Curl of Gradient is Zero

Definition
Let $\R^3 \left({x, y, z}\right)$ denote the real Cartesian space of $3$ dimensions..

Let $\left({\mathbf i, \mathbf j, \mathbf k}\right)$ be the standard ordered basis on $\R^3$.

Let $f \left({x, y, z}\right): \R^3 \to \R$ be a real-valued function on $\R^3$:

Then:
 * $\nabla \times \left({\nabla f}\right) = \mathbf 0$

where:
 * $\nabla f$ denotes the gradient of $f$
 * $\nabla \times \left({\nabla f}\right)$ denotes the curl of the gradient of $f$.

Proof
From Partial Differentiation Operator is Commutative for Continuous Functions:
 * $\dfrac {\partial^2 f} {\partial x \partial y} = \dfrac {\partial^2 f} {\partial y \partial x}$

and the same mutatis mutandis for the other partial derivatives.

The result follows.