Vector Operations & Operators

• Vectors are written with arrows: $\vec{A}, \vec{B}$
• Components of vectors are written as small letters: $a_i, b_i$ where $i = 1,2,3$
• Basis is $(\vec{e_1}, \vec{e_2}, \vec{e_3})$

Notation		Meaning	Expansion (Cartesian coordinates)
$\vec{\nabla} f$	vector	Gradient of scalar field $f$	$\vec{\nabla} f = \frac{\partial f}{\partial x} \hat{\mathbf{e_1}} + \frac{\partial f}{\partial y} \hat{\mathbf{e_2}} + \frac{\partial f}{\partial z} \hat{\mathbf{e_3}}$
$\vec{\nabla} \cdot \vec{A}$	scalar	Divergence of vector $\vec{A} = (a_1, a_2, a_3)$	$\vec{\nabla} \cdot \vec{A} = \frac{\partial a_1}{\partial x} + \frac{\partial a_2}{\partial y} + \frac{\partial a_3}{\partial z}$
$\vec{\nabla} \times \vec{A}$	vector	Curl of vector $\vec{A} = (a_1, a_2, a_3)$	$\vec{\nabla} \times \vec{A} = \big(\frac{\partial a_3}{\partial y} - \frac{\partial a_2}{\partial z}\big)\hat{\mathbf{e_1}} - \big(\frac{\partial a_3}{\partial x} - \frac{\partial a_1}{\partial z}\big)\hat{\mathbf{e_2}} + \big(\frac{\partial a_2}{\partial x} - \frac{\partial a_1}{\partial y}\big)\hat{\mathbf{e_3}}$
$\vec{A} \cdot \vec{B}$	scalar	Dot product	$\vec{A} \cdot \vec{B} = a_1 b_1 + a_2 b_2 + a_3 b_3$
$\vec{A} \times \vec{B}$	vector	Cross product	$\vec{A} \times \vec{B} = (a_2 b_3 - a_3 b_2)\hat{\mathbf{e_1}} - (a_1 b_3 - a_3 b_1)\hat{\mathbf{e_2}} + (a_1 b_2 - a_2 b_1)\hat{\mathbf{e_3}}$
$\vec{A} \wedge \vec{B}$	bivector	2-D Wedge (exterior) product	$\vec{A} \wedge \vec{B} = \det\begin{vmatrix}a_1 & a_2 \b_1 & b_2 \end{vmatrix} , \hat{\mathbf{e}}_1 \wedge \hat{\mathbf{e}}_2$
$\vec{A} \vec{B}$	scalar + bivector	Geometric product (Geometric Algebra)	$\vec{A}\vec{B} = \vec{A}\cdot\vec{B} + \vec{A}\wedge\vec{B}$