Mathematical Preliminary for EM

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Derivative gives us relationship between two small increments of variables.

If we have a one-dimensional function [math]f(x)[/math],

[math]df = \left(\frac{df}{dx}\right) dx[/math]

gives the relationship between [math]df[/math] and [math]dx[/math]


Gradient

[math] \nabla f = \cfrac{1}{h_i}{\partial f \over \partial q^i} \mathbf{e}^i [/math]
If we have a 3-dimensional function [math]f(x,y,z)[/math], the derivative will have to give us the relationship between [math]df[/math] and [math]d\vec{r} = (dx,dy,dz)[/math].
This kind of derivative is called gradient.
As we know [math]df[/math] can be represented as below using partial derivatives:
[math]df = \left(\frac{\partial f}{\partial x}\right)dx + \left(\frac{\partial f}{\partial y}\right)dy + \left(\frac{\partial f}{\partial z}\right)dz [/math]
So, in Cartesian coordinates, we can define gradient as follows
[math]\nabla f \equiv \left( \frac{\partial f}{\partial x}, \frac{\partial f}{\partial y}, \frac{\partial f}{\partial z} \right)[/math]
so we can write [math]df = \nabla f \cdot d\vec{r} [/math]

Cylindrical coordinates

[math]\nabla f={\partial f \over \partial \rho}\hat{\boldsymbol \rho} + {1 \over \rho}{\partial f \over \partial \varphi}\hat{\boldsymbol \varphi} + {\partial f \over \partial z}\hat{\mathbf z}[/math]

Spherical coordinates

[math]\nabla f={\partial f \over \partial r}\hat{\mathbf r}+ {1 \over r}{\partial f \over \partial \theta}\hat{\boldsymbol \theta}+ {1 \over r\sin\theta}{\partial f \over \partial \varphi}\hat{\boldsymbol \varphi}[/math]

Divergence

[math]\nabla\cdot\mathbf{A} = \cfrac{1}{\prod_j h_j} \frac{\partial }{\partial q^i}(A^i\prod_{j\ne i} h_j) [/math]
Or, more explicitly in 3-dim,
[math]\nabla\cdot\mathbf{A} = \cfrac{1}{h_1h_2h_3} \left[ {\partial\over\partial q_1}(A_1h_2h_3) + {\partial\over\partial q_2}(A_2h_3h_1) + {\partial\over\partial q_3}(A_3h_1h_2) \right][/math]

Cartesian coordinates

[math]\nabla\cdot\mathbf{A} = {\partial A_x \over \partial x} + {\partial A_y \over \partial y} + {\partial A_z \over \partial z}[/math]

Cylindrical coordinates

[math]\nabla\cdot\mathbf{A} = {1 \over \rho}{\partial \left( \rho A_\rho \right) \over \partial \rho} + {1 \over \rho}{\partial A_\varphi \over \partial \varphi} + {\partial A_z \over \partial z}[/math]

Spherical coordinates

[math]\nabla\cdot\mathbf{A} = {1 \over r^2}{\partial \left( r^2 A_r \right) \over \partial r} + {1 \over r\sin\theta}{\partial \over \partial \theta} \left( A_\theta\sin\theta \right) + {1 \over r\sin\theta}{\partial A_\varphi \over \partial \varphi}[/math]

Curl

[math] \nabla\times\mathbf{A} = \frac{1}{h_1h_2h_3} \mathbf{e}_i \epsilon_{ijk} h_i \frac{\partial (h_k A_k)}{\partial q^j} [/math]
where [math]\epsilon_{ijk}[/math] is the Levi-Civita symbol.
Or, more explicitly,
[math] \nabla\times\mathbf{A} = \frac{1}{h_1h_2h_3} \left| \begin{array}{ccc} h_1\textbf{e}_1 & h_2\textbf{e}_2 & h_3\textbf{e}_3 \\ {\partial\over\partial q_1} & {\partial\over\partial q_2} &{\partial\over\partial q_3} \\ h_1A_1 & h_2A_2 & h_3A_3 \end{array} \right|[/math]

Cartesian coordinates

[math]\nabla\times\mathbf{A} = \left(\frac{\partial A_z}{\partial y} - \frac{\partial A_y}{\partial z}\right) \hat{\mathbf x} + \left(\frac{\partial A_x}{\partial z} - \frac{\partial A_z}{\partial x}\right) \hat{\mathbf y} + \left(\frac{\partial A_y}{\partial x} - \frac{\partial A_x}{\partial y}\right) \hat{\mathbf z} [/math]

Cylindrical coordinates

[math]\nabla\times\mathbf{A} = \left( \frac{1}{\rho} \frac{\partial A_z}{\partial \varphi} - \frac{\partial A_\varphi}{\partial z} \right) \hat{\boldsymbol \rho} + \left( \frac{\partial A_\rho}{\partial z} - \frac{\partial A_z}{\partial \rho} \right) \hat{\boldsymbol \varphi} + \frac{1}{\rho} \left( \frac{\partial \left( \rho A_\varphi\right) }{\partial \rho} - \frac{\partial A_\rho}{\partial \varphi} \right) \hat{\mathbf z}[/math]

Spherical coordinates

[math]\nabla\times\mathbf{A} = \frac{1}{r\sin\theta} \left(\frac{\partial}{\partial \theta} \left(A_\varphi\sin\theta \right) - \frac{\partial A_\theta}{\partial \varphi} \right) \hat{\mathbf r} + \frac{1}{r} \left( \frac{1}{\sin\theta} \frac{\partial A_r}{\partial \varphi} - \frac{\partial}{\partial r} \left( r A_\varphi \right) \right) \hat{\boldsymbol \theta} {}+ \frac{1}{r} \left( \frac{\partial}{\partial r} \left( r A_{\theta} \right) - \frac{\partial A_r}{\partial \theta} \right) \hat{\boldsymbol \varphi}[/math]

Laplacian

[math]\nabla^2 f = \nabla\cdot\nabla f = \frac1{r^2}\frac{\partial}{\partial r}\left( r^2\frac{\partial f}{\partial r}\right)+\frac1{r^2\sin\theta}\frac{\partial}{\partial\theta}\left(\sin\theta\frac{\partial f}{\partial\theta}\right) + \frac1{r^2\sin^2\theta}\frac{\partial^2f}{\partial\phi^2}[/math]
[math]\nabla^2\mathbf{A} = \nabla\cdot\nabla \mathbf{A} = ?[/math]