### Differential Operators ### Gradient $\nabla a = \hat{x}\frac{ \partial a }{ \partial x }+\hat{y}\frac{ \partial a }{ \partial y }+\hat{z}\frac{ \partial a }{ \partial z }$ #### Divergence $\nabla\cdot \vec{A}=\frac{ \partial A_{x} }{ \partial x }+\frac{ \partial A_{y} }{ \partial y }+\frac{ \partial A_{z} }{ \partial z }$ #### Curl $\nabla \times \vec{A}=\begin{vmatrix}\hat{x} & \hat{y} & \hat{z} \\ \frac{ \partial }{ \partial x } & \frac{ \partial }{ \partial y } & \frac{ \partial }{ \partial z } \\ A_{x} & A_{y} & A_{z}\end{vmatrix}= \hat{x}\,\left(\frac{\partial A_{z}}{\partial y}-\frac{\partial A_{y}}{\partial z}\right)+\hat{y}\,\left(\frac{\partial A_{x}}{\partial z}-\frac{\partial A_{z}}{\partial x}\right)+\hat{z}\,\left(\frac{\partial A_{y}}{\partial x}-\frac{\partial A_{x}}{\partial y}\right)$ ### Laplacian $\nabla^{2}a = \nabla\cdot(\nabla a)=\frac{ \partial^{2} a }{ \partial x^{2} }+\frac{ \partial^{2} a }{ \partial y^{2} }+\frac{ \partial^{2} a }{ \partial z^{2} }$ same as above for vectors, component by component (Cartesian only) ### Theorems #### Gradient Theorem $ \int _{C}\nabla f \cdot dl = f(r_{b})-f(r_{a}) $ Basically the fundamental theorem of calculus for vector calculus. #### Divergence Theorem (Gauss) The integral of all the divergences within a volume fully cancel except for the flux out of the exterior surface $ \iiint_{V} \nabla\cdot \vec{A} ~dV = \oint _{S}\vec{A} \cdot d\vec{S} $ #### Stokes' Theorem The curls over a surface fully cancel except for the circulation around the boundary curve $ \iint_{S}(\nabla \times \vec{A})\cdot d\vec{S} = \oint _{C} \vec{A}\cdot d\vec{C} $ ### Identities Curls and divergences cancel each other: $\nabla \cdot (\nabla \times A) = 0$ $\nabla \times (\nabla f) = 0$ #### Product Rules For a scalar f and vector fields $A$ and $B$, we have $\nabla\cdot(fA)=f(\nabla\cdot A)+A\cdot \nabla f$ $\nabla \times(fA)=f(\nabla \times A)+\nabla f\times A$ $\nabla\cdot(A\times B)=B\cdot(\nabla\times A)-A\cdot(\nabla \times B)$ ### Scalar Triple Product $A\cdot(B\times C)=B\cdot(C\times A)=C\cdot(A\times B)$ ### Vector Triple Product (BAC-CAB) $A\times (B\times C) = B (A\cdot C)-C(A\cdot B)$ Often, we let $A=B=\nabla$ such that $\nabla \times (\nabla \times C) = \nabla(\nabla\cdot C)-\nabla^{2}C$