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| 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>\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> | ||
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| + | | 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> | ||
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| align="right" style="padding-right: 1em;" | place note here || <math> \nabla \left( f+g \right)= \nabla f+ \nabla g </math> | | align="right" style="padding-right: 1em;" | place note here || <math> \nabla \left( f+g \right)= \nabla f+ \nabla g </math> | ||
Revision as of 11:16, 19 November 2009
| Vector Identities and Operator Definitions | |
|---|---|
| Vector Identities | |
| place note here | $ \bold{x}\cdot \left(\bold{y}\times \bold{z}\right)= \left(\bold{x}\times \bold{y}\right)\cdot \bold{z} $ |
| place note here | $ \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) $ |
| place note here | $ \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) $ |
| place note here | $ \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} $ |
| place note here | $ \nabla \left( f+g \right)= \nabla f+ \nabla g $ |
| place note here | $ \nabla \left( f g \right)= f \nabla g+ g\nabla f $ |
| place note here | $ \nabla \cdot \left(\nabla\times \bold{x} \right)= 0 $ |
| place note here | $ \nabla \times \nabla \bold{x} = 0 $ |
| Vector Operators in Rectangular Coordinates | |
|---|---|
| place note here | $ \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} $ |
| Vector Operators in Spherical Coordinates | |
|---|---|
| place note here | $ \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} $ |
| Vector Operators in Cylindrical Coordinates | |
|---|---|
| place note here | $ \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} $ |
