Int. and Vector Fields

line int. over curve $C$ -> line int. of a vector field $F$ over curve $C$

相当于把一系列点串起来，此时形成了curve $C$。
然后限定 Field $F(x,y,z)$中xyz满足curve $C$的轨迹
把轨迹方程里面所有点的信息_输入_到F中, 方程替代自变量

$\vec{F}(\vec{r}(t)) = M(x,y)\hat{i} + N(x,y)\hat{j}$

forward direction is defined by $\vec{r}(t)$

Unit Tangent vector

imagine the definition in geometry. 
first is the distribution on s, which is scalar 
second is by definition in velocity, $v$(velocity) point out where I go next,
while $r$(track) only point out the position

$\vec{T(t)}= \frac{d\vec{r}}{ds}=\frac{v(t)}{||v(t)||}$

Field

1. first we give discription of force on the particle

we assign a value for each point

问? 为什么在二维空间中，i前面的系数M（x,y）会受到两个未知数影响？
答:设想x方向分力之手x影响，那么同一个x（一条竖直线）上的力全部相等，这个
不够一般，为了使得不同的y值有不同的分力，我们需要一个自变量。
所以得到以下的表达式

$\vec{F(x,y)} = M(x,y)\hat{i} + N(x,y)\hat{j}\\ \vec{F(x,y,z)} = M(x,y,z)\hat{i} + N(x,y,z)\hat{j}+P(x,y,z)\hat{k}$

2.conti.

if all the component fnc. are conti. then the field conti.

3.example

Gradiant Field
force field -- work-965  
velocity field -- flow and circulation-966  
gravity field
electric field
etc.

4.场上分量

w.r.t $dt$

if I want to know the magnitude of $\vec{u}$ on $\vec{v}$ derection, in another word-- $Proj_u(\vec{v})$
remember the dot product?
so we have the projection:

$\vec{F}(x,y,z)\cdot\vec{T}(x,y,z)$

is field $F$ on direction $\vec{T}$ i.e. on curve $C$ at point (x,y,z)

w.r.t $dx,dy,dz$

still take correspond part in r(t) into F(x,y,z)

then change all variable w.r.t $dt$

Flux and Circulation

turbulence water system, and you padding in the field. Imagine the force you give to exist.

Quick test of def.

• flow start and end at same point
• flow cross curve C/ outward at n direction
• curve not cross itself
• curve start and end at same point
• normal comoponenet/ unit normal vector
• inward/ outward

* flow start and end at same point --- "circulation"
* flow cross curve C/ outward at n direction --- "flux"
* curve not cross itself ---"simple"
* curve start and end at same point --- "close/ loop"
* normal comoponenet/ unit normal vector --- "n" in dim = 2, $n = T\times k$ 
* inward(clockwise)/ outward(counter-clockwise) --- "how C is tracversed as ⬆️"

16.3

Quick test of def.

• piecewise smooth
• Path Independence
• Conservative Fields
• Potential Functions
• connected reigeon
• conserve and loop property
• fundamental theorem of line integral
• conservative field are gradiant field
• differential form

* piecewise smooth --consist of infinite segment end to end  **cts. first partial derivatives**
* Path Independence
* Conservative Fields
* Potential Functions 
* connected reigeon --any two point a line in D connect them
* not simply connected --contain loop that connot contract to a point
* conserve and loop property
* fundamental theorem of line integral
* conservative field are gradiant field
* differential form

Path Independence

any path between 2 pts. on region D, have same value of line integral

Conservative Fields

A to B add B to A, any path, if the sum is 0, then it is conservative

saying that F is conservative on D 
is equivalent to saying that the integral of F 
around !!every!! closed path in D is zero.
三个等价：F is nabla f on D == F conservative == circulation int. = 0

TEOREM1- line int. in conservative fields

Fundamental Theorem of Line Integrals
矢量场沿任何路径之间的线积分等于这两个点处势函数值之差。

THEOREM 2-Conservative Fields are Gradient Fields

Let F = Mi + Nj + Pk
be a vector field whose components are continuous throughout an open connected
region D in space.
Then F is conservative if and only if F is a gradient field ∇ƒ for a
differentiable function ƒ

THEOREM 3—Loop Property of Conservative Fields

The following statements are equivalent.

