$\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 32, 11/10/2021. This page is for Section 1 only.
ACMS 20550: Applied Mathematics Method I
Instructor: Bei Hu, b1hu@nd.edu, Hurley 174A

Review.
Chapter 5.
  1. Double and triple integrals: Draw a picture and a small rectangle to determine the order of integration.
  2. Applications
  3. Polar coordinate system, cylindrical coordinate system, spherical coordinate system, Jacobian.
  4. Surface integrals.
Chapter 6.
  1. Scalar product (dot product). The result is a scalar.
    If $ \vec A = (A_1, A_2, A_3), \m \vec B = (B_1, B_2, B_3)$, then
    $\vec{A} \cdot \vec{B} = A_1B_1+ A_2B_2+A_3B_3$;


  2. Cross product. The result is a vector.
    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 $


  3. A bunch of formulas:
    1. Work $\vec F \cdot \vec d$;
    2. Torque $\vec r\times \vec F$;
    3. Angular velocity $\vec \omega \times \vec r$;
    4. Velocity, accelaration, and their magnitude;

  4. The formula involving $\nabla $
    1. $\fcolorbox{white}{yellow}{$\nabla = \Big(\frac\p{\p x}, \frac\p{\p y}, \frac\p{\p z}\Big) = \vec i \frac\p{\p x} + \vec j \frac\p{\p y} + \vec k \frac\p{\p z}$}$
    2. $\fcolorbox{white}{yellow}{Directional derivative of $\phi$ in the direction $\vec u$: $\m \nabla \phi \cdot \vec u\m$ (make sure to make $\vec u$ a unit vector!!!)}$
    3. $\fcolorbox{white}{yellow}{Normal vector for the surface $\phi(x,y,z)=0$: $\m \vec n = \frac{\nabla \phi}{|\nabla\phi|}$}$
    4. $\fcolorbox{white}{yellow}{Divergence}$. Let $\vec V =(V_1, V_2, V_3)$, then $\fcolorbox{white}{yellow}{ div $\vec V = \nabla \cdot \vec V = \frac{\p V_1}{\p x} + \frac{\p V_2}{\p y}+\frac{\p V_3}{\p z} $}$
    5. $\fcolorbox{white}{yellow}{Curl}.$ Let $\vec V =(V_1, V_2, V_3)$, then $\fcolorbox{white}{yellow}{ curl $\vec V = \nabla \times \vec V = \left|\begin{array}{ccc} \vec i & \vec j & \vec k \\ \frac\p{\p x} & \frac\p{\p y} & \frac\p{\p z} \\ V_1 & V_2 & V_3 \end{array}\right| $}$
    6. $\fcolorbox{white}{yellow}{ Laplacian. }$ $\fcolorbox{white}{yellow}{ $\nabla^2 \phi = \nabla \cdot\nabla \phi = $ div ( grad $\phi ) = \frac{\p^2 \phi}{\p x^2} + \frac{\p^2 \phi}{\p y^2}+\frac{\p^2 \phi}{\p z^2} $}$
  5. Conserved field:
    1. $\vec F$ is a conserved field if and only if
      curl$\vec F = 0$.
    2. If $\vec F = f(x,y) \vec i + g(x,y) \vec j$, then $\vec F$ is a conserved field if and only if
      $ \frac{\p f}{\p y}=\frac{\p g}{\p x}$.
    3. To find the potential $\phi$ in a conserved field ($\vec F = - \nabla\phi$):
      (a) Verify $\mm$ curl$\vec F = 0.\m$
      (b) Integrate on an easy path from $\vec 0$ to $(x,y,z)$, to obtain $W = \int\vec F\cdot d\vec r .\m$
      (c) $\phi = - W$.

  6. Green's Formula (in a plane): $\m\intd{A} \Big(\frac{\p Q}{\p x} -\frac{\p P}{\p y} \Big) dxdy = \oint_{\p A} (P dx + Q dy)$.
  7. Divergence Theorem (in 2D): $\m \intd{A} \text{div }\vec V dxdy = \oint_{\p A } \vec V\cdot \vec n dS$.
  8. Divergence Theorem (in 3D): $\m \int\hspace{-0.7em}\intd{\tau} \text{div }\vec V d\tau = \intd{\p\tau} \vec V\cdot \vec n d\sigma$.