Ethan's Notes

    Vector Identities

    #Math

    Topics

    Geometric Algebra

    Product TypeNotationReturnsExample$=$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 * 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_1e_2$2
    Geometric$ab$Scalar + Bivector$(e_1+e_2)e_2$$e_1e_2+e_2e_2 = e_1e_2 + 1$3

    Formulas

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

    • $\displaystyle f$ and $\displaystyle g$ are scalar fields

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

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

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

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

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

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

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

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

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

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