### Vector Indentities

Triple Products $$\vec{A}\cdot(\vec{B}\times\vec{C})=\vec{B}\cdot(\vec{C}\times\vec{A})=\vec{C}\cdot(\vec{A}\times\vec{B})$$$$\vec{A}\times(\vec{B}\times\vec{C})=\vec{B}(\vec{A}\cdot\vec{C})-\vec{C}(\vec{A}\cdot\vec{B})$$

Product Rules $$\nabla(fg) = f(\nabla g) + g (\nabla f)$$ $$\nabla(\vec{A}\cdot\vec{B})=\vec{A}\times(\nabla\times\vec{B})+\vec{B}\times(\nabla\times\vec{A})+(\vec{A}\cdot\nabla)\vec{B}+(\vec{B}\cdot\nabla)\vec{A}$$ $$\nabla\cdot(f\vec{A}) = f(\nabla\cdot\vec{A}) + \vec{A} \cdot (\nabla f)$$ $$\nabla\cdot(\vec{A}\times\vec{B})=\vec{B}\cdot(\nabla\times\vec{A})-\vec{A}\cdot(\nabla\times\vec{B})$$ $$\nabla\times(f\vec{A}) = f(\nabla\times\vec{A}) - \vec{A} \times(\nabla f)$$ $$\nabla\times(\vec{A}\times\vec{B})=(\vec{B}\cdot\nabla)\vec{A}-(\vec{A}\cdot\nabla)\vec{B}+\vec{A}(\nabla\cdot\vec{B})-\vec{B}(\nabla\cdot\vec{A})$$
Second Derivatives $$\nabla\cdot(\nabla\times\vec{A})=0$$ $$\nabla\times(\nabla f)=0$$ $$\nabla\times(\nabla\times\vec{A})=\nabla(\nabla\cdot\vec{A})-\nabla^{2}\vec{A}$$

Fundamental Theorems
Gradient Theorem: $$\displaystyle \int_a^b(\nabla f)\cdot d\vec{l} = f(b) - f(a)$$ Divergence Theorem: $$\int_V(\nabla\cdot\vec{A})d\vec{\tau}=\oint_{S} \! \vec{A}\cdot d\vec{a}$$ Curl Theorem: $$\int_S(\nabla\times\vec{A})\cdot d\vec{a}=\oint_{L} \! \vec{A}\cdot d\vec{l}$$