Example 2.2.4: Curl of a gradient

The gradient has components $(\nabla f{)}_x=\partial f/\partial x$, $(\nabla f{)}_y=\partial f/\partial y$, $(\nabla f{)}_z=\partial f/\partial z$. Inserting them in (2.2.8),

For a function with continuous second derivatives the order of differentiation does not matter (Schwarz's theorem), so every bracket vanishes:

The gradient of any scalar field is irrotational. Physically: the circulation of $\nabla f$ around a small loop is $\oint \nabla f\cdot d\ell =\oint df$, the total change of $f$ around the loop, which is zero because we come back to the starting value. (On the hill of Figure 2.2.2: a walk that returns to its starting point has gone up exactly as much as it has gone down.)