# Mathematical Analysis A Modern Approach To Advanced Calculus Homework

## Course Meeting Times

Lectures: Three sessions / week, 1 hour / session

## Prerequisites

The official prerequisites for this course are Analysis I (18.100B), and either Algebra I (18.700 or 18.701). Students who have taken Linear Algebra (18.06) may consider this course, though 18.700 or 18.701 are preferable.

## Textbooks

The main textbook for this course is:

Munkres, J. *Analysis on Manifolds*. Cambridge, MA: Perseus Publishing, 1991. ISBN: 0201510359, ISBN: 0201315963 (paperback).

Some material and homework problems will be taken from:

Spivak, M. *Calculus on Manifolds: A Modern Approach to Classical Theorems of Advanced Calculus*. Cambridge, MA: Perseus Publishing, 1965. ISBN: 0805390219.

## Homework

There are two kinds of homework assignments in this course: daily homework problems and graded assignments. The daily homework problems are not graded but will be used on the graded assignments. There will be a total of four graded assignments based on the daily homework.

## Exams

There is one midterm exam and one comprehensive final exam.

## Grading

ACTIVITIES | PERCENTAGES |
---|---|

Graded Assignments | 50% |

Midterm Exam | 20% |

Final Exam | 30% |

#### MATH 101: Calculus with Problem Solving

An introduction to the central ideas of calculus with review and practice of those skills needed for the continued study of calculus. Problem solving strategies will be emphasized. (Meets Monday through Friday).Prerequisites: Not open to students who have received credit for Math 111.*6 credit; Formal or Statistical Reasoning; offered Fall 2017*·

**Deanna Haunsperger**

#### MATH 111: Introduction to Calculus

An introduction to the differential and integral calculus. Derivatives, antiderivatives, the definite integral, applications, and the fundamental theorem of calculus.Prerequisites: Requires placement via the Calculus Placement Exam 1, see Mathematics web page. Not open to students who have received credit for Mathematics 101.*6 credit; Formal or Statistical Reasoning; offered Fall 2017, Winter 2018*·

**Owen D Biesel**,

**Rita B Thompson**

#### MATH 115: Statistics: Concepts and Applications

Introduction to statistical concepts with emphasis on understanding and interpretation of statistical information, especially in the context of media reports and scholarly articles. Examples taken from a wide-range of areas such as public policy, health and medicine, and the social and natural sciences. Computationally less intensive than Math 215. Students will learn how to use statistical software. Topics include: Uncertainty and variability, statistical graphs, types of studies, correlation and linear regression, two-way tables, and inference.

Prerequisites: Not open to students who have already received credit for Mathematics 211, Mathematics 215, Psychology 200/201, or Sociology/Anthropology 239*6 credit; Formal or Statistical Reasoning, Quantitative Reasoning Encounter; not offered 2017–2018*

#### MATH 120: Calculus 2

Inverse functions, integration by parts, improper integrals, modeling with differential equations, vectors, calculus of functions of two independent variables including directional derivatives and double integrals, Lagrange multipliers.

Prerequisites: Mathematics 101, 111, score of 4 or 5 on Calculus AB Exam, score of 5,6, or 7 on Mathematics IB exam or placement via a Carleton placement exam.*6 credit; Formal or Statistical Reasoning; offered Fall 2017, Winter 2018, Spring 2018*·

**Gail S Nelson**,

**Liz Sattler**,

**Rob Thompson**,

**Eric S Egge**,

**Owen D Biesel**,

**Stephen F Kennedy**

#### MATH 206: A Tour of Mathematics

A series of eight lectures intended for students considering a Mathematics major. The emphasis will be on presenting various striking ideas, concepts and results in modern mathematics, rather than on developing extensive knowledge or techniques in any particular subject area.*1 credit; S/CR/NC; Does not fulfill a curricular exploration requirement; offered Winter 2018*·

**Deanna Haunsperger**

#### MATH 210: Calculus 3

Vectors, curves, calculus of functions of three independent variables, including directional derivatives and triple integrals, cylindrical and spherical coordinates, line integrals, Green's theorem, sequences and series, power series, Taylor series.

