Difference between revisions of "VectorFormulas" - Rhea

Latest revision as of 10:58, 24 February 2015

keywords: vector, gradient, curl, laplacian

Vector Identities and Operator Definitions

(Used in ECE311)

Vector Identities and Operator Definitions
Vector Identities
Notes	Identity
	$\bold{x}\cdot \left(\bold{y}\times \bold{z}\right)= \left(\bold{x}\times \bold{y}\right)\cdot \bold{z}$
	$\bold{x}\times \left(\bold{y}\times \bold{z} \right)=\bold{y}\left(\bold{x} \cdot \bold{z} \right)-\bold{z} \left( \bold{x}\cdot\bold{y}\right)$
	$\left( \bold{x}\times \bold{y}\right)\cdot \left(\bold{z}\times \bold{w} \right)=\left( \bold{x}\cdot \bold{z}\right) \left(\bold{y} \cdot \bold{w} \right)- \left(\bold{x}\cdot\bold{w} \right) \left( \bold{y}\cdot\bold{z}\right)$
	$\nabla \left( \bold{x}\cdot \bold{y}\right)= \bold{y}\times \left(\nabla\times \bold{x}\right)+ \bold{x} \times \left(\nabla\times \bold{y} \right)+ \left(\bold{y}\cdot\nabla \right)\bold{x} + \left( \bold{x}\cdot\nabla\right) \bold{y}$
	$\nabla \left( f+g \right)= \nabla f+ \nabla g$
	$\nabla \left( f g \right)= f \nabla g+ g\nabla f$
	$\nabla \cdot \left( \bold{x}+\bold{y} \right)= \nabla \cdot \bold{x} + \nabla \cdot \bold{y}$
	$\nabla \cdot \left( f \bold{x}\right)= \bold{x} \cdot \nabla f + f \left( \nabla \cdot\bold{x} \right)$
	$\nabla \times \left( \bold{x} + \bold{y} \right)= \nabla \times \bold{x} + \nabla \times \bold{y}$
	$\nabla \times \left( u \bold{x} \right)= \left( \nabla u \right) \times \bold{x} + u \left( \nabla \times \bold{x} \right)$
	$\nabla \cdot \left( \bold{x}\times \bold{y}\right)= \bold{y} \cdot \left( \nabla \times \bold{x}\right) - \bold{x} \cdot \left( \nabla \times \bold{y}\right)$
	$\nabla \cdot \left(\nabla\times \bold{x} \right)= 0$
	$\nabla \times \left( \bold{x} \times \bold{y} \right) = \left( \nabla \cdot \bold{y} \right) \bold{x} - \left( \nabla \cdot \bold{x} \right) \bold{y} + \left( \bold{y} \cdot \nabla \right) \bold{x} - \left( \bold{x} \cdot \nabla \right) \bold{y}$
	$\nabla \times \nabla \bold{x} = 0$
	$\nabla ( \bold{C} \cdot \bold{r} ) = \bold{C} \qquad \text{where }\bold{C}\text{ is a constant (real or complex)}$
	$\nabla \times \left( \nabla \times \bold{x} \right) = \nabla \left( \nabla \cdot \bold{x} \right) - \nabla^2 \bold{x}$
	$\left( \bold{A} \cdot \nabla \right) \bold{B} = \hat{\bold{x}} ( \bold{A}_x \frac{\partial \bold{B}_x}{\partial x} + \bold{A}_y \frac{\partial \bold{B}_x}{\partial y} + \bold{A}_z \frac{\partial \bold{B}_x}{\partial z} ) + \hat{\bold{y}} ( \bold{A}_x \frac{\partial \bold{B}_y}{\partial x} + \bold{A}_y \frac{\partial \bold{B}_y}{\partial y} + \bold{A}_z \frac{\partial \bold{B}_y}{\partial z} ) + \hat{\bold{z}} ( \bold{A}_x \frac{\partial \bold{B}_z}{\partial x} + \bold{A}_y \frac{\partial \bold{B}_z}{\partial y} + \bold{A}_z \frac{\partial \bold{B}_z}{\partial z} )$
	$\frac{d \left( \bold{x} \cdot \bold{y} \right)}{d\sigma} =\frac{d \bold{y}}{d\sigma}\cdot \bold{x} + \frac{d \bold{x}}{d\sigma}\cdot \bold{y}$
	$\frac{d \left( \bold{x} \times \bold{y} \right)}{d\sigma} =\frac{d \bold{y}}{d\sigma}\times \bold{x} + \frac{d \bold{x}}{d\sigma}\times \bold{y}$
	$\frac {d ( u \bold{v} )}{d \sigma} = \frac {d u}{d \sigma} \bold{v} + u \frac{d \bold{v}}{d \sigma}$

