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\title{Advanced Calculus and \\Methods of Mathematical Physics}
\author{Review for Final Exam}
\date{Exam Date: May 25, 9:00--11:00}
\maketitle

\begin{enumerate}

\item Point-set topology: know the definitions of open, closed, and
compact sets, and their behavior under continuous maps.

See, e.g., Problem 1 from the Analysis II Review Sheet, Spring
Semester 2019.

\item Differentiation on normed vector spaces.

See, e.g., Question 5 on the Analysis II Final Exam, Spring Semester
2019; Question~4 on Mock Midterm Exam.

\item Inverse and implicit function theorem.

Homework 5, in particular Questions 1, 4; Mock Midterm Exam,
Question~5; Problem~1 from the Analysis II Review Sheet, Spring
Semester 2019.

\item Lagrange multipliers.

E.g., Problem 7 from the Analysis II Review Sheet, Spring
Semester 2019.

\item Power series, radius of convergence, Laurent series.

E.g., Homework 10 Problem 5, Problem 4 from the Analysis II Review
Sheet, Spring Semester 2019.

\item Change of variable formula.

E.g., use the change of variables formula to compute the area of the
image of the unit square under the map
\[
  \begin{pmatrix}
    x \\ y
  \end{pmatrix}
  \mapsto
  \begin{pmatrix}
    x+y \\ y^2-x
  \end{pmatrix} \,.
\]

\item Line and surface integrals; Green's theorem, the divergence
theorem, and Stokes' theorem; conservative vector fields.
\textbf{This topic area will be a major part of the exam.}

All of Homework 8, 9 and similar questions are highly relevant.

\item Fourier series: elementary properties.

Elementary questions only, e.g.\ Homework 10 Questions 1 and 2.

\item Complex variables.  Cauchy Riemann equations, setting up a
complex line integral, Cauchy theorem, residue theorem.

Elementary questions only, e.g.\ Homework 10 Questions 3 and 5.

\end{enumerate}
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