Prerequisites: Math 120. Not open to students who have received credit for Math 211*6 credit; Formal or Statistical Reasoning; offered Winter 2018, Spring 2018*·

**Gail S Nelson**,

**Liz Sattler**,

**Josh Davis**,

**Eric S Egge**

#### MATH 211: Introduction to Multivariable Calculus

Vectors, curves, partial derivatives, gradient, multiple and iterated integrals, line integrals, Green's theorem.Prerequisites: Mathematics 121 or placement via Calculus Placement Exam #3*6 credit; Formal or Statistical Reasoning; offered Fall 2017, Spring 2018*·

**Sam Patterson**,

**Rafe Jones**,

**Mark Krusemeyer**

#### MATH 215: Introduction to Statistics

Introduction to statistics and data analysis. Practical aspects of statistics, including extensive use of statistical software, interpretation and communication of results, will be emphasized. Topics include: exploratory data analysis, correlation and linear regression, design of experiments, basic probability, the normal distribution, randomization approach to inference, sampling distributions, estimation, hypothesis testing, and two-way tables. Students who have received credit for Mathematics 115 may petition the department to seek approval to register for Mathematics 215. Students who have taken Mathematics 211 are encouraged to consider the more advanced Mathematics 265-275 Probability-Statistics sequence.

Prerequisites: Not open to students who have already received credit for Math 115, Psychology 200/201, Sociology/Anthropology 239 or Math 275.*6 credit; Formal or Statistical Reasoning, Quantitative Reasoning Encounter; offered Fall 2017, Winter 2018, Spring 2018*·

**Katie St. Clair**,

**Laura M Chihara**,

**Andy Poppick**,

**Adam Loy**

#### MATH 232: Linear Algebra

Vector spaces, linear transformations, determinants, inner products and orthogonality, eigenvectors and eigenvalues.Prerequisites: Mathematics 210 or 211*6 credit; Formal or Statistical Reasoning; offered Fall 2017, Winter 2018, Spring 2018*·

**Mark Krusemeyer**,

**Sam Patterson**,

**Owen D Biesel**,

**Rafe Jones**

#### MATH 236: Mathematical Structures

Basic concepts and techniques used throughout mathematics. Topics include logic, mathematical induction and other methods of proof, problem solving, sets, cardinality, equivalence relations, functions and relations, and the axiom of choice. Other topics may include: algebraic structures, graph theory, and basic combinatorics.Prerequisites: Mathematics 232 or instructor permission*6 credit; Formal or Statistical Reasoning; offered Fall 2017, Winter 2018, Spring 2018*·

**Deanna Haunsperger**,

**Gail S Nelson**

#### MATH 241: Ordinary Differential Equations

An introduction to ordinary differential equations, including techniques for finding solutions, conditions under which solutions exist, and some qualitative analysis.Prerequisites: Mathematics 232 or instructor permission*6 credit; Formal or Statistical Reasoning; offered Winter 2018, Spring 2018*·

**Sam Patterson**,

**Rob Thompson**

#### MATH 244: Geometries

Euclidean geometry from an advanced perspective; projective, hyperbolic, inversive, and/or other geometries. Recommended for prospective secondary school teachers.Prerequisites: Mathematics 236*6 credit; Formal or Statistical Reasoning; offered Spring 2018*·

**Stephen F Kennedy**

#### MATH 245: Applied Regression Analysis

A second course in statistics covering simple linear regression, multiple regression and ANOVA, and logistic regression. Exploratory graphical methods, model building and model checking techniques will be emphasized with extensive use of statistical software to analyze real-life data.Prerequisites: Mathematics 215 (or equivalent) or 275*6 credit; Formal or Statistical Reasoning, Quantitative Reasoning Encounter; offered Fall 2017, Winter 2018, Spring 2018*·

**Andy Poppick**,

**Laura M Chihara**,

**Adam Loy**

#### MATH 251: Chaotic Dynamics

An exploration of the behavior of non-linear dynamical systems. Topics include one and two-dimensional dynamics, Sarkovskii's Theorem, chaos, symbolic dynamics,and the Hénon Map.

