Question

A function \(f(x, y, z)\) is called homogeneous of degree \(n\) if \(f(t x, t y, t z)=t^{n} f(x, y, z)\) For example, \(z^{2} \ln (x / y)\) is homogeneous of degree 2 since $$ (t z)^{2} \ln \frac{t x}{t y}=t^{2}\left(z^{2} \ln \frac{x}{y}\right) $$ Euler's theorem on homogeneous functions says that if \(f\) is homogeneous of degree \(n,\) then $$ x \frac{\partial f}{\partial x}+y \frac{\partial f}{\partial y}+z \frac{\partial f}{\partial z}=n f $$ Prove this theorem. Hints: Differentiate \(f(t x, t y, t z)=t^{n} f(x, y, z)\) with respect to \(t,\) and then let \(t=1 .\) It is convenient to call \(\partial f / \partial(t x)=f_{1}(\) that is, the partial derivative of \(f\) with respect to its first variable), \(f_{2}=\partial f / \partial(t y),\) and so on. Or, you can at first call \(t x=u, t y=v, t z=w\). (Both the definition and the theorem can be extended to any number of variables.)

Short answer

Differentiate \(f(tx, ty, tz) = t^n f(x, y, z)\) with respect to \(t\), then set \(t = 1\).

Step-by-step solution

Express the Function Using Homogeneity

Given that a function is homogeneous of degree \(n\), start by expressing it as follows: \(f(t x, t y, t z) = t^n f(x, y, z)\).

Differentiate with Respect to \(t\)

Differentiate both sides of the equation \(f(t x, t y, t z) = t^n f(x, y, z)\) with respect to \(t\). Use the chain rule for the left-hand side:\[ \frac{d}{dt} f(t x, t y, t z) = x \frac{\partial f}{\partial (t x)} + y \frac{\partial f}{\partial (t y)} + z \frac{\partial f}{\partial (t z)} \]and the power rule for the right-hand side:\[ \frac{d}{dt} (t^n f(x, y, z)) = n t^{n-1} f(x, y, z) \]

Set \(t = 1\)

After differentiating, set \(t = 1\) in the resulting equation. This simplifies to:\[ x \frac{\partial f}{\partial x} + y \frac{\partial f}{\partial y} + z \frac{\partial f}{\partial z} = n f(x, y, z) \]

Finalize the Proof

The resulting equation from Step 3 is exactly the statement of Euler's theorem for homogeneous functions, which proves the theorem.\[ x \frac{\partial f}{\partial x} + y \frac{\partial f}{\partial y} + z \frac{\partial f}{\partial z} = n f \]

Key concepts

Euler's Theorem on Homogeneous Functions

Euler's Theorem is a pivotal concept in the field of calculus when dealing with homogeneous functions. A function, say, of three variables, is termed homogeneous of degree if scaling every variable by a factor of results in multiplying the function by . This theorem provides a mathematical bridge to understand how changes scale when inputs are proportionately increased.
Formally, if a function \(f(x, y, z)\) is homogeneous of degree \(n\), it means that \(f(t x, t y, t z) = t^n f(x, y, z)\).
To prove Euler's theorem, we differentiate both sides of this equation with respect to \(t\). On the left, we apply the chain rule and on the right the power rule. The differentiation gives us:
a\[d/dt f(t x, t y, t z) = x (\frac{\partial f}{\partial (t x)}) + y (\frac{\partial f}{\partial (t y)}) + z (\frac{\partial f}{\partial (t z)}) = nt^{n-1}f(x,y,z)\]
We set \(t=1\) to simplify the equation:
\[x (\frac{\partial f}{\partial x}) + y (\frac{\partial f}{\partial y}) + z (\frac{\partial f}{\partial z}) = n f(x, y, z)\]
This equation is the essence of Euler's Theorem, illustrating the relationship between the partial derivatives of \(f\) and the function itself.

Introduction to Partial Differentiation

Partial differentiation is a technique used when we have functions of multiple variables. It helps us understand how a function changes along one variable while keeping the others constant.
If we have a function \(f(x, y, z)\), we take the partial derivative of \(f\) with respect to one variable (say \(x\)), while treating \(y\) and \(z\) as constants. The notation for this is \(\frac{\partial f}{\partial x}\).
For example, if \(f(x, y, z) = x^2 + y^2 + z^2\), then:

• \(\frac{\partial f}{\partial x} = 2x\)
• \(\frac{\partial f}{\partial y} = 2y\)
• \(\frac{\partial f}{\partial z} = 2z\)

In the context of Euler's Theorem, we use partial differentiation to express how the function scales with each variable. Applying the chain rule allows us to link these derivatives together, demonstrating how they sum up to give the degree of homogeneity.

Understanding the Degree of Homogeneity

The degree of homogeneity of a function is a measure of how the function scales when its inputs are scaled. For a function \( f(x, y, z) \) that is homogeneous of degree \( n \), it means:
\[ f(t x, t y, t z) = t^n f(x, y, z) \]
This tells us that if all inputs are scaled by a factor of \( t \), the output of the function scales by \( t^n \).
For example, consider the function \( z^2 \, ln(x/y) \). When we scale \(x\), \(y\), and \(z\) by \(t\), we get:
\[ (t z)^2 \, ln(t x / t y) = t^2 \, z^2 \, ln(x / y) \]
Thus, \( f(x, y, z) = z^2 \, ln(x / y) \) is homogeneous of degree 2.
The degree of homogeneity helps in simplifying complex functions and gives insights into their scaling properties, making it easier to analyze and work with them mathematically.