For a vector field \(\bm{F} = L\mathbf{i} + M\mathbf{j} + N\mathbf{k}\) the divergence of \(\bm{F}\) and the curl of \(\bm{F},\) denoted \(\operatorname{div}\bm{F}\) and \(\operatorname{curl}\bm{F}\) respectively, are calculated as \[ \operatorname{div}\bm{F} = \nabla \cdot \bm{F} = \frac{\partial L}{\partial x} + \frac{\partial M}{\partial y} + \frac{\partial N}{\partial z} \qquad \operatorname{curl}\bm{F} = \nabla \times \bm{F} = \operatorname{det}\!\begin{pmatrix} \mathbf{i} & \mathbf{j} & \mathbf{k} \\ \frac{\partial}{\partial x} & \frac{\partial}{\partial y} & \frac{\partial}{\partial z} \\ L & M & N \end{pmatrix} = \bigg(\frac{\partial N}{\partial y}\!-\!\frac{\partial M}{\partial z}\bigg)\mathbf{i} + \bigg(\frac{\partial L}{\partial z}\!-\!\frac{\partial N}{\partial x}\bigg)\mathbf{j} + \bigg(\frac{\partial M}{\partial x}\!-\!\frac{\partial L}{\partial y}\bigg)\mathbf{k} \,. \]
Oversimplifying things, we can say divergence measures a vector field’s “flow density” at a given point, and curl measures a vector field’s “circulation” at a given point. Just like conservative vector fields are those that are the gradient of some function \(f,\) solenoidal vector fields are those that are the curl of some vector field \(\bm{F}.\) Just like for a conservative vector field \(\bm{F} = \nabla f\) we call \(f\) the scalar potential of the field \(\bm{F},\) for a solenoidal vector field \({\bm{F} = \operatorname{curl}\bm{A}}\) we call \(\bm{A}\) the vector potential of the field \(\bm{F}.\)
Conservative vector fields are curl-free: a vector field \(\bm{F}\) is conservative if and only if \(\operatorname{curl}\bm{F} = \mathbf{0}.\) In particular, for any function \(f\) with continuous second-order partial derivatives, \(\operatorname{curl}\bigl(\nabla f\bigr) = \mathbf{0}.\) Analogously solenoidal vector fields are divergence-free: a vector field \(\bm{F}\) is solenoidal if and only if \(\operatorname{div}\bm{F} = 0.\) In particular, for any vector field \(\bm{A}\) whose components have continuous second-order partial derivatives, \(\operatorname{div}\bigl(\operatorname{curl} \bm{A}\bigr) = 0.\)
For a planar region \(R\) with boundary \(C = \partial R\) and a vector field \(\bm{F},\) Green’s theorem can be interpreted in terms of curl or divergence: \[ \iint_R \big(\operatorname{curl}\bm{F}\big) \cdot \mathbf{k} \,\mathrm{d}A = \oint_C \bm{F}\cdot \mathbf{T} \,\mathrm{d}s = \oint_C \bm{F}\cdot\mathrm{d}\bm{r} \qquad\qquad \iint_R \operatorname{div}\bm{F} \,\mathrm{d}A = \oint_C \bm{F}\cdot \mathbf{N} \,\mathrm{d}s %= \oint_C \bm{F}\times\mathrm{d}\bm{r} \]