keywords: vector, gradient, curl, laplacian
Vector Identities and Operator Definitions
(Used in ECE311)
| Vector Identities and Operator Definitions | |
|---|---|
| Vector Identities | |
| Notes | Identity |
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$ \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) $ | ||
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$ \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 |
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$ \oint_S \bold{A} \cdot d \bold{a} = \int_V \nabla \cdot \bold{A} d \tau \qquad \text{(Divergence therorem)} $ |
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$ \oint_C \bold{A} \cdot d \bold{s} = \int_S ( \nabla \times \bold{A} ) \cdot d \bold{a} \qquad \textrm{(Stokes' therorem)} $ |
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$ \oint_S u d \bold{a} = \int_V \nabla u d \tau $ |
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$ \oint_S \bold{A} \times d \bold{a} = - \int_V ( \nabla \times \bold{A} ) d \tau $ |
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$ \oint_C u d \bold{s} = - \int_S \nabla u \times d \bold{a} $ |
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$ \oint_S u \bold{A} \cdot d \bold{a} = \int_V [ \bold{A} \cdot ( \nabla u ) + u ( \nabla \cdot \bold{A} ) ]d \tau $ |
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$ \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 |
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$ \frac{\partial f ( \bold{R} )}{\partial x} = - \frac{\partial f ( \bold{R} )}{\partial x^{'}} $ |
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$ \nabla f ( \bold{R} ) = - \nabla^{'} f ( \bold{R} ) $ |
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$ \nabla \cdot \bold{A} ( \bold{R} ) = - \nabla^{'} \cdot \bold{A} ( \bold{R} ) $ |
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$ \nabla \times \bold{A} ( \bold{R} ) = - \nabla^{'} \times \bold{A} ( \bold{R} ) $ |
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$ \nabla^2 f ( \bold{R} )= \nabla^{'2} f ( \bold{R} ) $ |
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$ \nabla R = - \nabla^{'} R = \frac{\bold{R}}{R} = \hat{\bold{R}} $ |
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$ \nabla ( \frac{1}{R} ) = - \nabla^{'} ( \frac{1}{R} ) = - \frac{\hat{\bold{R}}}{R^2} = - \frac{\bold{R}}{R^3} $ |
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$ \nabla^2 ( \frac{1}{R} )= \nabla^{'2} ( \frac{1}{R} ) = 0 \qquad ( \ R \neq 0 \ ) $ |
