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\begin{document}


\section{Ma 227 -- Fall 2009}

\subsubsection{Matrices and Systems of Linear differential equations (Nagle,
Chap. 9)\protect\vspace{1pt}\protect\vspace{1pt}}

\vspace{1pt}

Systems of linear equations

\qquad row reduction

Matrix algebra

\qquad Addition

\qquad Multiplication by scalar

\qquad Multiplication of matrices

\qquad Inverse

\qquad Inverse by row operations

Calculus of matrices

Linear systems of differential equations

\qquad Normal form

\qquad Conversion of nth order d.e. to linear system

\qquad Representation of solutions

\qquad \qquad Homogeneous equation

\qquad \qquad \qquad Fundamental solutions set and fundamental solution
matrix\qquad \qquad 
\[
x_{h}=c_{1}\mathbf{x}_{1}(t)+c_{2}\mathbf{x}_{2}(t)+...+c_{n}\mathbf{x}%
_{n}(t)=\mathbf{X}(t)\mathbf{c} 
\]

\qquad \qquad Non-homogeneous equation

\[
\mathbf{x}(t)=\mathbf{x}_{h}(t)+\mathbf{x}_{p}(t) 
\]

\qquad Solutions for d.e. with constant coefficients

\qquad \qquad Homogeneous d.e.

\qquad \qquad \qquad Eigenvalues and eigenvectors

\qquad \qquad \qquad \qquad Characteristic equation

\qquad \qquad \qquad \qquad Linear independence of eigenvectors

\qquad \qquad \qquad \qquad Form of solution(s) in real \& complex cases

\vspace{1pt}\qquad \qquad Non-homogeneous d.e.

\qquad \qquad \qquad Undetermined coefficients

\qquad \qquad \qquad Variations of parameters (omitted from course)

\qquad \qquad Matrix exponential function

\qquad \qquad \qquad From fundamental solution matrix

\qquad \qquad \qquad Nil-potent matrix (one distinct eigenvalue)

\qquad \qquad \qquad Generalized eigenvectors (omitted from course)

\subsection{\protect\vspace{1pt}Multiple integrals (Stewart, Chap. 12)}

\vspace{1pt}

Double integrals

\qquad Limits of iterated integral from region of integration

\qquad Region of integration from limits of integration

\qquad Interchange of order of integration

\qquad Equivalent iterated integrals in rectangular and polar coordinates

\[
\diint\limits_{R}f(x,y)dA=\diint\limits_{R}f\left( x,y\right)
dydx=\diint\limits_{R}f\left( x,y\right) dxdy=\diint\limits_{R}f\left( r\cos
\theta ,r\sin \theta \right) rdrd\theta 
\]

\vspace{1pt}

Triple integrals

\qquad Limits of iterated integral from region of integration

\qquad Region of integration from limits of integration

\qquad Interchange of order of integration

\qquad Equivalent iterated integrals in rectangular, cylindrical and
spherical coordinates

\begin{eqnarray*}
\diiint\limits_{R}f\left( x,y,z\right) dV &=&\diiint\limits_{R}f\left(
x,y,z\right) dzdydx=... \\
&=&\diiint\limits_{R}f\left( r\cos \theta ,r\sin \theta ,z\right)
rdzdrd\theta \\
&=&\diiint\limits_{R}f\left( \rho \cos \theta \sin \phi ,\rho \sin \theta
\sin \phi ,\rho \cos \phi \right) \rho ^{2}\sin \phi d\rho d\phi d\theta
\end{eqnarray*}

\vspace{1pt}

Application of double integral to surface area

\qquad Graph of a function, \ $z=f(x,y)$

\[
A(S)=\diint\limits_{R}\sqrt{1+f_{x}^{2}+f_{y}^{2}}dA_{xy} 
\]

\vspace{1pt}\qquad Parametric representation \ \ $\mathbf{r}(u,v)=x\left(
u,v\right) \mathbf{i}+y\left( u,v\right) \mathbf{j}+z\left( u,v\right) 
\mathbf{k}$%
\[
A\left( s\right) =\diint\limits_{D}\left\vert \mathbf{r}_{u}\times \mathbf{r}%
_{v}\right\vert dA_{uv} 
\]%
\pagebreak

\subsection{Vector calculus (Stewart, Chap. 13)}

\vspace{1pt}

Operations relating to vector functions (fields) \ [$\mathbf{F}=P\mathbf{i}+Q%
\mathbf{j}+R\mathbf{k}$]

\qquad Gradient \ $\mathbf{F}=\bigtriangledown f=\frac{\partial f}{\partial x%
}\mathbf{i}+\frac{\partial f}{\partial y}\mathbf{j}+\frac{\partial f}{%
\partial z}\mathbf{k}$ \ (input scalar, output vector)

