$\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 27, 10/31/2022. 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. Gradient, Directional derivatives
    1. Gradient of $\phi = \nabla \phi = $ grad $\phi = \Big(\frac{\p \phi}{\p x}, \frac{\p \phi}{\p y}, \frac{\p \phi}{\p z}\Big) = \fcolorbox{white}{pink}{the direction where $\phi$ increases most rapidly}$
    2. Directional derivative in the direction $\vec u$:
      $ \nabla \phi \cdot \vec u$
      Make sure $\vec u$ is a unit vector.
    3. Normal vector for the surface $\phi(x,y,z)=$ constant:
      $ \vec n = \frac{\nabla \phi }{|\nabla \phi |} $
    4. If normal vector is $\vec n = (n_1, n_2, n_3)$, then
      Tangent plane at $(x_0,y_0,z_0)$ is $\mm n_1(x-x_0) + n_2 (y-y_0)+ n_3(z-z_0)=0 $

      Normal line at $(x_0,y_0,z_0)$ is $\mm \frac{x-x_0}{n_1} = \frac{y-y_0}{n_2}= \frac{z-z_0}{n_3} $


  2. Other Expressions 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}$} $
      Cylindircal coordinate: $\nabla = \vec e_r \frac{\p f}{\p r} + \vec e_\theta \frac1r\frac{\p f}{\p \theta}+ \vec e_z \frac{\p f}{\p z} $
      Spherical coordinate: $\nabla = \vec e_r \frac{\p f}{\p r} + \vec e_\theta \frac1r\frac{\p f}{\p \theta}+ \vec e_\phi \frac1{r\sin\phi} \frac{\p f}{\p \phi} $
    2. $\fcolorbox{white}{yellow}{Divergence}$. 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} $}$
    3. $\fcolorbox{white}{yellow}{Curl}.$ 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| $}$
    4. $\fcolorbox{white}{yellow}{ Laplacian. }$ $\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} $}$