Prerequisites: Mathematics 232 or instructor permission*6 credit; Formal or Statistical Reasoning; not offered 2017–2018*

#### MATH 255: Introduction to Sampling Techniques

Covers sampling design issues beyond the basic simple random sample: stratification, clustering, domains, and complex designs like two-phase and multistage designs. Inference and estimation techniques for most of these designs will be covered and the idea of sampling weights for a survey will be introduced. We may also cover topics like graphing complex survey data and exploring relationships in complex survey data using regression and chi-square tests.

Prerequisites: Mathematics 215 or 275*6 credit; Formal or Statistical Reasoning, Quantitative Reasoning Encounter; not offered 2017–2018*

#### MATH 261: Functions of a Complex Variable

Algebra and geometry of complex numbers, analytic functions, complex integration, series, residues, applications. Not open to students who have already received credits for Mathematics 361.Prerequisites: Mathematics 210 or 211*6 credit; Formal or Statistical Reasoning; not offered 2017–2018*

#### MATH 265: Probability

Introduction to probability and its applications. Topics include discrete probability, random variables, independence, joint and conditional distributions, expectation, limit laws and properties of common probability distributions.Prerequisites: Mathematics 210 or 211*6 credit; Formal or Statistical Reasoning; offered Fall 2017, Winter 2018*·

**Adam Loy**,

**Josh Davis**,

**Andy Poppick**

#### MATH 275: Introduction to Statistical Inference

Introduction to modern mathematical statistics. The mathematics underlying fundamental statistical concepts will be covered as well as applications of these ideas to real-life data. Topics include: resampling methods (permutation tests, bootstrap intervals), classical methods (parametric hypothesis tests and confidence intervals), parameter estimation, goodness-of-fit tests, regression, and Bayesian methods. The statistical package R will be used to analyze data sets.Prerequisites: Mathematics 265*6 credit; Formal or Statistical Reasoning, Quantitative Reasoning Encounter; offered Winter 2018, Spring 2018*·

**Adam Loy**,

**Katie St. Clair**

#### MATH 280: Statistical Consulting

Students will apply their statistical knowledge by analyzing data problems solicited from the Northfield community. Students will also learn basic consulting skills, including communication and ethics.Prerequisites: Mathematics 245 and instructor permission*2 credit; S/CR/NC; Formal or Statistical Reasoning, Quantitative Reasoning Encounter; offered Fall 2017, Winter 2018, Spring 2018*·

**Katie St. Clair**

#### MATH 285: Introduction to Data Science

This course will cover the computational side of data analysis, including data acquisition, management and visualization tools. Topics may include: data scraping, clean up and manipulation, data visualization using packages such as ggplots, understanding and visualizing spatial and network data, and supervised and unsupervised classification methods. We will use the statistics software R in this course.

Prerequisites: Mathematics 215 or Mathematics 275*6 credit; Formal or Statistical Reasoning, Quantitative Reasoning Encounter; offered Winter 2018*·

**Katie St. Clair**

#### MATH 295: Coding Theory

This course is an introduction to error-correcting codes. The course will cover topics including linear codes, Hamming codes and cyclic codes. Additional topics may include low-density parity-check codes and perfect codes.

