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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" 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%;" | Vector Identities and Operator Definitions |
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| − | ! 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" 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;" | Vector Identities |
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| align="right" style="padding-right: 1em;" | Notes | | align="right" style="padding-right: 1em;" | Notes | ||
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{| | {| | ||
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| − | ! 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" 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;" | Vector Operators in Rectangular Coordinates |
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| align="right" style="padding-right: 1em;" | Notes | | align="right" style="padding-right: 1em;" | Notes | ||
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{| | {| | ||
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| − | ! 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" 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;" | Vector Operators in Cylindrical Coordinates |
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| align="right" style="padding-right: 1em;" | Notes | | align="right" style="padding-right: 1em;" | Notes | ||
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| align="right" style="padding-right: 1em;" | | | align="right" style="padding-right: 1em;" | | ||
| − | | <math>\nabla \cdot \bold{v} =</math> | + | | <math>\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}</math> | ||
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| align="right" style="padding-right: 1em;" | | | align="right" style="padding-right: 1em;" | | ||
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| − | ! 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" 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;" | Vector Operators in Spherical Coordinates |
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| align="right" style="padding-right: 1em;" | Notes | | align="right" style="padding-right: 1em;" | Notes | ||
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+ {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>]] | ||
| + | |- | ||
| + | | align="right" style="padding-right: 1em;" | | ||
| + | | <math>\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}</math> | ||
| + | |- | ||
| + | | align="right" style="padding-right: 1em;" | | ||
| + | | | ||
| + | |- | ||
| + | | align="right" style="padding-right: 1em;" | | ||
| + | | | ||
|} | |} | ||
Revision as of 10:12, 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 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} $ | |