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
	$\nabla f(\rho,\phi,z) = {\partial f \over \partial \rho}\boldsymbol{\hat \rho} + {1 \over \rho}{\partial f \over \partial \phi}\boldsymbol{\hat \phi} + {\partial f \over \partial z}\boldsymbol{\hat z}$
	$\nabla \cdot \bold{v} = \frac{1}{\rho} \frac{\partial \rho v_{\rho}}{\partial \rho} + \frac{1}{\rho} \frac{\partial v_{\phi}}{\partial \phi} + \frac{\partial v_z}{\partial z}$
	$\nabla \times \bold{v} = \boldsymbol{\hat \rho} ( \frac{1}{\rho} \frac{\partial \bold{v}_z }{\partial \phi} - \frac{\partial \bold{v}_\phi}{\partial z} ) + \boldsymbol{\hat \phi} ( \frac{\partial \bold{v}_\rho}{\partial z} - \frac{\partial \bold{v}_z}{\partial \rho} ) + \hat{\bold{z}} ( \frac{1}{\rho} \frac{\partial ( \rho \bold{v}_\phi )}{\partial \rho} - \frac{1}{\rho} \frac{\partial \bold{v}_\rho}{\partial \phi} )$
	$\nabla^2 f = \frac{1}{\rho} \frac{\partial }{\partial \rho} \left( \rho \frac{\partial f}{\partial \rho}\right) + \frac{1}{\rho^2} \frac{\partial^2 f}{\partial \phi^2} + \frac{\partial^2 f}{\partial z^2}$

Vector Operators in Spherical Coordinates
Notes	Operator
	$\nabla f(x,y,z) = {\partial f \over \partial r}\boldsymbol{\hat r} + {1 \over r}{\partial f \over \partial \theta}\boldsymbol{\hat \theta} + {1 \over r\sin\theta}{\partial f \over \partial \phi}\boldsymbol{\hat \phi}$
	$\nabla \cdot \bold{v} = \frac{1}{r^2} \frac{\partial r^2 v_r}{\partial r} + \frac{1}{r\sin\theta} \frac{\partial \sin\theta v_{\theta}}{\partial \theta} + \frac{1}{r\sin\theta} \frac{\partial v_{\phi}}{\partial \phi}$
	$\nabla \times \bold{v} = \frac{\boldsymbol{\hat r } }{r \sin \theta} [ \frac{\partial ( \sin \theta \bold{v}_\phi )}{\partial \theta} - \frac{\partial \bold{v}_\theta}{\partial \phi} ] + \frac { \boldsymbol{\hat \theta} }{r} [ \frac{1}{\sin \theta} \frac{\partial \bold{v}_r}{\partial \phi} - \frac{\partial ( r \bold{v}_\phi )}{\partial r} ] + \frac {\boldsymbol{\hat \phi} }{r} [ \frac{\partial ( r \bold{v}_\theta )}{\partial r} - \frac{\partial \bold{v}_r}{\partial \theta} ]$
	$\nabla^2 f = \frac{1}{r^2} \frac{\partial }{\partial r} \left( r^2 \frac{\partial f}{\partial r}\right) + \frac{1}{r^2 \sin \theta} \frac{\partial }{\partial \theta} \left(\sin \theta \frac{\partial f}{\partial \theta} \right) + \frac{1}{r^2 \sin^2 \theta}\frac{\partial^2 f}{\partial \phi^2}$

