Vector Identities

Topics

• Dot Product
• Cross Product
• Outer Product
• Double Product
• Hadamard Product
• Vector Triple Product
• Kronecker Product
• Triple Product
• Poisson’s Equation

Geometric Algebra

Product Type	Notation	Returns	Example	$=$	Priority in Order of Operations
Dot	$a\cdot b$	Scalar	$(1,2)\cdot (2,3)$	$8$	1
Cross	$a\times b$	Vector	$(1,2,3)\times (4,5,6)$	$(-3,6,-3)$	2
Complex	$Z\ast W$	Complex	$(1-2i)(3+4i)$	$11-2i$	N/A
Wedge	$a\wedge b$	Bivector	$e_1\wedge e_2$	$-e_2\wedge e_1=e_1{e}_2$	2
Geometric	$ab$	Scalar + Bivector	$(e_1+e_2)e_2$	$e_1{e}_2+e_2{e}_2=e_1{e}_2+1$	3

Formulas

$\nabla (fg)=f\nabla g+g\nabla f$

• $f$ and $g$ are scalar fields

$\nabla (\vec{A}\cdot \vec{B})=\vec{A}\times (\nabla \times \vec{B})+\vec{B}\times (\nabla \times \vec{A})+(\vec{A}\cdot \nabla )\vec{B}+(\vec{B}\cdot \nabla )\vec{A}$

• $\vec{A}$ and $\vec{B}$ are vector fields

$\nabla \cdot f\vec{A}=f(\nabla \cdot \vec{A})+\vec{A}\cdot \nabla f$

$\nabla \cdot (\vec{A}\times \vec{B})=\vec{B}\cdot (\nabla \times \vec{A})-\vec{A}\cdot (\nabla \times \vec{B})$

$\nabla \times f\vec{A}=f(\nabla \times \vec{A})-\vec{A}\times \nabla f$

$\nabla \times (\nabla V)=0$

• $V$ is a scalar field
• Curl of the gradient of a scalar field is 0
• Given by equality of mixed partials (e.g. ${\partial }_x{\partial }_{y}V-{\partial }_y{\partial }_{x}V=0$)

$\nabla \times (\nabla \times \vec{A})=\nabla (\nabla \cdot \vec{A})-{\nabla }^2\vec{A}$

• Gradient of divergence - divergence of gradient
• GDDG, gold diggers dig gold

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