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'''This [[Collective Table of Formulas|Collective table of formulas]] is proudly sponsored'''<br> '''by the [http://www.facebook.com/hkn.beta Nice Guys of Eta Kappa Nu].''' <br><br> Visit us at the HKN Lounge in EE24 for hot coffee and fresh bagels only $1 each!
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'''This [[Collective Table of Formulas|Collective table of formulas]] is proudly sponsored'''<br> '''by the [http://www.facebook.com/hkn.beta Nice Guys of Eta Kappa Nu].''' <br><br> Visit us at the [[HKN|HKN Lounge]] in EE24 for hot coffee and fresh bagels only $1 each!
  
 
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Revision as of 06:55, 24 April 2012

This Collective table of formulas is proudly sponsored
by the Nice Guys of Eta Kappa Nu.

Visit us at the HKN Lounge in EE24 for hot coffee and fresh bagels only $1 each!

                                         HKNlogo.jpg
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 \times \left( \bold{x} + \bold{y} \right)= \nabla \times \bold{x} + \nabla \times \bold{y} $
$ \nabla \times \left( u \bold{x} \right)= \left( \nabla u \right) \times \bold{x} + u \left( \nabla \times \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 \left( \bold{x} \times \bold{y} \right) = \left( \nabla \cdot \bold{y} \right) \bold{x} - \left( \nabla \cdot \bold{x} \right) \bold{y} + \left( \bold{y} \cdot \nabla \right) \bold{x} - \left( \bold{x} \cdot \nabla \right) \bold{y} $
$ \nabla \times \nabla \bold{x} = 0 $
$ \nabla ( \bold{C} \cdot \bold{r} ) = \bold{C} \qquad \text{where} \ \bold{C} = \text{const} $
$ \nabla \times \left( \nabla \times \bold{x} \right) = \nabla \left( \nabla \cdot \bold{x} \right) - \nabla^2 \bold{x} $
$ \left( \bold{A} \cdot \nabla \right) \bold{B} = \hat{\bold{x}} ( \bold{A}_x \frac{\partial \bold{B}_x}{\partial x} + \bold{A}_y \frac{\partial \bold{B}_x}{\partial y} + \bold{A}_z \frac{\partial \bold{B}_x}{\partial z} ) + \hat{\bold{y}} ( \bold{A}_x \frac{\partial \bold{B}_y}{\partial x} + \bold{A}_y \frac{\partial \bold{B}_y}{\partial y} + \bold{A}_z \frac{\partial \bold{B}_y}{\partial z} ) + \hat{\bold{z}} ( \bold{A}_x \frac{\partial \bold{B}_z}{\partial x} + \bold{A}_y \frac{\partial \bold{B}_z}{\partial y} + \bold{A}_z \frac{\partial \bold{B}_z}{\partial z} ) $
$ \frac{d \left( \bold{x} \cdot \bold{y} \right)}{d\sigma} =\frac{d \bold{y}}{d\sigma}\cdot \bold{x} + \frac{d \bold{x}}{d\sigma}\cdot \bold{y} $
$ \frac{d \left( \bold{x} \times \bold{y} \right)}{d\sigma} =\frac{d \bold{y}}{d\sigma}\times \bold{x} + \frac{d \bold{x}}{d\sigma}\times \bold{y} $
$ \frac {d ( u \bold{v} )}{d \sigma} = \frac {d u}{d \sigma} \bold{v} + u \frac{d \bold{v}}{d \sigma} $
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} = \hat\bold{\rho} ( \frac{1}{\rho} \frac{\partial \bold{v}_z }{\partial \phi} - \frac{\partial \bold{v}_\phi}{\partial z} ) + \hat\bold{\phi} ( \frac{\partial \bold{v}_\rho}{\partial z} - \frac{\partial \bold{v}_z}{\partial \rho} ) + \hat\bold{z} ( \frac{1}{\rho} \frac{\partial ( \rho \bold{v}_\phi )}{\partial \rho} - \frac{1}{\rho} \frac{\partial \bold{v}_\rho}{\partial \phi} ) $
$ \nabla^2 f = \frac{1}{\rho} \frac{\partial }{\partial \rho} \left( \rho \frac{\partial f}{\partial \rho}\right) + \frac{1}{\rho^2} \frac{\partial^2 f}{\partial \phi^2} + \frac{\partial^2 f}{\partial z^2} $


