| Line 24: | Line 24: | ||
! colspan="2" style="background: #eee;" | Vector Operators in Rectangular Coordinates | ! colspan="2" style="background: #eee;" | Vector Operators in Rectangular Coordinates | ||
|- | |- | ||
| − | | align="right" style="padding-right: 1em;" | place note here || <math>\nabla f(x,y,z) = \ | + | | 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> |
|- | |- | ||
|} | |} | ||
| Line 33: | Line 33: | ||
! colspan="2" style="background: #eee;" | Vector Operators in Spherical Coordinates | ! colspan="2" style="background: #eee;" | Vector Operators in Spherical Coordinates | ||
|- | |- | ||
| − | | align="right" style="padding-right: 1em;" | place note here ||[[Formula_contributed_by_Anshita| <math>\nabla f( | + | | 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} |
+ {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> ]] | ||
Revision as of 15:20, 4 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( 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) = $ |