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
	$\frac{\partial f ( \bold{R} )}{\partial x} = - \frac{\partial f ( \bold{R} )}{\partial x^{'}}$
	$\nabla f ( \bold{R} ) = - \nabla^{'} f ( \bold{R} )$
	$\nabla \cdot \bold{A} ( \bold{R} ) = - \nabla^{'} \cdot \bold{A} ( \bold{R} )$
	$\nabla \times \bold{A} ( \bold{R} ) = - \nabla^{'} \times \bold{A} ( \bold{R} )$
	$\nabla^2 f ( \bold{R} )= \nabla^{'2} f ( \bold{R} )$
	$\nabla R = - \nabla^{'} R = \frac{\bold{R}}{R} = \hat{\bold{R}}$
	$\nabla ( \frac{1}{R} ) = - \nabla^{'} ( \frac{1}{R} ) = - \frac{\hat{\bold{R}}}{R^2} = - \frac{\bold{R}}{R^3}$
	$\nabla^2 ( \frac{1}{R} )= \nabla^{'2} ( \frac{1}{R} ) = 0 \qquad ( \ R \neq 0 \ )$

Back to Collective Table

@@ Line 1: / Line 1: @@
+[[Category:Formulas]]
+keywords: vector, gradient, curl, laplacian
+<center><font size= 4>
+'''[[Collective_Table_of_Formulas|Collective Table of Formulas]]'''
+</font size>
+'''Vector Identities and Operator Definitions'''
+(Used in [[ECE311]])
+</center>
+----
 {|
 |-
@@ Line 29: / Line 45: @@
 |-
 | align="right" style="padding-right: 1em;" |
-| <math> \nabla \cdot \left( \bold{x}+\bold{y} \right)= \nabla \cdot \bold{x}+ \nabla \cdot \bold{y}  </math>
+| <math> \nabla \cdot \left( \bold{x}+\bold{y} \right)= \nabla \cdot \bold{x} + \nabla \cdot \bold{y}  </math>
 |-
 | align="right" style="padding-right: 1em;" |
 | <math> \nabla \cdot \left( f \bold{x}\right)= \bold{x} \cdot \nabla f + f \left( \nabla \cdot\bold{x} \right) </math>
+|-
+| align="right" style="padding-right: 1em;" |
+| <math> \nabla \times \left( \bold{x} + \bold{y} \right)= \nabla \times \bold{x} + \nabla \times \bold{y} </math>
+|-
+| align="right" style="padding-right: 1em;" |
+| <math> \nabla \times \left( u \bold{x} \right)= \left( \nabla u \right) \times \bold{x} + u \left( \nabla \times \bold{x} \right) </math>
 |-
 | align="right" style="padding-right: 1em;" |
@@ Line 38: / Line 60: @@
 |-
 | align="right" style="padding-right: 1em;" |
 | <math> \nabla \cdot \left(\nabla\times \bold{x} \right)= 0 </math>
 |-
 | align="right" style="padding-right: 1em;" |
+| <math> \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}</math>
+|-
+| align="right" style="padding-right: 1em;" |
 | <math> \nabla \times \nabla \bold{x} = 0 </math>
+|-
+| align="right" style="padding-right: 1em;" |
+| <math> \nabla ( \bold{C} \cdot \bold{r} ) = \bold{C}  \qquad  \text{where }\bold{C}\text{ is a constant (real or complex)}</math>
+|-
+| align="right" style="padding-right: 1em;" |
+| <math> \nabla \times \left( \nabla \times \bold{x} \right) = \nabla \left( \nabla \cdot \bold{x} \right) - \nabla^2 \bold{x}</math>
+|-
+| align="right" style="padding-right: 1em;" |
+| <math> \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} )</math>
+|-
+| align="right" style="padding-right: 1em;" |
+| <math> \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}</math>