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} $
$ \nabla \times \bold{v} = \frac{\hat \bold{r}}{r \sin \theta} [ \frac{\partial ( \sin \theta \bold{v}_\phi )}{\partial \theta} - \frac{\partial \bold{v}_\theta}{\partial \phi} ] + \frac {\hat \bold{\theta}}{r} [ \frac{1}{\sin \theta} \frac{\partial \bold{v}_r}{\partial \phi} - \frac{\partial ( r \bold{v}_\phi )}{\partial r} ] + \frac {\hat \bold{\phi}}{r} [ \frac{\partial ( r \bold{v}_\theta )}{\partial r} - \frac{\partial \bold{v}_r}{\partial \theta} ] $
$ \nabla^2 f = \frac{1}{r^2} \frac{\partial }{\partial r} \left( r^2 \frac{\partial f}{\partial r}\right) + \frac{1}{r^2 \sin \theta} \frac{\partial }{\partial \theta} \left(\sin \theta \frac{\partial f}{\partial \theta} \right) + \frac{1}{r^2 \sin^2 \theta}\frac{\partial^2 f}{\partial \phi^2} $


Vector Integral formulas
Notes Operator

$ \oint_S \bold{A} \cdot d \bold{a} = \int_V \nabla \cdot \bold{A} d \tau \qquad \text{(Divergence therorem)} $

$ \oint_C \bold{A} \cdot d \bold{s} = \int_S ( \nabla \times \bold{A} ) \cdot d \bold{a} \qquad \textrm{(Stokes'\ therorem)} $

$ \oint_S u d \bold{a} = \int_V \nabla u d \tau $

$ \oint_S \bold{A} \times d \bold{a} = - \int_V ( \nabla \times \bold{A} ) d \tau $

$ \oint_C u d \bold{s} = - \int_S \nabla u \times d \bold{a} $

$ \oint_S u \bold{A} \cdot d \bold{a} = \int_V [ \bold{A} \cdot ( \nabla u ) + u ( \nabla \cdot \bold{A} ) ]d \tau $

$ \oint_S \bold{B} ( \bold{A} \cdot d \bold{a} ) = \int_V [( \bold{A} \cdot \nabla ) \bold{B} + \bold{B} ( \nabla \cdot \bold{A}) ] d \tau $


Formulas Involving Relative Coordinates
Notes Operator

$ \frac{\partial f ( \bold{R} )}{\partial x} = - \frac{\partial f ( \bold{R} )}{\partial x^'} $

$ \nabla f ( \bold{R} ) = - \nabla^' f ( \bold{R} ) $

$ \nabla \cdot \bold{A} ( \bold{R} ) = - \nabla^' \cdot \bold{A} ( \bold{R} ) $

$ \nabla \times \bold{A} ( \bold{R} ) = - \nabla^' \times \bold{A} ( \bold{R} ) $

$ \nabla^2 f ( \bold{R} )= \nabla^{'2} f ( \bold{R} ) $

$ \nabla R = - \nabla^' R = \frac{\bold{R}}{R} = \hat{\bold{R}} $

$ \nabla ( \frac{1}{R} ) = - \nabla^' ( \frac{1}{R} ) = - \frac{\hat{\bold{R}}}{R^2} = - \frac{\bold{R}}{R^3} $

$ \nabla^2 ( \frac{1}{R} )= \nabla^{'2} ( \frac{1}{R} ) = 0 \qquad ( \ R \neq 0 \ ) $

Go to Relevant Course Page: ECE 311

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