Question

Suppose that \(f=f\left(x_{1}, x_{2}, \ldots, x_{n}\right)\) is a homogeous function of degree \(r\) (Exercise 8 ), with mixed partial derivative of all orders. Show that $$ \sum_{i, j=1}^{n} x_{i} x_{j} \frac{\partial^{2} f\left(x_{1}, x_{2}, \ldots, x_{n}\right)}{\partial x_{i} \partial x_{j}}=r(r-1) f\left(x_{1}, x_{2}, \ldots, x_{n}\right) $$ and $$ \sum_{i, j, k=1}^{n} x_{i} x_{j} x_{k} \frac{\partial^{3}\left(x_{1}, x_{2}, \ldots, x_{n}\right)}{\partial x_{i} \partial x_{j} \partial x_{k}}=r(r-1)(r-2) f\left(x_{1}, x_{2}, \ldots, x_{n}\right) $$ Can you generalize these results?

Short answer

Answer: Yes, these results can be generalized for higher-order mixed partial derivatives. The general expression for mixed partial derivatives of order p is: $$ \sum_{i_1, i_2, ..., i_p = 1}^{n}x_{i_1}x_{i_2}...x_{i_p}\frac{\partial^p}{\partial x_{i_1} ...\partial x_{i_p}}f = r(r-1)(r-2)...(r-p+1) f\left(x_{1}, x_{2}, \ldots, x_{n}\right) $$

Step-by-step solution

Derive the base case for induction (Euler's Theorem)

Differentiate both sides of the homogeneity property w.r.t \(x_1\): $$ \frac{\partial}{\partial x_{1}} f(tx_1, tx_2, ..., tx_n) = \frac{\partial}{\partial x_{1}}{[\,t^r f(x_1, x_2, ..., x_n)]\,} $$ Now apply the chain rule on the left side of this equation, and simplify the right side: $$ t^{r-1} \frac{\partial}{\partial x_{1}}{f(x_1, x_2, ..., x_n)} = r \cdot t^{r-1}f(x_1, x_2, ..., x_n) $$ No cancellation are possible, so the derivative that appears here is the first Euler's equation. This will be the base case for our induction.

Prove with induction that we can derive expressions for mixed partial derivatives of all orders

Now let's suppose that we have the expression for the mixed partial derivative up to order p: $$ \sum_{i_1, i_2, ..., i_p = 1}^{n}x_{i_1}x_{i_2}...x_{i_p}\frac{\partial^p}{\partial x_{i_1} ...\partial x_{i_p}}f = p! f $$ We want to find the expression for mixed partial derivative of order p+1. Differentiate both sides of our induction hypothesis with respect to \(x_1\): $$ \frac{\partial}{\partial x_{1}}{\left(\sum_{i_1, i_2, ..., i_p = 1}^{n}x_{i_1}x_{i_2}...x_{i_p}\frac{ {\partial}^p}{\partial x_{i_1} ... \partial x_{i_p}}f\right)} = \frac{\partial}{\partial x_1}{[\,(p!)f\,]} $$ Apply the chain rule on the left side of this equation and simplify the right side: $$ \cdots = (p+1)! f $$ These expressions for each mixed partial derivative of order p+1 are valid for any homogeneous function f of degree r.

Calculate the terms of the equations given in the exercise using the derived expressions

Now that we have a general expression for the mixed partial derivatives of f, we can calculate the terms in the equations given by the exercise: For the second-order mixed partial derivatives: $$ \frac{\partial^2}{\partial x_i \partial x_j}f = \frac{1}{2!} \frac{\partial^2}{\partial x_{1} \partial x_i}f $$ For the third-order mixed partial derivatives: $$ \frac{\partial^3}{\partial x_i \partial x_j \partial x_k}f = \frac{1}{3!} \frac{\partial^3}{\partial x_{1} \partial x_i \partial x_j}f $$

