[Mathematics][MIT 18.02]Detailed discussions about 2-D and 3-D integral and their connections

• 2019 年 10 月 21 日
• 笔记

　　Since it is just a sort of discussion, I will just give the formula and condition without proving them or leaving examples.

General:

• Line integral(Work and in the plane)

　　　　$displaystyle int_{C}vec{F}cdot mathrm{d}vec{r} = int_{C}Mmathrm{d}x+Nmathrm{d}y$, in which $vec{F} = <M,N>$

　　　　　　Method: Express $x$ and $y$ in a single variable (OR means parameterization).

• Gradient fields & path-independence

Condition:

　　$curl(vec{F}) = 0$ and $vec{F}$ is defined in a simple-connected region,

　　　　　　　　in which $displaystyle curl(vec{F}) = N_{x} – M_{y}$ if $vec{F} = <M,N>$ AND $displaystyle curl(vec{F}) = nablatimesvec{F}$(namely$displaystyle begin{vmatrix}hat{i} & hat{j} & hat{k} \frac{partial}{partial x} & frac{partial}{partial y} & frac{partial}{partial z} \P & Q & Rend{vmatrix}) $,if $vec{F} = <P,Q,R>$

　　　　then $vec{F} = nabla f$, or $vec{F}$ is the partial derivative vector of some vector field.

The method of finding the potential:

　　Method 1. Do line integral. Integral along the x-axis and y-axis and z-axis, if they exist. (Using path-independence)

　　Method 2. Integral one component of $vec{F}$ and then differential it over another variable and compare. (…)

• Flux in plane & space

in the plane:

　　$hat{n} = hat{T}$ rotated 90 degrees clockwise $=<mathrm{d}y,-mathrm{d}x>$

　　$displaystyle int_{C}vec{F}cdothat{n}mathrm{d}s = int_{C}Pmathrm{d}y-Qmathrm{d}x$, in which $vec{F} = <P,Q>$

in the space(or specifically, surface):

　　$displaystyle iint_{S}vec{F}cdothat{n}mathrm{d}S = iint_{S}vec{F}cdot(<-f_{x},-f_{y},1>mathrm{d}xmathrm{d}y)$, if we use $z = f(x,y)$ to describe the surface.

　　$displaystyle =iint_{S}vec{F}cdot(pmfrac{vec{N}}{vec{N}cdothat{k}}mathrm{d}xmathrm{d}y)$, if we are given the normal vector of the surface,or specifically, $g(x,y,z) = 0$

Association:

Work(line integral):

• 2-D:

　　$displaystyle oint_{C}vec{F}cdotmathrm{d}vec{r} = iint_{R}curl(vec{F})mathrm{d}A$

• 3-D:

　　$displaystyle oint_{C}vec{F}cdotmathrm{d}vec{r} = iint_{S}curl(vec{F})hat{n}mathrm{d}S$,in which $S$ means any surface bounded by this curve and $curl(vec{F})=nablatimesvec{F}$.

Flux:

• 2-D:

　　$displaystyle oint_{C}vec{F}cdothat{n}mathrm{d}s = iint_{R}div(vec{F})mathrm{d}A$,in which $vec{F} = <P,Q>$ and $div(vec{F}) = P_{x} + Q_{y}$.

• 3-D:

　　$displaystyleoiint_{S}vec{F}cdothat{n}mathrm{d}S = iiint_{R}div(vec{F})mathrm{d}V$, in which $vec{F} = <P,Q,R>$ and $div(vec{F}) = P_{x} + Q_{y} + R_{z}$.