State and prove Euler’s theorem for homogeneous function.

Euler’s theorem is a fundamental concept in the field of mathematics, particularly in the study of homogeneous functions. To grasp this theorem, we first need to understand what a homogeneous function is. A function \( f(x_1, x_2, \ldots, x_n) \) is said to be homogeneous of degree \( k \) if, for any scalar \( t \), the following holds true:

f(tx_1, tx_2, \ldots, tx_n) = t^k f(x_1, x_2, \ldots, x_n)

In simpler terms, if you scale all the inputs of the function by a factor of \( t \), the output is scaled by \( t^k \). Now, let’s delve into Euler’s theorem for homogeneous functions.

Statement of Euler’s Theorem

Euler’s theorem states that if \( f(x_1, x_2, \ldots, x_n) \) is a homogeneous function of degree \( k \), then:

x_1 \frac{\partial f}{\partial x_1} + x_2 \frac{\partial f}{\partial x_2} + \ldots + x_n \frac{\partial f}{\partial x_n} = k f(x_1, x_2, \ldots, x_n)

This equation essentially expresses that the weighted sum of the partial derivatives of the function, multiplied by their respective variables, equals \( k \) times the function itself.

Proof of Euler’s Theorem

To prove this theorem, we will use the definition of a homogeneous function and apply the concept of differentiation. Let’s consider a homogeneous function \( f(x_1, x_2, \ldots, x_n) \) of degree \( k \). We will differentiate \( f(tx_1, tx_2, \ldots, tx_n) \) with respect to \( t \) and then evaluate it at \( t = 1 \).

Starting with the definition of homogeneity, we have:

f(tx_1, tx_2, \ldots, tx_n) = t^k f(x_1, x_2, \ldots, x_n)

Now, differentiating both sides with respect to \( t \):

\(\frac{d}{dt} f(tx_1, tx_2, \ldots, tx_n) = k t^{k-1} f(x_1, x_2, \ldots, x_n)\)

Using the chain rule on the left side, we get:

\(\sum_{i=1}^{n} \frac{\partial f}{\partial x_i} \cdot \frac{d(tx_i)}{dt} = \sum_{i=1}^{n} \frac{\partial f}{\partial x_i} \cdot x_i\)

Thus, we can rewrite the equation as:

\(\sum_{i=1}^{n} x_i \frac{\partial f}{\partial x_i} = k f(x_1, x_2, \ldots, x_n)\)

Now, evaluating this at \( t = 1 \) gives us the desired result:

x_1 \frac{\partial f}{\partial x_1} + x_2 \frac{\partial f}{\partial x_2} + \ldots + x_n \frac{\partial f}{\partial x_n} = k f(x_1, x_2, \ldots, x_n)

Example for Clarity

Let’s consider a simple example to illustrate Euler’s theorem. Take the function:

f(x, y) = x^2 + y^2

This function is homogeneous of degree 2 because:

f(tx, ty) = (tx)^2 + (ty)^2 = t^2(x^2 + y^2) = t^2 f(x, y)

Now, applying Euler’s theorem:

x \frac{\partial f}{\partial x} + y \frac{\partial f}{\partial y}

Calculating the partial derivatives:

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

Substituting these into Euler’s formula:

x(2x) + y(2y) = 2x^2 + 2y^2 = 2f(x, y)

This confirms that Euler’s theorem holds true for this function, as expected.

In summary, Euler’s theorem provides a powerful tool for analyzing homogeneous functions, linking their structure to their derivatives in a meaningful way. Understanding this theorem not only deepens your grasp of homogeneous functions but also enhances your overall mathematical insight.