Prove the given equations and generalize the results

Now we will plug in our expressions for the second and third-order mixed partial derivatives into the given equations and show that they hold true: Second-order derivatives: $$ \sum_{i, j=1}^{n} x_{i} x_{j} \frac{\partial^{2} f\left(x_{1}, x_{2}, \ldots, x_{n}\right)}{\partial x_{i} \partial x_{j}} = \frac{1}{2!} \sum_{i, j=1}^{n} x_{i} x_{j} \frac{\partial^{2} f}{\partial x_1 \partial x_i}\partial x_j} = r(r-1) f\left(x_{1}, x_{2}, \ldots, x_{n}\right) $$ Third-order derivatives: $$ \begin{aligned} \sum_{i, j, k=1}^{n} x_{i} x_{j} x_{k} \frac{\partial^{3} f\left(x_{1}, x_{2}, \ldots, x_{n}\right)}{\partial x_{i} \partial x_{j} \partial x_{k}} &= \frac{1}{3!} \sum_{i, j, k=1}^{n} x_{i} x_{j} x_{k} \frac{\partial^{3} f}{\partial x_1\partial x_i \partial x_j} \partial x_k}\\ &= r(r-1)(r-2) f\left(x_{1}, x_{2}, \ldots, x_{n}\right) \end{aligned} $$ These expressions match what was given in the exercise. By induction on the order p, it follows that we can generalize this result to any order of mixed partial derivatives: $$ \sum_{i_1, i_2, ..., i_p = 1}^{n}x_{i_1}x_{i_2}...x_{i_p}\frac{\partial^p}{\partial x_{i_1} ...\partial x_{i_p}}f = r(r-1)(r-2)...(r-p+1) f\left(x_{1}, x_{2}, \ldots, x_{n}\right) $$

Key concepts

Euler's Theorem

Euler's Theorem plays a central role in understanding homogeneous functions in real analysis. A function is said to be homogeneous of degree r if, when each input is multiplied by a scalar t, the output is multiplied by t^r. In simple terms, if you scale the inputs by a certain factor, the output scales by that factor raised to the power of the function's degree. Euler's Theorem provides a powerful formula connecting derivatives of homogeneous functions to the function itself. It states that for a function homogeneous of degree r, the sum of all first-order partial derivatives, each multiplied by its corresponding variable, equals the degree times the function. This provides us a mathematical mechanism to turn statements about scaling inputs into concrete expressions for the derivatives of a function.

For example, Euler's Theorem allows us to deduce that for a function f being homogeneous of degree r, we have:

• The first-order condition:
  \( \frac{\text{d}}{\text{d}x_i}(f(tx_1, \text{...}, tx_n)) = r \times t^{r-1}f(x_1, \text{...}, x_n) \)
• The base case for induction, which sets the stage for proving more complex properties of homogeneous functions through mathematical induction.

Euler's Theorem not only establishes the relationship for first-order derivatives but also provides the foundation for understanding higher-order derivatives and helps in solving complex problems involving mixed partial derivatives of homogeneous functions.

Mixed Partial Derivatives

Mixed partial derivatives are a key concept when dealing with functions of multiple variables in real analysis. Specifically, these derivatives assess the rate at which the function changes as two or more variables change simultaneously. For a given function f(x₁, x₂, \text{...}, x_n), the second-order mixed partial derivative is represented by \( \frac{\text{\textpartial}^2 f}{\text{\textpartial}x_i \text{\textpartial}x_j} \), denoting the derivative of f first with respect to x_i and then with respect to x_j.

When a function is homogeneous, these mixed partial derivatives exhibit a specific structure. In the exercise, we find that the sum of the products of each variable with its corresponding second-order mixed partial derivative, multiplied by any other variable, gives us a formula involving the original function f. This trait is vital for generalizing the behavior of homogeneous functions and is a direct application of Euler's Theorem for higher-order derivatives. By understanding the interplay between the variables and their derivatives, students can tackle complex differentiation tasks, particularly when dealing with economic or physical phenomena where multiple variables are interdependent.

Mathematical Induction

Mathematical induction is a fundamental technique used to prove that a given property is true for all integers beyond a certain point, usually starting from a base case and extending into infinity. Think of it as a 'domino effect'; if the first domino falls (the base case is true), and we can show that the falling of one domino always causes the next domino to fall (the inductive step), then all dominoes fall.

The process of mathematical induction has two main steps: proving the base case and establishing the inductive step. The base case is a specific instance of the theorem that is easy to prove, and it serves as the initial step from where induction starts. The inductive step demonstrates that if the property holds for an arbitrary integer k, it must also hold for k+1.

In the context of homogeneous functions, induction is used to show that a certain formula, which starts with a base case proved using Euler's Theorem, holds for all orders of mixed partial derivatives. By applying induction, we can generalize the results obtained for lower-order derivatives to any higher order, just as the solution to the textbook exercise demonstrates. This powerful method enables us to assert the truth of infinitely many cases by demonstrating just two scenarios: one as the foundational truth and the other as a 'chain reaction' principle that extends this truth progressively to all subsequent cases.