+|-
+| align="right" style="padding-right: 1em;" |
+| <math> \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}</math>
+|-
+| align="right" style="padding-right: 1em;" |
+| <math> \frac {d ( u \bold{v} )}{d \sigma} = \frac {d u}{d \sigma} \bold{v} + u \frac{d \bold{v}}{d \sigma}</math>
 |}
@@ Line 77: / Line 120: @@
 |-
 | align="right" style="padding-right: 1em;" |
-| [[Formula contributed by Anshita|<math>\nabla f(\rho,\phi,z) = {\partial f \over \partial \rho}\boldsymbol{\hat \rho}
+| <math>\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}</math> ]]
+   + {\partial f \over \partial z}\boldsymbol{\hat z}</math>
 |-
 | align="right" style="padding-right: 1em;" |
@@ Line 85: / Line 128: @@
 \frac{1}{\rho} \frac{\partial \rho v_{\rho}}{\partial \rho}
 +
 \frac{1}{\rho} \frac{\partial  v_{\phi}}{\partial \phi}
 +
@@ Line 91: / Line 133: @@
 |-
 | align="right" style="padding-right: 1em;" |
-| <math>\nabla \times \bold{v} =</math>
+| <math>\nabla \times \bold{v} = \boldsymbol{\hat \rho}  ( \frac{1}{\rho} \frac{\partial \bold{v}_z }{\partial \phi} - \frac{\partial \bold{v}_\phi}{\partial z} ) + \boldsymbol{\hat \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} )</math>
 |-
 | align="right" style="padding-right: 1em;" |
@@ Line 114: / Line 156: @@
 |-
 | align="right" style="padding-right: 1em;" | <br>
-| [[Formula contributed by Anshita|<math>\nabla f(x,y,z) = {\partial f \over \partial r}\boldsymbol{\hat r}
+| <math>\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}</math>]]
+   + {1 \over r\sin\theta}{\partial f \over \partial \phi}\boldsymbol{\hat \phi}</math>
 |-
 | align="right" style="padding-right: 1em;" |
@@ Line 128: / Line 170: @@
 |-
 | align="right" style="padding-right: 1em;" |
-| <math>\nabla \times \bold{v} =</math>
+| <math>\nabla \times \bold{v} = \frac{\boldsymbol{\hat r } }{r \sin \theta} [ \frac{\partial ( \sin \theta \bold{v}_\phi )}{\partial \theta} - \frac{\partial \bold{v}_\theta}{\partial \phi} ] + \frac { \boldsymbol{\hat \theta} }{r} [ \frac{1}{\sin \theta} \frac{\partial \bold{v}_r}{\partial \phi} - \frac{\partial ( r \bold{v}_\phi )}{\partial r} ] + \frac {\boldsymbol{\hat \phi} }{r} [ \frac{\partial ( r \bold{v}_\theta )}{\partial r} - \frac{\partial \bold{v}_r}{\partial \theta} ]</math>
 |-
 | align="right" style="padding-right: 1em;" |
@@ Line 141: / Line 183: @@
 \frac{1}{r^2 \sin^2 \theta}\frac{\partial^2  f}{\partial \phi^2}</math>
 |}
+<br>
-----
+{|
-This list of formulas is useful for ECE 311, an electro-magnetics course.
+|-
+! style="background: rgb(238, 238, 238) none repeat scroll 0% 0%; -moz-background-clip: -moz-initial; -moz-background-origin: -moz-initial; -moz-background-inline-policy: -moz-initial;" colspan="2" | Vector Integral formulas
+|-
+| align="right" style="padding-right: 1em;" | Notes
+| Operator
+|-
+| align="right" style="padding-right: 1em;" | <br>
