$\newcommand{\dis}{\displaystyle} \newcommand{\m}{\hspace{1em}} \newcommand{\mm}{\hspace{2em}} \newcommand{\x}{\vspace*{1ex}} \newcommand{\xx}{\vspace*{2ex}} \let\limm\lim \renewcommand{\lim}{\dis\limm} \let\fracc\frac \renewcommand{\frac}{\dis\fracc} \let\summ\sum \renewcommand{\sum}{\dis\summ} \let\intt\int \renewcommand{\int}{\dis\intt} \newcommand{\sech}{\text{sech}} \newcommand{\csch}{\text{csch}} \newcommand{\Ln}{\text{Ln}} \newcommand{\p}{\partial} \newcommand{\intd}[1]{\int\hspace{-0.7em}\int\limits_{\hspace{-0.7em}{#1}}} $

Lecture 31, 11/9/2022. This page is for Section 1 only.
ACMS 20550: Applied Mathematics Method I
Instructor: Bei Hu, b1hu@nd.edu, Hurley 174A

Chapter 6: Vector Analysis

  1. Divergence Theorem (in 3D):
    Divergence Theorem : $\m \int\hspace{-0.7em}\intd{\tau} \text{div }\vec V d\tau = \intd{\p\tau} \vec V\cdot \vec n d\sigma$.
  2. The curl and Stoke's theorem:
    Stoke's theorem: $\m\dis \intd{\sigma} (\nabla\times \vec V)\cdot \vec n d \sigma= \oint_{\p \sigma} \vec V\cdot d\vec r $,
    here $\sigma$ is a surface in 3-d.


    The following are equivalent:
    1. curl $\vec F = 0$;
    2. $\dis\oint \vec F\cdot d\vec r =0$ for any simple closed curve;
    3. $\vec F$ is conservative, i.e., $\dis\int_A^B \vec F\cdot d\vec r$ is independent of path;
    4. $\vec F = $ grad $W$ for some $W$;
    We also say: $\m \vec F$ is irrotational.