Skip to main content\(\require{cancel}\newcommand{\highlight}[1]{{\color{blue}{#1}}}
\newcommand{\ds}{\displaystyle}
\newcommand{\fp}{f'}
\newcommand{\fpp}{f''}
\newcommand{\lz}[2]{\frac{d#1}{d#2}}
\newcommand{\lzn}[3]{\frac{d^{#1}#2}{d#3^{#1}}}
\newcommand{\lzo}[1]{\frac{d}{d#1}}
\newcommand{\lzoo}[2]{{\frac{d}{d#1}}{\left(#2\right)}}
\newcommand{\lzon}[2]{\frac{d^{#1}}{d#2^{#1}}}
\newcommand{\lzoa}[3]{\left.{\frac{d#1}{d#2}}\right|_{#3}}
\newcommand{\plz}[2]{\frac{\partial#1}{\partial#2}}
\newcommand{\plzoa}[3]{\left.{\frac{\partial#1}{\partial#2}}\right|_{#3}}
\newcommand{\inflim}[1][n]{\lim\limits_{#1 \to \infty}}
\newcommand{\infser}[1][1]{\sum_{n=#1}^\infty}
\newcommand{\Fp}{F\primeskip'}
\newcommand{\Fpp}{F\primeskip''}
\newcommand{\yp}{y\primeskip'}
\newcommand{\gp}{g\primeskip'}
\newcommand{\dx}{\Delta x}
\newcommand{\dy}{\Delta y}
\newcommand{\ddz}{\Delta z}
\newcommand{\thet}{\theta}
\newcommand{\norm}[1]{\left\lVert#1\right\rVert}
\newcommand{\vnorm}[1]{\left\lVert\vec #1\right\rVert}
\newcommand{\snorm}[1]{\left|\left|\,#1\,\right|\right|}
\newcommand{\la}{\left\langle}
\newcommand{\ra}{\right\rangle}
\newcommand{\dotp}[2]{\vec #1 \cdot \vec #2}
\newcommand{\proj}[2]{\text{proj}_{\,\vec #2}{\,\vec #1}}
\newcommand{\crossp}[2]{\vec #1 \times \vec #2}
\newcommand{\veci}{\vec i}
\newcommand{\vecj}{\vec j}
\newcommand{\veck}{\vec k}
\newcommand{\vecu}{\vec u}
\newcommand{\vecv}{\vec v}
\newcommand{\vecw}{\vec w}
\newcommand{\vecx}{\vec x}
\newcommand{\vecy}{\vec y}
\newcommand{\vrp}{\vec r\hskip0.75pt '}
\newcommand{\vrpp}{\vec r\hskip0.75pt ''}
\newcommand{\vsp}{\vec s\hskip0.75pt '}
\newcommand{\vrt}{\vec r(t)}
\newcommand{\vst}{\vec s(t)}
\newcommand{\vvt}{\vec v(t)}
\newcommand{\vat}{\vec a(t)}
\newcommand{\px}{\partial x}
\newcommand{\py}{\partial y}
\newcommand{\pz}{\partial z}
\newcommand{\pf}{\partial f}
\newcommand{\unittangent}{\vec{{}T}}
\newcommand{\unitnormal}{\vec{N}}
\newcommand{\unittangentprime}{\vec{{}T}\hskip0.75pt '}
\newcommand{\R}{mathbb{R}}
\newcommand{\mathN}{\mathbb{N}}
\newcommand{\surfaceS}{\mathcal{S}}
\newcommand{\zerooverzero}{\ds \raisebox{8pt}{\text{``\ }}\frac{0}{0}\raisebox{8pt}{\textit{ ''}}}
\newcommand{\deriv}[2]{\myds\frac{d}{dx}\left(#1\right)=#2}
\newcommand{\myint}[2]{\myds\int #1\, dx= {\ds #2}}
\newcommand{\primeskip}{\hskip.75pt}
\newcommand{\abs}[1]{\left\lvert #1\right\rvert}
\newcommand{\sech}{\operatorname{sech}}
\newcommand{\csch}{\operatorname{csch}}
\newcommand{\curl}{\operatorname{curl}}
\newcommand{\divv}{\operatorname{div}}
\newcommand{\Hess}{\operatorname{Hess}}
\newcommand{\lt}{<}
\newcommand{\gt}{>}
\newcommand{\amp}{&}
\definecolor{fillinmathshade}{gray}{0.9}
\newcommand{\fillinmath}[1]{\mathchoice{\colorbox{fillinmathshade}{$\displaystyle \phantom{\,#1\,}$}}{\colorbox{fillinmathshade}{$\textstyle \phantom{\,#1\,}$}}{\colorbox{fillinmathshade}{$\scriptstyle \phantom{\,#1\,}$}}{\colorbox{fillinmathshade}{$\scriptscriptstyle\phantom{\,#1\,}$}}}
\)
APEX Calculus
Gregory Hartman, Ph.D.
Department of Applied Mathematics
Virginia Military Institute
Sean Fitzpatrick, Ph.D., Editor
Department of Mathematics and Computer Science
University of Lethbridge
Alex Jordan, Ph.D., Editor
Department of Mathematics
Portland Community College
Carly Vollet, M.S., Editor
Department of Mathematics
Portland Community College
October 7, 2024
