The divergence of a vector field $\mathbf{F} = (P, Q, R)$ is the scalar field $\operatorname{div} \mathbf{F} = \nabla \cdot \mathbf{F} = \frac{\partial P}{\partial x} + \frac{\partial Q}{\partial y} + \frac{\partial R}{\partial z}$ Physically, $\operatorname{div} \mathbf{F}(\mathbf{a})$ measures the net outward flux of $\mathbf{F}$ per unit volume at $\mathbf{a}$: $\operatorname{div} \mathbf{F}(\mathbf{a}) = \lim_{V \to 0} \frac{1}{|V|} \oiint_{\partial V} \mathbf{F} \cdot d\mathbf{S}$

The curl of a vector field $\mathbf{F} = (P, Q, R)$ is the vector field $\operatorname{curl} \mathbf{F} = \nabla \times \mathbf{F} = \det \begin{pmatrix} \mathbf{i} & \mathbf{j} & \mathbf{k} \\ \partial_x & \partial_y & \partial_z \\ P & Q & R \end{pmatrix}$ The curl measures the infinitesimal rotation of $\mathbf{F}$: the component $(\nabla \times \mathbf{F}) \cdot \hat{\mathbf{n}}$ gives the circulation density of $\mathbf{F}$ about the axis $\hat{\mathbf{n}}$: $(\nabla \times \mathbf{F})(\mathbf{a}) \cdot \hat{\mathbf{n}} = \lim_{A \to 0} \frac{1}{|A|} \oint_{\partial A} \mathbf{F} \cdot d\mathbf{r}$

The Laplacian of a scalar field $f$ is $\nabla^2 f = \Delta f = \nabla \cdot (\nabla f) = \frac{\partial^2 f}{\partial x^2} + \frac{\partial^2 f}{\partial y^2} + \frac{\partial^2 f}{\partial z^2}$ A function satisfying $\nabla^2 f = 0$ is called harmonic. Harmonic functions arise in electrostatics, heat conduction, and fluid dynamics.