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Vector Identities and Operator Definitions
Vector Identities
Notes Identity

$ \bold{x}\cdot \left(\bold{y}\times \bold{z}\right)= \left(\bold{x}\times \bold{y}\right)\cdot \bold{z} $

$ \bold{x}\times \left(\bold{y}\times \bold{z} \right)=\bold{y}\left(\bold{x} \cdot \bold{z} \right)-\bold{z} \left( \bold{x}\cdot\bold{y}\right) $
$ \left( \bold{x}\times \bold{y}\right)\cdot \left(\bold{z}\times \bold{w} \right)=\left( \bold{x}\cdot \bold{z}\right) \left(\bold{y} \cdot \bold{w} \right)- \left(\bold{x}\cdot\bold{w} \right) \left( \bold{y}\cdot\bold{z}\right) $
$ \nabla \left( \bold{x}\cdot \bold{y}\right)= \bold{y}\times \left(\nabla\times \bold{x}\right)+ \bold{x} \times \left(\nabla\times \bold{y} \right)+ \left(\bold{y}\cdot\nabla \right)\bold{x} + \left( \bold{x}\cdot\nabla\right) \bold{y} $
$ \nabla \left( f+g \right)= \nabla f+ \nabla g $
$ \nabla \left( f g \right)= f \nabla g+ g\nabla f $
$ \nabla \cdot \left( \bold{x}+\bold{y} \right)= \nabla \cdot \bold{x}+ \nabla \cdot \bold{y} $
$ \nabla \cdot \left( f \bold{x}\right)= \bold{x} \cdot \nabla f + f \left( \nabla \cdot\bold{x} \right) $
$ \nabla \cdot \left( \bold{x}\times \bold{y}\right)= \bold{y} \cdot \left( \nabla \times \bold{x}\right) - \bold{x} \cdot \left( \nabla \times \bold{y}\right) $
$ \nabla \cdot \left(\nabla\times \bold{x} \right)= 0 $
$ \nabla \times \nabla \bold{x} = 0 $
Vector Operators in Rectangular Coordinates
Notes Operator
$ \nabla f(x,y,z) = \mathbf{\hat x} \frac{\partial f}{\partial x}+\mathbf{\hat y}\frac{\partial f}{\partial y}+\mathbf{\hat z} \frac{\partial f}{\partial z} $
$ \nabla \cdot \bold{v} = \frac{\partial v_x}{\partial x}+\frac{\partial v_y}{\partial y}+ \frac{\partial v_z}{\partial z} $
$ \nabla \times \bold{v} = \mathbf{\hat x} \left( \frac{\partial v_z}{\partial y}-\frac{\partial v_y}{\partial z} \right) + \mathbf{\hat y} \left( \frac{\partial v_x}{\partial z}-\frac{\partial v_z}{\partial x} \right) + \mathbf{\hat z} \left( \frac{\partial v_y}{\partial x}-\frac{\partial v_x}{\partial y} \right) $

$ \nabla^2 f = \frac{\partial^2 f}{\partial x^2}+\frac{\partial^2 f}{\partial y^2}+ \frac{\partial^2 f}{\partial z^2} $


Vector Operators in Cylindrical Coordinates
Notes Operator
$ \nabla f(\rho,\phi,z) = {\partial f \over \partial \rho}\boldsymbol{\hat \rho} + {1 \over \rho}{\partial f \over \partial \phi}\boldsymbol{\hat \phi} + {\partial f \over \partial z}\boldsymbol{\hat z} $
$ \nabla \cdot \bold{v} = \frac{1}{\rho} \frac{\partial \rho v_{\rho}}{\partial \rho} + \frac{1}{\rho} \frac{\partial v_{\phi}}{\partial \phi} + \frac{\partial v_z}{\partial z} $
$ \nabla \times \bold{v} = $
$ \nabla^2 f = $


Vector Operators in Spherical Coordinates
Notes Operator

$ \nabla f(x,y,z) = {\partial f \over \partial r}\boldsymbol{\hat r} + {1 \over r}{\partial f \over \partial \theta}\boldsymbol{\hat \theta} + {1 \over r\sin\theta}{\partial f \over \partial \phi}\boldsymbol{\hat \phi} $
$ \nabla \cdot \bold{v} = \frac{1}{r^2} \frac{\partial r^2 v_r}{\partial r} + \frac{1}{r\sin\theta} \frac{\partial \sin\theta v_{\theta}}{\partial \theta} + \frac{1}{r\sin\theta} \frac{\partial v_{\phi}}{\partial \phi} $

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Ryne Rayburn