\qquad Divergence $\bigtriangledown \bullet \mathbf{F=}\frac{\partial P}{%
\partial x}\mathbf{i}+\frac{\partial Q}{\partial y}\mathbf{j}+\frac{\partial
R}{\partial z}\mathbf{k}$\ (input vector, output scalar)

\qquad Curl $\bigtriangledown \times \mathbf{F=}\left\vert 
\begin{array}{ccc}
\mathbf{i} & \mathbf{j} & \mathbf{k} \\ 
\frac{\partial }{\partial x} & \frac{\partial }{\partial y} & \frac{\partial 
}{\partial z} \\ 
P & Q & R%
\end{array}%
\right\vert $\ (input vector, output vector)

\vspace{1pt}

Line integral \ \ $\int_{C}\mathbf{F}\bullet d\mathbf{r=}\int_{C}Pdx+Qdy+Rdz$%
\ 

\qquad Parametrize C

\qquad Set up and evaluate in 2 or 3 dimensions

\qquad Independence of path: $\mathbf{F}=\bigtriangledown \phi $

\qquad \qquad Test: $curl\mathbf{F}=\nabla \times \mathbf{F}=\mathbf{0}$

\qquad \qquad Determination of $\phi $

\vspace{1pt}

Surface integral of scalar function \ $\iint\limits_{S}fdS=\iint%
\limits_{S}fds$

\qquad Surface given explicitly by $z=g(x,y)$ \ : $dS=\sqrt{%
1+g_{x}^{2}+g_{y}^{2}}dA_{xy}$

\ \ \ \ \ \ \ \ \ \ and the region of integration is the domain of $g(x,y).$

\vspace{1pt}

\qquad Surface given parametrically \ $\mathbf{r=}\left\langle
x,y,z\right\rangle =x\mathbf{i}+y\mathbf{j}+z\mathbf{k}=\left\langle x\left(
u,v\right) ,y\left( u,v\right) ,z\left( u,v\right) \right\rangle $ :

\qquad \qquad $dS=\left\vert \mathbf{r}_{u}\times \mathbf{r}_{v}\right\vert
dA_{uv}$ and the region of integration is the set of values of the
parameters u and v which define the surface.

\vspace{1pt}

Surface integral of vector function \ $\iint\limits_{S}\mathbf{F}\bullet d%
\mathbf{S}=\iint\limits_{S}\mathbf{F}\bullet \mathbf{n}dS=\iint\limits_{D}%
\mathbf{F}\bullet \left( \mathbf{r}_{u}\times \mathbf{r}_{v}\right) dA_{uv}$

\qquad Here we use the parametric representation and that $\mathbf{r}%
_{u}\times \mathbf{r}_{v}$ is normal to the surface.

\vspace{1pt}

\vspace{1pt}\pagebreak

\textbf{Equivalencies}

\vspace{1pt}

Green's Theorem \ (in the plane)

\vspace{1pt}

\qquad $\iint\limits_{S}\left( \frac{\partial Q}{\partial x}-\frac{\partial P%
}{\partial y}\right) dA=\oint\limits_{\partial S}Pdx+Qdy$ \ with the
direction around the boundary so that the surface is to the left.

\vspace{1pt}

Stoke's Theorem (in 3D space)

\vspace{1pt}

\qquad $\iint\limits_{S}curl\mathbf{F}\bullet d\mathbf{S=}%
\iint\limits_{S}\left( \nabla \times \mathbf{F}\right) \bullet \mathbf{n}%
dS=\int\limits_{\partial S}\mathbf{F}\bullet d\mathbf{r}$ \ where the
direction around the boundary and of the normal must be matched.

\vspace{1pt}

Divergence Theorem (in 3D space)

\vspace{1pt}

\qquad $\iiint\limits_{E}\left( \nabla \bullet \mathbf{\ F}\right)
dV=\iint\limits_{\partial E}\mathbf{F}\bullet d\mathbf{S=}%
\iint\limits_{\partial E}\mathbf{F}\bullet \mathbf{n}dS$ and the direction
of the normal must be outward from the solid.

\vspace{1pt}

\vspace{1pt}

\vspace{1pt}

Texts: R. Kent Nagle, Edward B. Saff and Arthur David Snider: \textit{%
Fundamentals of Differential Equations and Boundary Value Problems, Fourth
Ed.}

James Stewart: \textit{Calculus -- Concepts and Contexts, Third Ed.}

\vspace{1pt}

General review - Monday, 21 December 2009 - K 228

\vspace{1pt}

1:00 - 1:50 \ Systems of DEs \ (Prof. Levine)

\ 2:00 - 2:50 \ Multiple Integration (Prof. Levine)

\ 3:00 - 3:50 \ Gradient, Divergence, Curl, Line Integrals \& Green's Theorem

\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ (Prof. Brady)

\ 4:00 - 4:50 \ Surface Integrals, Stokes's Theorem, Divergence Theorem

\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ (Prof. Brady)

\end{document}
