# Equality and Inequality of Mixed Partial Derivatives of Functions from Rn to Rm

On the Higher Order Partial Derivatives of Functions from Rn to Rm we defined the notion of higher order partial derivatives for a function. We will now focus our attention on the equality / inequality of mixed partial derivatives of a function.

Let $f : \mathbb{R}^2 \to \mathbb{R}$ be defined by $f(x, y) = xy^2$. Then the partial derivatives of $f$ are:

(1)The second order partial derivatives of $f$ are:

(2)Notice that $D_{1, 2} f(x, y) = D_{2, 1} f(x, y)$. One might ask if this is true in general. The answer unfortunately is not.

For example, consider the function $f : \mathbb{R}^2 \to \mathbb{R}$ defined by:

(3)Then the first order partial derivatives of $f$ are:

(4)We now compute the mixed partial derivatives of $f$ at $0$. We have that:

(6)And also:

(7)Therefore $D_{1, 2} f(0, 0) \neq D_{2, 1} f(0, 0)$!

So mixed partial derivatives of a function need not be equal. We will now state a sufficient condition for the equality of mixed partial derivatives.

Theorem 1: Let $S \subseteq \mathbb{R}^n$ be open, $\mathbf{c} \in S$, and $\mathbf{f} : S \to \mathbb{R}^m$. If:1) $D_j \mathbf{f}$ and $D_k \mathbf{f}$ exist on an open ball centered at $\mathbf{c}$, $B(\mathbf{c}, r)$.2) $D_j \mathbf{f}$ and $D_k \mathbf{f}$ are differentiable at $\mathbf{c}$.Then $D_{j, k} \mathbf{f}(\mathbf{c}) = D_{k, j} \mathbf{f} (\mathbf{c})$. |