Prerequisites: Mathematics 232*6 credit; Formal or Statistical Reasoning; not offered 2017–2018*

#### MATH 295: Differential Forms and Vector Calculus

Differential Forms provide a modern approach to a classical topic: Vector Calculus. They find applications in such diverse fields as geometry, algebra, engineering, electricity and magnetism, and general relativity. This course will rigorously develop differential forms then apply them to classical topics including divergence, gradient, and curl. A primary focus of the course will be the proof of the generalized Stokes' Theorem which is a general n-dimensional form of the familiar Fundamental Theorem of Calculus. Modern treatments of other topics from advanced calculus will be considered as time permits.

Prerequisites: Mathematics 236 or instructor permission*6 credit; Formal or Statistical Reasoning; offered Fall 2017*·

**Sam Patterson**

#### MATH 295: Numerical Analysis

Methods of mathematical approximation and applications to scientific computing. Topics include optimization, interpolation, numerical linear algebra, solution of differential equations, and Fourier methods. Both theory and implementation of numerical algorithms will be emphasized.

Prerequisites: Mathematics 232*6 credit; Formal or Statistical Reasoning; offered Winter 2018*·

**Rob Thompson**

#### MATH 295: Seminar in Low-dimensional Topology

A combinatorial introduction to the study of manifolds in dimensions less than four, including selected topics in knot theory.

Prerequisites: Mathematics 236*6 credit; Formal or Statistical Reasoning; not offered 2017–2018*

#### MATH 295: Seminar in Set Theory

Introduction to set-theoretic foundations of mathematics. The axiom system of Zermelo-Fraenkel, cardinal and ordinal numbers, and the Axiom of Choice. As time permits, additional topics may include construction of the real number, transfinite induction, or consistency/independence proofs.

Prerequisites: Mathematics 236 or instructor permission*6 credit; Formal or Statistical Reasoning; offered Spring 2018*·

**Gail S Nelson**

#### MATH 297: Assessment and Communication of External Mathematical Activity

An independent study course intended for students who have completed an external activity related to the mathematics major (for example, an internship or an externship) to communicate (both in written and oral forms) and assess their mathematical learning from that activity.Prerequisites: Permission of department chair and homework in advance of the external mathematical activity*1 credit; S/CR/NC; Does not fulfill a curricular exploration requirement; offered Fall 2017, Winter 2018, Spring 2018*·

**Eric S Egge**,

**Laura M Chihara**,

**Rob Thompson**

#### MATH 312: Elementary Theory of Numbers

Properties of the integers. Topics include the Euclidean algorithm, classical unsolved problems in number theory, prime factorization, Diophantine equations, congruences, divisibility, Euler's phi function and other multiplicative functions, primitive roots, and quadratic reciprocity. Other topics may include integers as sums of squares, continued fractions, distribution of primes, integers in extension fields, p-adic numbers.Prerequisites: Mathematics 236 or consent of the instructor.*6 credit; Formal or Statistical Reasoning; not offered 2017–2018*

#### MATH 315: Topics Probability/Statistics: Time Series Analysis

Models and methods for characterizing dependence in data that are ordered in time. Emphasis on univariate, quantitative data observed over evenly spaced intervals. Topics include perspectives from both the time domain (e.g., autoregressive and moving average models, and their extensions) and the frequency domain (e.g., periodogram smoothing and parametric models for the spectral density).

Prerequisites: Mathematics 245 and 275. Exposure to matrix algebra may be helpful but is not required*6 credit; Formal or Statistical Reasoning, Quantitative Reasoning Encounter; offered Fall 2017*·

**Andy Poppick**

#### MATH 321: Real Analysis I

A systematic study of concepts basic to calculus, such as topology of the real numbers, limits, differentiation, integration, convergence of sequences, and series of functions.Prerequisites: Mathematics 236 or permission of the instructor*6 credit; Formal or Statistical Reasoning; offered Fall 2017, Spring 2018*·

**Liz Sattler**,

**Gail S Nelson**

#### MATH 331: Real Analysis II

Further topics in analysis such as measure theory, Lebesgue integration or Banach and Hilbert spaces.Prerequisites: Mathematics 321 or instructor permission*6 credit; Formal or Statistical Reasoning; offered Winter 2018*·

