$\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}}} $

Chap 6 Sections 1 - 8 Summary

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
    $\fcolorbox{white}{yellow}{$\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
    $\fcolorbox{white}{yellow}{$ \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. $\fcolorbox{white}{yellow}{Work $\vec F \cdot \vec d$}$;
    2. $\fcolorbox{white}{yellow}{Torque $\vec r\times \vec F$}$;
    3. $\fcolorbox{white}{yellow}{Angular velocity $\vec \omega$, then rotational velocity $\vec v = \vec \omega \times \vec r$}$;
    4. $\fcolorbox{white}{yellow}{Velocity, accelaration, and their magnitudes}$;

  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}$}$, $\fcolorbox{white}{yellow}{$\nabla\phi = $ grad $\phi = \Big(\frac{\p\phi}{\p x}, \frac{\p\phi}{\p y}, \frac{\p\phi}{\p z}\Big)$}$.
    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 $\phi(x,y,z)=0$: $\m \vec n = \frac{\nabla \phi}{|\nabla\phi|}$}$ Associated with $\fcolorbox{white}{yellow}{Tangent Plane and Normal Line}$.
    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. $\fcolorbox{white}{yellow}{$\vec F$ is a conserved field if and only if}$
      $\fcolorbox{white}{yellow}{curl$\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. $\fcolorbox{white}{yellow}{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) $\phi = - W$.