+| <math>\oint_S \bold{A} \cdot d \bold{a} = \int_V \nabla \cdot \bold{A} d \tau \qquad \text{(Divergence therorem)}</math>
+|-
+| align="right" style="padding-right: 1em;" | <br>
+| <math>\oint_C \bold{A} \cdot d \bold{s} = \int_S ( \nabla \times \bold{A} ) \cdot d \bold{a} \qquad \textrm{(Stokes'  therorem)}</math>
+|-
+| align="right" style="padding-right: 1em;" | <br>
+| <math>\oint_S u d \bold{a} = \int_V \nabla u d \tau</math>
+|-
+| align="right" style="padding-right: 1em;" | <br>
+| <math>\oint_S \bold{A} \times d \bold{a} = - \int_V ( \nabla \times \bold{A} ) d \tau </math>
+|-
+| align="right" style="padding-right: 1em;" | <br>
+| <math>\oint_C u d \bold{s} = - \int_S \nabla u \times d \bold{a} </math>
+|-
+| align="right" style="padding-right: 1em;" | <br>
+| <math>\oint_S u \bold{A} \cdot d \bold{a} = \int_V [ \bold{A} \cdot ( \nabla u ) + u ( \nabla \cdot \bold{A} ) ]d \tau </math>
+|-
+| align="right" style="padding-right: 1em;" | <br>
+| <math>\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 </math>
+|-
-[[ECE311| Go to ECE 311]]
+|}
+<br>
-[[Collective_Table_of_Formulas|Back to Collective Table]]
+{|
+|-
-[[Category:Formulas]]
+! style="background: rgb(238, 238, 238) none repeat scroll 0% 0%; -moz-background-clip: -moz-initial; -moz-background-origin: -moz-initial; -moz-background-inline-policy: -moz-initial;" colspan="2" | Formulas Involving Relative Coordinates
+|-
+| align="right" style="padding-right: 1em;" | Notes
+| Operator
+|-
+| align="right" style="padding-right: 1em;" | <br>
+| <math> \frac{\partial f ( \bold{R} )}{\partial x} = - \frac{\partial f ( \bold{R} )}{\partial x^{'}}</math>
+|-
+| align="right" style="padding-right: 1em;" | <br>
+|<math> \nabla f ( \bold{R} ) = - \nabla^{'} f ( \bold{R} )</math>
+|-
+| align="right" style="padding-right: 1em;" | <br>
+| <math>\nabla \cdot \bold{A} ( \bold{R} ) = - \nabla^{'} \cdot \bold{A} ( \bold{R} )</math>
+|-
+| align="right" style="padding-right: 1em;" | <br>
+| <math>\nabla \times \bold{A} ( \bold{R} ) = - \nabla^{'} \times \bold{A} ( \bold{R} )</math>
+|-
+| align="right" style="padding-right: 1em;" | <br>
+| <math>\nabla^2 f ( \bold{R} )= \nabla^{'2} f ( \bold{R} ) </math>
+|-
+| align="right" style="padding-right: 1em;" | <br>
+| <math>\nabla R = - \nabla^{'} R = \frac{\bold{R}}{R} = \hat{\bold{R}}</math>
+|-
+| align="right" style="padding-right: 1em;" | <br>
+| <math>\nabla ( \frac{1}{R} ) = - \nabla^{'} ( \frac{1}{R} ) = - \frac{\hat{\bold{R}}}{R^2} = - \frac{\bold{R}}{R^3}</math>
+|-
+| align="right" style="padding-right: 1em;" | <br>
+| <math>\nabla^2 ( \frac{1}{R} )= \nabla^{'2} ( \frac{1}{R} ) = 0 \qquad ( \  R \neq 0 \  )</math>
+|-
+|}
+----
+[[Collective_Table_of_Formulas|Back to Collective Table]]

Difference between revisions of "VectorFormulas" - Rhea

Latest revision as of 10:58, 24 February 2015

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