Vector Calculus (14) Gradient

35. Gradient Operator (Vector Differential Operator)

 

Scalar Function $ \phi (x_{1},x_{2},x_{3}) $

$ \phi' (x_{1}',x_{2}',x_{3}')=\phi (x_{1},x_{2},x_{3}) $: Coordinate Transformation

$ \frac{\partial \phi '}{\partial x_{i}'}=\sum_{j}^{}\frac{\partial \phi }{\partial x_{j}}\frac{\partial x_{j}}{\partial x_{i}'} $: Chain Rule

 

$ x_{j}=\sum_{k}^{}\lambda _{kj}x_{k}' $: Inverse Coordinate Transformation

$ \frac{\partial x_{j}}{\partial x_{i}'}=\frac{\partial}{\partial x_{i}'}(\sum_{k}^{}\lambda _{kj}x_{k}')=\sum_{k}^{}\lambda _{kj}(\frac{\partial x_{k}'}{\partial x_{i}'})=\sum_{k}^{}\lambda _{kj}\delta _{ik}=\lambda _{ij} $

$ \Rightarrow \frac{\partial \phi '}{\partial x_{i}'}=\sum_{j}^{}\lambda _{ij}\frac{\partial \phi }{\partial x_{j}} $: Correct Transformation Equation

 

$ \frac{\partial \phi }{\partial x_{j}} $: jth component of a gradient of function $ \phi $

$ \vec{grad}=\vec{\bigtriangledown }=\sum_{i}^{}\hat{e_{i}}\frac{\partial }{\partial x_{i}} $ ($ \vec{\bigtriangledown } $: del)

$ (\vec{grad})_{i}=\vec{\bigtriangledown }_{i}=\frac{\partial }{\partial x_{i}} $

 

 

36. Gradient, Divergence, Curl, Laplacian

 

$ \vec{grad}\phi =\vec{\bigtriangledown }\phi =\sum_{i}^{}\hat{e_{i}}\frac{\partial \phi }{\partial x_{i}} $: Vector

$ div\vec{A} =\vec{\bigtriangledown }\cdot \vec{A} =\sum_{i}^{}\frac{\partial A_{i}}{\partial x_{i}} $: Scalar

$ \vec{curl}\vec{A} =\vec{\bigtriangledown }\times \vec{A} =\sum_{i,j,k}^{}\epsilon _{ijk}\frac{\partial \vec A_{k}}{\partial x_{j}}\hat{e_{i}} $: Vector

1) $ \vec{\bigtriangledown }\phi  $ is normal to the lines or surfaces for which $ \phi $=constant
2) $ \vec{\bigtriangledown }\phi  $ has the direction of the maximun change in $ \phi $
3) rate of change of $ \phi $ in the direction of $ \vec{n} $ (directional derivative of $ \phi $) can be found from $ \vec{n}\cdot \vec{\bigtriangledown }\phi \equiv \frac{\partial \phi }{\partial n} $

 

Laplacian: Successive Operation of the Gradient Operator

$ \vec{\bigtriangledown }\cdot \vec{\bigtriangledown }=\sum_{i}^{}\frac{\partial }{\partial x_{i}}\frac{\partial }{\partial x_{i}}=\sum_{i}^{}\frac{\partial^2 }{\partial x_{i}^2}=\vec{\bigtriangledown }^{2} $

$ \vec{\bigtriangledown }^{2}\psi =\sum_{i}^{}\frac{\partial^2 \psi }{\partial x_{i}^2} $

 

 

37. Rectangular Coordinate (Gradient)

 

$ \vec{\bigtriangledown }\phi =\hat{e_{1}}\frac{\partial \phi }{\partial x_{1}}+\hat{e_{2}}\frac{\partial \phi }{\partial x_{2}}+\hat{e_{3}}\frac{\partial \phi }{\partial x_{3}}=\sum_{i}^{}\hat{e_{i}}\frac{\partial \phi }{\partial x_{i}} $

$ \vec{\bigtriangledown }\cdot \vec{A}=\frac{\partial A_{1}}{\partial x_{1}}+\frac{\partial A_{2}}{\partial x_{2}}+\frac{\partial A_{3}}{\partial x_{3}}=\sum_{i}^{}\frac{\partial A_{i}}{\partial x_{i}} $

$ \vec{\bigtriangledown }\times \vec{A}=\begin{vmatrix}
\hat{e_{1}} & \hat{e_{2}} & \hat{e_{3}}\\ 
\frac{\partial }{\partial x_{1}} & \frac{\partial }{\partial x_{2}} & \frac{\partial }{\partial x_{3}}\\ 
A_{1} & A_{2} & A_{3}
\end{vmatrix}=\hat{e_{1}}(\frac{\partial A_{3}}{\partial x_{2}}-\frac{\partial A_{2}}{\partial x_{3}})+\hat{e_{2}}(\frac{\partial A_{1}}{\partial x_{3}}-\frac{\partial A_{3}}{\partial x_{1}})+\hat{e_{3}}(\frac{\partial A_{2}}{\partial x_{1}}-\frac{\partial A_{1}}{\partial x_{2}}) $

