$\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} $

Lecture 13, 9/21/2022. This page is for Section 1 only.
ACMS 20550: Applied Mathematics Method I
Instructor: Bei Hu, b1hu@nd.edu, Hurley 174A

  1. Partial Derivatives
    1. $\fcolorbox{white}{yellow}{Keep all other variable fixed while taking derivative} $, e.g., $\frac{\p z}{\p x}, \frac{\p^2 z}{\p x\p y}$.
    2. Notation: $\Big(\frac{\p z}{\p r}\Big)_x$ means taking derivatives with respect to $r$ while keeping $x$ fixed.
      Steps: (a) $\fcolorbox{white}{yellow}{Wirte $z$ as a function of $x$ and $r$,}$ (b) differentiate
  2. Taylor series
    $ f(x,y) = \sum_{n=0}^\infty \frac1{n!} \Big( h \frac{\p}{\p x}+ k \frac{\p}{\p y}\Big)^n f(a,b), \mm h =x-a, \m h= y-b$.