$\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 22, 10/12/2022. This page is for Section 1 only.
ACMS 20550: Applied Mathematics Method I
Instructor: Bei Hu, b1hu@nd.edu, Hurley 174A

  1. Change of variables; Jacobians
    1. Spherical coordinate:$\fcolorbox{white}{yellow}{ $x=r\sin\theta\cos\phi, \m y=r\sin\theta\sin\phi, \m z= r\cos\theta$}$
      Volume : $ dV= r^2\sin\theta drd\theta d\phi$.
      $\mm$
      Surface area at $r=a$: $\m dA = a^2 \sin\theta d\theta d\phi$

      Arc length: $\m ds^2 = dr^2+r^2 d\theta^2+r^2\sin^2\theta d\phi^2, \mm $
    2. Jacobians: $\fcolorbox{white}{yellow}{ If $y=f(x)$, then $dy = f'(x) dx$}$.
      If $x=x(s,t), \m y=y(s,t)$, then $\fcolorbox{white}{yellow}{$dA = dxdy = |J| dsdt$}$, where $J=$ Jacobian:
      $\mm$
      $J = \frac{\p (x,y)}{\p(s, t)} = \left| \begin{array}{cc} \frac{\p x}{\p s} & \frac{\p x}{\p t} \\ \frac{\p y}{\p s} & \frac{\p y}{\p t} \end{array}\right|$

      3-d formula is similar: $\m$ If $u=u(r,s,t), \m v=v(r,s,t), \m w = w(r,s,t)$, then
      $\mm$ $\fcolorbox{white}{yellow}{$dV = dudvdw = |J| dr dsdt$}$, where $J=$ Jacobian:
      $\mm$
      $J = \frac{\p (u,v,w)}{\p(r, s, t)} = \left| \begin{array}{ccc} \frac{\p u}{\p r} & \frac{\p u}{\p s} & \frac{\p u}{\p t} \\ \frac{\p v}{\p r} & \frac{\p v}{\p s} & \frac{\p v}{\p t} \\ \frac{\p w}{\p r} & \frac{\p w}{\p s} & \frac{\p w}{\p t} \end{array}\right|$