   $ =\sum_{i,j,k}^{}\epsilon _{ijk}\frac{\partial A_{k}}{\partial x_{j}}\hat{e_{i}} $

$ \vec{\bigtriangledown }^{2}\phi =(\frac{\partial^2 }{\partial x_{1}^2}+\frac{\partial^2 }{\partial x_{2}^2}+\frac{\partial^2 }{\partial x_{3}^2})\phi  $

 

 

38. Cylindrical Coordinate (Gradient)

 

$ \vec{\bigtriangledown }=\hat{e_{r}}\frac{\partial }{\partial r}+\hat{e_{\Theta }}\frac{1}{r}\frac{\partial }{\partial \Theta }+\hat{e_{z}}\frac{\partial }{\partial z} $

($ d\vec{r}=dr\hat{e_{r}}+rd\Theta \hat{e_{\Theta }}+dz\hat{e_{z}} $

 $ d\phi =\frac{\partial \phi }{\partial r}dr+\frac{\partial \phi }{\partial \Theta }d\Theta +\frac{\partial \phi }{\partial z}dz $: Total Derivative)

 

$ \vec{\bigtriangledown }\cdot \vec{A}=\frac{1}{r}\frac{\partial }{\partial r}(rA_{r})+\frac{1}{r}\frac{\partial A_{\Theta }}{\partial \Theta }+\frac{\partial A_{z}}{\partial z} $

$ \vec{\bigtriangledown }\times \vec{A}=\hat{e_{r}}(\frac{1}{r}\frac{\partial A_{z}}{\partial \Theta }-\frac{\partial A_{\Theta }}{\partial z})+\hat{e_{\Theta }}(\frac{\partial A_{r}}{\partial z}-\frac{\partial A_{z}}{\partial r})+\hat{e_{z}}(\frac{1}{r}\frac{\partial }{\partial r}(rA_{\Theta })-\frac{1}{r}\frac{\partial A_{r}}{\partial \Theta }) $
$ \vec{\bigtriangledown }^{2}\phi =\frac{1}{r}\frac{\partial }{\partial r}(r\frac{\partial \phi }{\partial r})+\frac{1}{r^{2}}\frac{\partial^2 \phi }{\partial \Theta ^{2}}+\frac{\partial^2 \phi }{\partial z^{2}} $

 

 

39. Spherical Coordinate (Gradient)

 

$ \vec{\bigtriangledown }=\hat{e_{r}}\frac{\partial }{\partial r}+\hat{e_{\Theta }}\frac{1}{r}\frac{\partial }{\partial \Theta }+\hat{e_{\phi }}\frac{1}{r\sin \Theta }\frac{\partial }{\partial \phi } $

($ d\vec{r}=dr\hat{e_{r}}+rd\Theta \hat{e_{\Theta }}+r\sin \Theta d\phi \hat{e_{\phi }} $

 $ d\psi =\frac{\partial \psi }{\partial r}dr+\frac{\partial \psi }{\partial \Theta }d\Theta +\frac{\partial \psi }{\partial \phi }d\phi $: Total Derivative)

 

$ \vec{\bigtriangledown }\cdot \vec{A}=\frac{1}{r^{2}}\frac{\partial }{\partial r}(r^{2}A_{r})+\frac{1}{r\sin \Theta }\frac{\partial }{\partial \Theta }(\sin \Theta A_{\Theta })+\frac{1}{r\sin \Theta }\frac{\partial A_{\phi }}{\partial \phi } $

$ \vec{\bigtriangledown }\times \vec{A}=\hat{e_{r}}\frac{1}{r\sin \Theta }[\frac{\partial }{\partial \Theta }(\sin \Theta A_{\phi })-\frac{\partial A_{\Theta }}{\partial \phi }]+\hat{e_{\Theta }}[\frac{1}{r\sin \Theta }\frac{\partial A_{r}}{\partial \phi }-\frac{1}{r}\frac{\partial }{\partial r}(rA_{\phi })]+\hat{e_{\phi }}\frac{1}{r}[\frac{\partial }{\partial r}(rA_{\Theta })-\frac{\partial A_{r}}{\partial \Theta }] $

$ \vec{\bigtriangledown }^{2}\psi =\frac{1}{r^{2}}\frac{\partial }{\partial r}(r^{2}\frac{\partial \psi }{\partial r})+\frac{1}{r^{2}\sin \Theta }\frac{\partial }{\partial \Theta }(\sin \Theta \frac{\partial \psi }{\partial \Theta })+\frac{1}{r^{2}\sin^{2} \Theta }\frac{\partial^2 \varphi }{\partial \phi ^2} $