Vector Analysis
Review Triple product
- Vector triple product \[ A \times (B\times C) = B(A \cdot C) - C (A \cdot B) \]
Proved by expand \(ABC\)
- Scalar triple product \[ A \cdot (B \times C) = B \cdot(C \times A) \]
- 轮换
Q1. Show that \[ (a \times b) \times (a\times c) = (a \cdot(b\times c))a \]
\[ (a \times b) \cdot (c\times d) = (a \cdot c)(b \cdot d) - (a \cdot d)(b \cdot c) \]
- Relation of divergences \[ \nabla \cdot (fA) = (\nabla f) \cdot A + f(\nabla \cdot A) \]
\(\nabla\) operate on f, that’s gradient, from the gradient definition.
- Relation of curls \[ \nabla \times (fA) = (\nabla f) \times A + f(\nabla \times A) \]
\(\nabla\) 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
\[ A \times (B\times C) = B(A \cdot C) - (A \cdot B) C \]
当\(A \cdot B\) 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。
有二重导可以想想上面几个公式。
- Note: Extra for Laplace,\(\nabla^2\):
- scalar: \(\nabla^2f = \nabla \cdot (\nabla f)\)
- vector: \(\nabla^2 A \equiv (\nabla \cdot \nabla)A = \nabla (\nabla \cdot A) - \nabla \times (\nabla \times A)\)
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: \[ \begin{matrix} \nabla^2 E = \mu_0\epsilon_0\frac{\partial^2 E}{\partial^2 t} \\ \nabla^2 B = \mu_0\epsilon_0\frac{\partial^2 B}{\partial^2 t} \\ \end{matrix} \]
Solve:
Cross product to dot product: \(\nabla \times (\nabla \times A) = \nabla(\nabla \cdot A) - \nabla^2 A\)
\[ \nabla \times (\nabla \times E)=-\mu_0 \epsilon_0 \frac{\partial^2 E}{ \partial t^2} \]
So
\[ \nabla^2E= \mu_0 \epsilon_0 \frac{\partial^2 E}{ \partial t^2} + \nabla(\nabla \cdot E) \]
While the divergence of \(E\) is \(0\), because:
\[ \begin{aligned} \nabla \times \frac{\hat{r}}{r^2} &= \nabla \times \frac{\bold{r}}{r^3} \\ &=\nabla \times \left(\nabla \frac{1}{r}\right)\\ &=0 \end{aligned} \]
- NOTE: How to understand \(\nabla^2 A\)? \(\Rightarrow\) \((\nabla \cdot \nabla)E\)
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 (or path) integrals
- Surface integrals (or flux)
- Volume 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): \[ \int_{a \mathcal{P}}^b \bold{v} \cdot d \bold{l} =\int_{a \mathcal{P'}}^b \bold{v} \cdot d \bold{l} \]
不同路径的值是一样的
Surface Integral
\[ \int_{\mathcal{S}} v \cdot dS \]
Flux
- \(dS\) surface define a direction
- \(v\) or \(E\) has a direction
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
\[ \int_{aP}^{b}\nabla f \cdot d\bold{l}=f(b)-f(a) \] \[ \oint_{P} \nabla \mathcal{f} \cdot d\bold{l}=0 \]
\[ \int_\mathcal{V}(\nabla \cdot \bold{v} )dV = \oint_\mathcal{S} \bold{v} \cdot d \bold{S} \]
\[ \int_{\mathcal{S}} (\nabla \times \bold{v}) \cdot d \mathcal{S} = \oint_{\mathcal{P}} \bold{v} \cdot d \bold{l} \]
Proofs:
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:
\[ \int_\mathcal{V}(\nabla \cdot \bold{v} )dV = \oint_\mathcal{S} \bold{v} \cdot d \bold{S} \]
先看通量，右边
- 对一个面\(v\) has 3 values
- \(v_x(x,y,z)\)
- \(v_y(x,y,z)\)
- \(v_z(x,y,z)\)
- Surface vector points out forward of a small volume
\[ \begin{aligned} \bold{v} \cdot d\bold{S} &= [v_x(x_2) - v_x(x_1)]dydz\\ &+ [v_y(y_2) -v_y(y_1)] dxdz\\ &+ [v_z(z_2) -v_z(z_1)] dxdy\\ &= \left[\frac{\partial v_x}{ \partial x} + \frac{\partial v_y}{ \partial y} + \frac{\partial v_z}{ \partial z}\right] dxdydz\\ &= \nabla v \cdot dV \end{aligned} \]
这也是散度这么定义的来源
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:
- 和\(x\) 夹角\(\phi\)
- \(r\)投影到\(x-y\) 平面长度
- z \[ \begin{aligned} dl_s &= ds = h_1 ds\\ dl_\phi &= s d\phi =h_2 d\phi\\ dl_z &= d z = h_3 dz \end{aligned} \]