For a vector field \(\bm{F} = P\mathbf{i} + Q\mathbf{j} + R\mathbf{k}\) the curl of \(\bm{F},\) denoted \(\operatorname{curl}\bm{F}\) is defined to be \[ \operatorname{curl}\bm{F} = \nabla \times \bm{F} = \det\begin{pmatrix} \mathbf{i} & \mathbf{j} & \mathbf{k} \\ \frac{\partial}{\partial x} & \frac{\partial}{\partial y} & \frac{\partial}{\partial z} \\ P & Q & R \end{pmatrix} = \bigg(\frac{\partial R}{\partial y}-\frac{\partial Q}{\partial z}\bigg)\mathbf{i} + \bigg(\frac{\partial P}{\partial z}-\frac{\partial R}{\partial x}\bigg)\mathbf{j} + \bigg(\frac{\partial Q}{\partial x}-\frac{\partial P}{\partial y}\bigg)\mathbf{k} \]
If \(f\) has continuous second-order partial derivatives, then \(\operatorname{curl} \nabla f = \mathbf{0}.\)
If \(\bm{F}\) is defined on all of \(\mathbf{R}^3\) and if the components functions of \(\bm{F}\) have continuous second-order partial derivatives, and \(\operatorname{curl} \bm{F} = \mathbf{0},\) then \(\bm{F}\) is a conservative vector field.
For a vector field \(\bm{F} = P\mathbf{i} + Q\mathbf{j} + R\mathbf{k}\) the divergence of \(\bm{F},\) denoted \(\operatorname{div}\bm{F}\) is defined to be \[ \operatorname{div}\bm{F} = \nabla \cdot \bm{F} = \frac{\partial P}{\partial x} + \frac{\partial Q}{\partial y} + \frac{\partial R}{\partial z} \]
If \(\bm{F} = P\mathbf{i} + Q\mathbf{j} + R\mathbf{k}\) is defined on all of \(\mathbf{R}^3\) and if the components functions of \(\bm{F}\) have continuous second-order partial derivatives, then \(\operatorname{div}\operatorname{curl} \bm{F} = 0.\)
Laplace operator \(\nabla^2 = \nabla \cdot \nabla\)
Green’s theorem in vector form \[ \oint\limits_C \bm{F}\cdot\mathrm{d}r = \oint\limits_C \bm{F}\cdot \mathbf{T}_r \,\mathrm{d}\ell = \iint\limits_R \big(\operatorname{curl}\bm{F}\big) \cdot \mathbf{k} \,\mathrm{d}A \] or alternatively \[ = \oint\limits_C \bm{F}\cdot \mathbf{n}_r \,\mathrm{d}\ell = \iint\limits_R \operatorname{div}\bm{F}(x,y)\,\mathrm{d}A \]