$\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 24, 10/24/2022. This page is for Section 1 only.
ACMS 20550: Applied Mathematics Method I
Instructor: Bei Hu, b1hu@nd.edu, Hurley 174A

Chapter 5.

  1. Review: Surface Area $z =f(x,y)$:
    $ \intd{} dA = \intd{} \sec \gamma \; dxdy \mm $ where $\m \sec\gamma = \sqrt{1+\Big(\frac{\p f}{\p x}\Big)^2+\Big(\frac{\p f}{\p y}\Big)^2 }$

Chapter 6: Vector Analysis

  1. Vector multiplications.
    1. Scalar product (dot product, inner product): $\vec{A} \cdot \vec{B} = |\vec{A}|\; |\vec{B}| \; \cos\theta$, where $\theta$ is the angle betwen $\vec A$ and $\vec B$.
      If $\vec A = (A_1, A_2, A_3), \m \vec B = (B_1, B_2, B_3)$, then
      $\vec{A} \cdot \vec{B} = A_1B_1+ A_2B_2+A_3B_3$;

      $\fcolorbox{white}{yellow}{ $\vec A \perp \vec B$ if and only if $\vec{A} \cdot \vec{B} = 0$. }$


    2. Cross product: $\vec A \times \vec B = \vec C$:
      (a) $ |\vec C | = |\vec A|\; |\vec B| \sin \theta;\m $ (b) $ \vec C\perp \vec A; \m \vec C \perp \vec B; \m $ (c) right-hand rule;

      $\fcolorbox{white}{yellow}{ $\vec A \times \vec B = 0$ if and only if $\vec A // \vec B$}$


  2. Triple product = Volume.
    $\vec A \cdot \Big(\vec B\times \vec C\Big) =$ Volume.
    $\fcolorbox{white}{yellow}{Warning: $(\vec A \cdot \vec B)\times \vec C $ does not make sense.}$

    1. Several useful formulas:
      (a) $\vec A \cdot \vec B = \vec B \cdot \vec A; \mm $ (b) $\vec A \times \vec B = - \vec B \times \vec A$;


      (c) If $\vec B=(B_1, B_2, B_3)$ and $ \vec C =(C_1,C_2, C_3)$, then
      $ \vec B\times \vec C = \left|\begin{array}{ccc} \vec i & \vec j &\vec k \\ B_1 & B_2 & B_3 \\ C_1 & C_2 & C_3 \end{array}\right| = \left|\begin{array}{cc} B_2 & B_3 \\ C_2 & C_3 \end{array}\right| \vec i + \left|\begin{array}{cc} B_3 & B_1 \\ C_3 & C_1 \end{array}\right| \vec j + \left|\begin{array}{cc} B_1 & B_2 \\ C_1 & C_2 \end{array}\right| \vec k $


    2. $(\vec A \times \vec B)\cdot \vec C = \vec A \cdot( \vec B \times \vec C) = \vec C \cdot (\vec A \times \vec B) = ( - \vec A \times \vec C)\cdot \vec B$

      $ \vec A\cdot(\vec B\times \vec C) = \left|\begin{array}{ccc} A_1& A_2 & A_3 \\ B_1 & B_2 & B_3 \\ C_1 & C_2 & C_3 \end{array}\right| $
    3. $ \vec A\times (\vec B\times \vec C) = (\vec A\cdot \vec C) \vec B - (\vec A\cdot \vec B)\vec C$
      $\fcolorbox{white}{yellow}{Warning: $ \vec A\times (\vec B\times \vec C) \neq (\vec A\times \vec B) \times \vec C$}$.