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Vector Analysis 1

Tue, 2021-09-14
Vector Analysis 1
本作品采用知识共享署名-非商业性使用-相同方式共享 4.0 国际许可协议进行许可。

Vector Analysis

Review Triple product

Proved by expand \(ABC\)

Q1. Show that \[ (a \times b) \times (a\times c) = (a \cdot(b\times c))a \]

\[ (a \times b) \cdot (c\times d) = (a \cdot c)(b \cdot d) - (a \cdot d)(b \cdot c) \]

\(\nabla\) operate on f, that’s gradient, from the gradient definition.

\(\nabla\) operate on f, that’s still gradient

Second derivatives

div grad f
curl grad f
\(\nabla \cdot (\nabla f) = \nabla^2 f = \Delta f\)
\(\nabla \times (\nabla f) =0\) (可以看成两个相同的矢量相乘)
grad div A \(\nabla (\nabla \cdot A)\)
div curl A
curl curl A
\(\nabla \cdot (\nabla \times A) =0\)
\(\nabla \times (\nabla \times A) = \nabla(\nabla \cdot A) - \nabla^2 A\)

\(\nabla \times (\nabla \times A) = \nabla(\nabla \cdot A) - \nabla^2 A\)的证明似乎也要展开,而不能使用\(A \times(B \times C)\)?不是的, \[ A \times (B\times C) = B(A \cdot C) - C (A \cdot B) \] 也可以写成

\[ A \times (B\times C) = B(A \cdot C) - (A \cdot B) C \]

\(A \cdot B\) 是数的时候,两个可以互换,但当他们是算符的时候就会丧失一定的一般性。


Q2. From Maxwell’s equations \[ \left\{\begin{matrix} \nabla \cdot E = 0; \nabla \times E = - \frac{\partial B}{\partial t} \\ \nabla \cdot B = 0; \nabla \times B = \mu_0 \epsilon_0 \frac{\partial E}{\partial t} \\ \end{matrix}\right. \]

Derive the wave equations: \[ \begin{matrix} \nabla^2 E = \mu_0\epsilon_0\frac{\partial^2 E}{\partial^2 t} \\ \nabla^2 B = \mu_0\epsilon_0\frac{\partial^2 B}{\partial^2 t} \\ \end{matrix} \]


Cross product to dot product: \(\nabla \times (\nabla \times A) = \nabla(\nabla \cdot A) - \nabla^2 A\)

\[ \nabla \times (\nabla \times E)=-\mu_0 \epsilon_0 \frac{\partial^2 E}{ \partial t^2} \]


\[ \nabla^2E= \mu_0 \epsilon_0 \frac{\partial^2 E}{ \partial t^2} + \nabla(\nabla \cdot E) \]

While the divergence of \(E\) is \(0\), because:

\[ \begin{aligned} \nabla \times \frac{\hat{r}}{r^2} &= \nabla \times \frac{\bold{r}}{r^3} \\ &=\nabla \times \left(\nabla \frac{1}{r}\right)\\ &=0 \end{aligned} \]

Integral Calculus

Newton-Leibniz formula: 1D

\[ \int_{x_1}^{x_2} f(x) dx = F(x)|_{x_1}^{x_2} \]

Meaning: Only care about the value on two side point. What about 3D? \[ \int_{?}^{?}f(x,y,z)?dxdydz \]

In 3D, three types of integrals

line integrals

\[ \int_{a \mathcal{P}}^{b}v \cdot dl \]

For a closed loop: \[ \oint_\mathcal{P} \bold{v} \cdot d \bold{l} = 0 \]

Conservative(保守场): \[ \int_{a \mathcal{P}}^b \bold{v} \cdot d \bold{l} =\int_{a \mathcal{P'}}^b \bold{v} \cdot d \bold{l} \]


Surface Integral

\[ \int_{\mathcal{S}} v \cdot dS \]


Surface integral for a closed surface \[ \oint_{\mathcal{S}} \bold{v}\cdot d\bold{S} \]

Volume Integrals

\[ \int_{\mathcal{V}}TdV \]

Fundamental theorem of gradient, divergence and curl

Like Newton-Leibniz equation, we only cares about the two side of, and there is no need to care about things in.

Meaning: 降维

\[ \int_{aP}^{b}\nabla f \cdot d\bold{l}=f(b)-f(a) \] \[ \oint_{P} \nabla \mathcal{f} \cdot d\bold{l}=0 \]

\[ \int_\mathcal{V}(\nabla \cdot \bold{v} )dV = \oint_\mathcal{S} \bold{v} \cdot d \bold{S} \]

\[ \int_{\mathcal{S}} (\nabla \times \bold{v}) \cdot d \mathcal{S} = \oint_{\mathcal{P}} \bold{v} \cdot d \bold{l} \]


Fundamental theorem of gradient:

From the beginning \(\nabla f \cdot dl \equiv df\) \[ \begin{aligned} \int df \cdot dl =& \int_{r(x,y,z)}^{r'(x,y,z)} \frac{\partial f}{ \partial x} dx + \frac{\partial f}{ \partial y} dy + \frac{\partial f}{ \partial z} dz\\ =& \int \frac{\partial f}{ \partial x} dx + \ldots \\ =& f(x_1,y_0,z_0) - f(x_0,y_0,z_0) \\ &+f(x_1,y_1,z_0) - f(x_1,y_0,z_0)\\ &+ f(x_1,y_1,z_1)-f(x_1,y_0,z_0)\\ =& f(x_1,y_1,z_1) - f(x_0,y_0,z_0) \end{aligned} \] 最后一步的图像类似:


Fundamental theorem of divergence:

\[ \int_\mathcal{V}(\nabla \cdot \bold{v} )dV = \oint_\mathcal{S} \bold{v} \cdot d \bold{S} \]



\[ \begin{aligned} \bold{v} \cdot d\bold{S} &= [v_x(x_2) - v_x(x_1)]dydz\\ &+ [v_y(y_2) -v_y(y_1)] dxdz\\ &+ [v_z(z_2) -v_z(z_1)] dxdy\\ &= \left[\frac{\partial v_x}{ \partial x} + \frac{\partial v_y}{ \partial y} + \frac{\partial v_z}{ \partial z}\right] dxdydz\\ &= \nabla v \cdot dV \end{aligned} \]


Why \(v(x)\) from left to right: \[ v = v_x \hat{x} + v_y\hat{y} + v_z\hat{z} \]

Fundamental theorem of curl:

\[ \int_\mathcal{S} (\nabla \times \bold{v}) \cdot d \bold{S} = \oint_P \bold{v} \cdot d \bold{l} \]


同样从右边看,看一个小的环路的积分,微元加起来,重复的边界 cancel 掉了

Curvilinear Coordinates

Spherical polar coordinates (SPC):

\[ \begin{aligned} x &= r\sin \theta \cos \phi\\ y &= r \sin \theta \sin\phi\\ z &= r \cos \theta \end{aligned} \]


Direction of \(\theta\) is increase of theta

Unlike Descartes(笛卡尔) coordinates, the SPC coordinates changes with \(r\) ,\(\theta\) \(\phi\),坐标轴会变。

If we want to calculate the value of infinite small volume, because \(\theta\) \(\phi\) are not length, we need to 要算体积微元, \(dl\) \(dl_\theta\) \(dl_\phi\)

\[ \begin{aligned} dl_r &= dr = h_1 dr\\ dl_\theta &= r d\theta =h_2 d\theta\\ dl_\phi &= r \sin \theta d \phi = h_3 d\phi \end{aligned} \]

Here \(h_1,h_2,h_3\) are geometrical factors

For Cylindrical coordinates: