\[ \mathbf{rot}(\mathbf{grad} f)=\mathbf{rot}\left(\frac{\partial f}{\partial x_1},\frac{\partial f}{\partial x_2},\frac{\partial f}{\partial x_3}\right)= \] \[ \left(\frac{\partial}{\partial x_2}\left(\frac{\partial f}{\partial x_3}\right)- \frac{\partial}{\partial x_3}\left(\frac{\partial f}{\partial x_2}\right), \frac{\partial}{\partial x_3}\left(\frac{\partial f}{\partial x_1}\right)- \frac{\partial}{\partial x_1}\left(\frac{\partial f}{\partial x_3}\right), \frac{\partial}{\partial x_1}\left(\frac{\partial f}{\partial x_2}\right)- \frac{\partial}{\partial x_2}\left(\frac{\partial f}{\partial x_1}\right) \right) \] \[=(0,0,0)\] om \(f\) är av klass \(C^2\), eftersom dom blandade derivatorna då är lika.