$\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 26, 10/28/2022. 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. Fields: a collection of values at different points in space.
    1. Scalor field: a collection of scalors associated with every point. Ex: the height of a mountain on every point of a map.
    2. Vector field: a collection of vectors associated with every point. Ex: the current flow (magnitude and direction) in a lake at every point
  2. 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$
      $\fcolorbox{white}{yellow}{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 |} $