1. loop integral dr = 0 around every loop (that is, closed curve C) in D.
2. The field F is conservative on D

The test for a vector field being conservative – component test

involves the equivalence of certain first
partial derivatives of the field components
三个组合，分母互换

Potential Functions

definition

If F is a vector field defined on D and F = ∇ƒ for some scalar
function ƒ on D, then ƒ is called a potential function for F
some suppliment

在数学中，势函数是描述矢量场潜力的标量函数。
它与梯度概念密切相关，梯度表示标量场的定向变化。
势函数的梯度指向势能最陡峭上升的方向，其幅度代表该方向上的变化率。

势函数通常用于矢量分析来解决涉及保守矢量场的问题。
在这样的场中，矢量场沿任何路径之间的**线积分**等于这两个点处**势函数值之差**。

differential form

微分形式（differential form），也称为外形式（exterior form），是数学中用于表达和分析高维微积分概念的强大工具。它们提供了一种简洁优雅的方式来表示和操作多线性函数，即以多个矢量场作为输入的函数

typical question and hint

• the field is gradiant of fnc. find work done
• work done by conservative field along the smooth curve
• how to test whether it is conservative
  • component test
  • test assume it is simply connected
  • 三个等价：F is nabla f on D == F conservative == circulation int. = 0
• if it is conservative how to find the potential function
  • three step
  • peel the onion, reveal the truth, from x to z
• show differential from is exact
  • component test for exactness of $M dx+N dy+ P dz$
• for line integral in dim 3 field, better choose to find $\nabla f$, then use f to find the value

16.4 Green’s Theorem in the plane

通俗格林

intro

• WHY we need?
  computing a work or flux
  integral over a closed curve C in the plane when the field F is not conservative
• HOW it works?
  convert the line integral into a double integral over the region enclosed by C
• velocity fields of fluid flows
• two new ideas for Green’s Theorem: circulation density around an axis perpendicular to the plane and divergence (or flux density)

Spin Around an Axis: The k-Component of Curl

倒反天罡circulation density FTds

the circulation density of a vector field $F = Mi + Nj$ at the point (x,y) is the scalar expression

$(curl \bold{\_F})\cdot k = \frac{\partial{N}}{\partial{x}}-\frac{\partial{M}}{\partial{y}}$

• find circulation density of vector field (倒反天罡：交换式子，反转符号)
  e.g. of circulation type

1. uniform expansion or compression(向外向内爆炸)--0
2. uniform rotation --with M or N component
3. shearing flow --constant, negative clockwise 
4. whirlpool effect --0

正正常常flux density Fnds

The divergence (flux density) of a vector field $F = Mi + Nj$ at the point (x, y) is

$div \bold{\_F} = \frac{\partial{M}}{\partial{x}}+\frac{\partial{N}}{\partial{y}}$

• find flux density of vector field (正正常常：前后不变，符号照常)
  e.g. of flux type

1. uniform expansion or compression(向外向内爆炸)--with M or N component(c)
2. unchange --0

Two Forms for Green’s Theorem

we need: a piecewise smooth, simple, closed curve, enclosing R in plane.
M N should have cts. partial derivative

• counterclockwise is +
  
  然后面对写成这样的式子$\oint_{L^{+}}(Mdx+Ndy)$，先写个$\iint \limits_{D}()dxdy$先

$如果在曲线包围的区域D的内任何一点都可以让每个偏导数(也就是\frac{\partial M}{\partial y}和\frac{\partial N}{\partial x})连续\\ 那么:\oint_{L^{+}}(Mdx+Ndy)=\iint \limits_{D}(\frac{\partial N}{\partial x}-\frac{\partial P}{\partial y})dxdy$

注意

• flux左相减，右相加，正常序（正正常常）
• circulation左相加，右相减，互换序（倒反天罡）

typical question and hint

1. calculate circulation and det it meaning
2. calculate flux and det it meaning
3. use 2 type Green’s theorem
   1. calculate parts
   2. take that part into, treat dt as a number

16.5 Surface and area(calculation)

curves in the plane

three way to express:

1. explicit
2. implicit
3. parametric vector form

parametrize surface

area cal

implicit surface $F(x,y,z) = 0$

area cal

1. on bdd. plane region R
2. p is normal to R and p is one of the three basis vector
3. grad F is gradient of surface

explicit surface $z = f(x,y)$

面积微元$d \sigma$：根号下，方方1，dxdy

typical question and hint

1. how to parametrize a surface, a body
2. use double integral for calculation based on parametrization.
3. how we test the smoothness of the para. surface
   1. $r_u,r_v$ cts., u and v are two component\basis
   2. $r_u\times r_v \not = 0$ in interior of parameter domain
4. how we define the area of the smooth surface
   1. what is the form of surface area differential for para. surface
5. how we calculate three type of surface