Topic: Second Order Partial Derivatives and a Theorem on Their Symmetry

1. Definitions:

Let $f : D \subseteq \mathbb{R}^n \to \mathbb{R}$ be a function. The definition in the image outlines how second-order partial derivatives are formed.

1. First partial derivative:

   • If $\frac{\partial f}{\partial x_i}$ exists at point $a$ with respect to variable $x_i$ and is differentiable with respect to another variable $x_j$, then the second-order partial derivative of $f$ is denoted as:

   $\frac{\partial}{\partial x_j} \left( \frac{\partial f}{\partial x_i} \right)(a)$

2. Second-order partial derivatives:

   • The notation for second-order partial derivatives depends on whether $i \neq j$ or $i = j$: $\frac{\partial^2 f}{\partial x_j \partial x_i} \quad \text{if} \quad j \neq i$ $\frac{\partial^2 f}{\partial x_i^2} \quad \text{if} \quad j = i$

These represent the second-order partial derivatives of $f$ with respect to the variables $x_i$ and $x_j$.

2. Theorem:

If the second-order mixed partial derivatives of a function $f : D \subseteq \mathbb{R}^n \to \mathbb{R}$, namely: $\frac{\partial^2 f}{\partial x_j \partial x_i} \quad \text{and} \quad \frac{\partial^2 f}{\partial x_i \partial x_j}$ exist in an open neighborhood containing $a$ and are continuous at $a$, then these mixed derivatives are equal at point $a$: $\frac{\partial^2 f}{\partial x_i \partial x_j} (a) = \frac{\partial^2 f}{\partial x_j \partial x_i} (a)$

Explanation and Example:

This result, often referred to as Clairaut's theorem (or Schwarz's theorem in some cases), guarantees that for smooth enough functions (where partial derivatives exist and are continuous), the order of differentiation doesn't matter when taking mixed partial derivatives.

Example:

Consider a simple function $f(x,y) = x^2y + 3xy^2$.

1. First partial derivatives: $\frac{\partial f}{\partial x} = 2xy + 3y^2, \quad \frac{\partial f}{\partial y} = x^2 + 6xy$

2. Second-order partial derivatives:

   • Mixed partial derivatives: $\frac{\partial^2 f}{\partial y \partial x} = \frac{\partial}{\partial y}(2xy + 3y^2) = 2x + 6y$ $\frac{\partial^2 f}{\partial x \partial y} = \frac{\partial}{\partial x}(x^2 + 6xy) = 2x + 6y$ As we can see, $\frac{\partial^2 f}{\partial y \partial x} = \frac{\partial^2 f}{\partial x \partial y}$, confirming the theorem.

Summary Notes:

• Second-order partial derivatives generalize the concept of curvature and changes in a function in multiple dimensions.
• Clairaut's theorem guarantees symmetry of mixed partials under certain smoothness conditions, specifically continuity of partial derivatives.

Expanding Questions:

1. How does Clairaut's theorem apply when partial derivatives are not continuous?
2. What is the geometric interpretation of second-order partial derivatives?
3. Can you provide an example where second-order partial derivatives are not equal (i.e., function fails to meet the theorem conditions)?
4. What are the conditions for Clairaut's theorem to hold?
5. How does one compute third-order partial derivatives, and do they have a similar symmetry property?

Tip:

Always check the continuity of mixed partial derivatives when applying Clairaut's theorem to ensure that the conditions are met for equality.