Math Insight

The curl of a gradient is zero

Let $f(x,y,z)$ be a scalar-valued function. Then its gradient $$\nabla f(x,y,z) = \left(\pdiff{f}{x}(x,y,z),\pdiff{f}{y}(x,y,z),\pdiff{f}{z}(x,y,z)\right)$$ is a vector field, which we denote by $\dlvf = \nabla f$. We can easily calculate that the curl of $\dlvf$ is zero.

We use the formula for $\curl\dlvf$ in terms of its components $$\curl \dlvf = \left(\pdiff{\dlvfc_3}{y}-\pdiff{\dlvfc_2}{z}, \pdiff{\dlvfc_1}{z} - \pdiff{\dlvfc_3}{x}, \pdiff{\dlvfc_2}{x} - \pdiff{\dlvfc_1}{y} \right).$$ Since each component of $\dlvf$ is a derivative of $f$, we can rewrite the curl as $$\curl \nabla f = \left(\frac{\partial^2 f}{\partial y \partial z} -\frac{\partial^2 f}{\partial z \partial y}, \frac{\partial^2 f}{\partial z \partial x} -\frac{\partial^2 f}{\partial x \partial z}, \frac{\partial^2 f}{\partial x \partial y} -\frac{\partial^2 f}{\partial y \partial x}\right).$$

If $f$ is twice continuously differentiable, then its second derivatives are independent of the order in which the derivatives are applied. All the terms cancel in the expression for $\curl \nabla f$, and we conclude that $\curl \nabla f=\vc{0}.$