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{|
 
{|
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! colspan="2" style="background: #e4bc7e; font-size: 110%;" | Vector Identities and Operator Definitions
 
 
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|-
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! colspan="2" style="background: #eee;" | Vector Identities
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! style="background: rgb(228, 188, 126) none repeat scroll 0% 0%; -moz-background-clip: -moz-initial; -moz-background-origin: -moz-initial; -moz-background-inline-policy: -moz-initial; font-size: 110%;" colspan="2" | Vector Identities and Operator Definitions
 
|-
 
|-
−
| align="right" style="padding-right: 1em;" | place note here || <math>\bold{x}\cdot \left(\bold{y}\times \bold{z}\right)= \left(\bold{x}\times \bold{y}\right)\cdot \bold{z}</math>
+
! 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 Identities
−
|-  
+
−
| align="right" style="padding-right: 1em;" | place note here || <math>\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) </math>
+
−
|-  
+
−
| align="right" style="padding-right: 1em;" | place note here || <math>\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) </math>
+
−
|-  
+
−
| align="right" style="padding-right: 1em;" | place note here || <math>\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} </math>
+
 
|-
 
|-
−
| align="right" style="padding-right: 1em;" | place note here || <math> \nabla \left( f+g \right)= \nabla f+ \nabla g </math>
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| align="right" style="padding-right: 1em;" | Notes
 +
| Identity
 
|-
 
|-
−
| align="right" style="padding-right: 1em;" | place note here || <math> \nabla \left( f g \right)= f \nabla g+ g\nabla f </math>
+
| align="right" style="padding-right: 1em;" |  
 +
|  
 +
<math>\bold{x}\cdot \left(\bold{y}\times \bold{z}\right)= \left(\bold{x}\times \bold{y}\right)\cdot \bold{z}</math>  
 +
 
 
|-
 
|-
−
| align="right" style="padding-right: 1em;" | place note here || <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>\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) </math>
 
|-
 
|-
−
| align="right" style="padding-right: 1em;" | place note here || <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>\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) </math>
 
|-
 
|-
−
| align="right" style="padding-right: 1em;" | place note here || <math> \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) </math>
+
| align="right" style="padding-right: 1em;" |  
 +
| <math>\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} </math>
 
|-
 
|-
−
| align="right" style="padding-right: 1em;" | place note here || <math> \nabla \cdot \left(\nabla\times \bold{x} \right)= 0 </math>
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| align="right" style="padding-right: 1em;" |  
 +
| <math> \nabla \left( f+g \right)= \nabla f+ \nabla g </math>
 
|-
 
|-
−
| align="right" style="padding-right: 1em;" | place note here || <math> \nabla \times \nabla \bold{x} = 0 </math>
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| align="right" style="padding-right: 1em;" |  
 +
| <math> \nabla \left( f g \right)= f \nabla g+ g\nabla f </math>
 
|-
 
|-
 +
| align="right" style="padding-right: 1em;" |
 +
| <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 \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) </math>
 +
|-
 +
| 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 \nabla \bold{x} = 0 </math>
 
|}
 
|}
 +
 
{|
 
{|
 
|-
 
|-
−
! colspan="2" style="background: #eee;" | Vector Operators in Rectangular Coordinates
+
! 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 Operators in Rectangular Coordinates
 
|-
 
|-
−
| align="right" style="padding-right: 1em;" | place note here || <math>\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}</math>
+
| align="right" style="padding-right: 1em;" | Notes
 +
| Operator
 +
|  
 
|-
 
|-
−
| align="right" style="padding-right: 1em;" | place note here || <math>\nabla \cdot \bold{v} =  \frac{\partial v_x}{\partial x}+\frac{\partial v_y}{\partial y}+ \frac{\partial v_z}{\partial z}</math>  
+
| align="right" style="padding-right: 1em;" |  
 +
| <math>\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}</math>
 
|-
 
|-
−
| align="right" style="padding-right: 1em;" |  place note here || <math>\nabla \times \bold{v} =  \mathbf{\hat x} \left( \frac{\partial v_z}{\partial y}-\frac{\partial v_y}{\partial z} \right) +
+
| align="right" style="padding-right: 1em;" |  
 +
| <math>\nabla \cdot \bold{v} = \frac{\partial v_x}{\partial x}+\frac{\partial v_y}{\partial y}+ \frac{\partial v_z}{\partial z}</math>
 +
|-
 +
| align="right" style="padding-right: 1em;" |  
 +
| <math>\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 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)  
+
\mathbf{\hat z} \left( \frac{\partial v_y}{\partial x}-\frac{\partial v_x}{\partial y} \right) </math>
−
</math>
+
 
|-
 
|-
−
| align="right" style="padding-right: 1em;" | place note here || <math>\nabla^2 f = \frac{\partial^2 f}{\partial x^2}+\frac{\partial^2 f}{\partial y^2}+ \frac{\partial^2 f}{\partial z^2}  
+
| align="right" style="padding-right: 1em;" | <br>
−
</math>  
+
| <math>\nabla^2 f = \frac{\partial^2 f}{\partial x^2}+\frac{\partial^2 f}{\partial y^2}+ \frac{\partial^2 f}{\partial z^2}</math>
−
|-
+
 
|}
 
|}
  
 +
<br>
  
 
{|
 
{|
 
|-
 
|-
−
! colspan="2" style="background: #eee;" | Vector Operators in Spherical Coordinates
+
! 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 Operators in Spherical Coordinates
 
|-
 
|-
−
| align="right" style="padding-right: 1em;" | place note here ||[[Formula_contributed_by_Anshita| <math>\nabla f(\rho,\phi,z) = {\partial f \over \partial \rho}\boldsymbol{\hat \rho}  
+
| align="right" style="padding-right: 1em;" | Notes
 +
| Operator
 +
|-
 +
| align="right" style="padding-right: 1em;" |
 +
| [[Formula contributed by Anshita|<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}  
 
   + {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;" |
 +
| <math>\nabla \cdot \bold{v} =</math>
 +
|-
 +
| align="right" style="padding-right: 1em;" |
 +
| <math>\nabla \times \bold{v} =</math>
 +
|-
 +
| align="right" style="padding-right: 1em;" |
 +
| <math>\nabla^2 f =</math>
 
|}
 
|}
  
 +
<br>
  
 
{|
 
{|
 
|-
 
|-
−
! colspan="2" style="background: #eee;" | Vector Operators in Cylindrical Coordinates
+
! 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 Operators in Cylindrical Coordinates
 
|-
 
|-
−
| align="right" style="padding-right: 1em;" | place note here ||[[Formula_contributed_by_Anshita| <math>\nabla f(x,y,z) = {\partial f \over \partial r}\boldsymbol{\hat r}  
+
| align="right" style="padding-right: 1em;" | Notes
 +
| Operator
 +
|-
 +
| 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}  
 
   + {1 \over r}{\partial f \over \partial \theta}\boldsymbol{\hat \theta}  
 
   + {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>]]
−
|-
+
 
|}
 
|}
  
 
----
 
----
−
[[MegaCollectiveTableTrial1|Back to Collective Table]]
+
 
 +
[[MegaCollectiveTableTrial1|Back to Collective Table]]  
 +
 
 
[[Category:Formulas]]
 
[[Category:Formulas]]

Revision as of 10:01, 2 April 2010

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 Spherical 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} = $
$ \nabla \times \bold{v} = $
$ \nabla^2 f = $


Vector Operators in Cylindrical 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} $

Back to Collective Table

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Alumni Liaison

Ph.D. 2007, working on developing cool imaging technologies for digital cameras, camera phones, and video surveillance cameras.

Buyue Zhang