apply differentiation via the chain rule to express how \( f \) behaves as the variables are scaled, giving insights into the internal scaling dynamics.
• Derive expressions for both sides of the equation, ensuring they map directly to the concept of homogeneity by relating them back to the function's form.
• Specifically testing at \( t = 1 \) verifies consistency across all positive \( t \), confirming the theorem.

Such pieces construct a calculus proof, where deductions rest on transformation properties combined with partial derivatives, encapsulating the theorem’s essence within foundational concepts of calculus.