**Gail S Nelson**

#### MATH 332: Advanced Linear Algebra

Selected topics beyond the material of Mathematics 232. Topics may include the Cayley-Hamilton theorem, the spectral theorem, factorizations, canonical forms, determinant functions, estimation of eigenvalues, inner product spaces, dual vector spaces, unitary and Hermitian matrices, operators, infinite-dimensional spaces, and various applications.Prerequisites: Mathematics 236 or instructor permission*6 credit; Formal or Statistical Reasoning; offered Fall 2017*·

**Eric S Egge**

#### MATH 333: Combinatorial Theory

The study of structures involving finite sets. Counting techniques, including generating functions, recurrence relations, and the inclusion-exclusion principle; existence criteria, including Ramsey's theorem and the pigeonhole principle. Some combinatorial identities and bijective proofs. Other topics may include graph and/or network theory, Hall's ("marriage") theorem, partitions, and hypergeometric series.Prerequisites: Mathematics 236 or instructor permission*6 credit; Formal or Statistical Reasoning; not offered 2017–2018*

#### MATH 341: Fourier Series and Boundary Value Problems

Fourier series and their applications to boundary value problems in partial differential equations. Topics include separation of variables, orthogonal sets of functions, representations of functions in series of orthogonal functions, Sturm-Liouville theory, and Fourier transforms.Prerequisites: Mathematics 241*6 credit; Formal or Statistical Reasoning; offered Spring 2018*·

**Rob Thompson**

#### MATH 342: Abstract Algebra I

Introduction to algebraic structures, including groups, rings, and fields. Homomorphisms and quotient structures, polynomials, unique factorization. Other topics may include applications such as Burnside's counting theorem, symmetry groups, polynomial equations, or geometric constructions.Prerequisites: Mathematics 236 or instructor permission*6 credit; Formal or Statistical Reasoning; offered Winter 2018*·

**Rafe Jones**

#### MATH 344: Differential Geometry

Local and global theory of curves, Frenet formulas. Local theory of surfaces, normal curvature, geodesics, Gaussian and mean curvatures, Theorema Egregium.Prerequisites: Mathematics 236 or permission of the instructor.*6 credit; Formal or Statistical Reasoning; not offered 2017–2018*

#### MATH 345: Advanced Statistical Modeling

Many data sets violate a central assumption underlying modeling via multiple regression, namely that the observations be independent. For example, longitudinal studies of test scores of children at different ages, analysis of birthweights of pups from the same litters, and electrical activity on different parts of the brain measured on a sample of patients all involve observations that are correlated. In this course, we will earn methods to address this problem; we will also learn about general linear models of which logistic and Poisson regression are special cases.

Prerequisites: Math 245 and Math 275 or permission of instructor. Familiarity with matrix algebra helpful but not required*6 credit; Formal or Statistical Reasoning, Quantitative Reasoning Encounter; offered Spring 2018*·

**Laura M Chihara**

#### MATH 349: Methods of Teaching Mathematics

Methods of teaching mathematics in grades 7-12. Issues in contemporary mathematics education. Regular visits to school classrooms and teaching a class are required.Prerequisites: Junior or senior standing and instructor permission*6 credit; Does not fulfill a curricular exploration requirement; not offered 2017–2018*

#### MATH 352: Topics in Abstract Algebra

An intensive study of one or more of the types of algebraic systems studied in Mathematics 342.Prerequisites: Mathematics 342*6 credit; Formal or Statistical Reasoning; offered Spring 2018*·

**Mark Krusemeyer**

#### MATH 354: Topology

An introduction to the study of topological spaces. We develop concepts from point-set and algebraic topology in order to distinguish between different topological spaces up to homeomorphism. Topics include methods of construction of topological spaces; continuity, connectedness, compactness, Hausdorff condition; fundamental group, homotopy of maps.

Prerequisites: Mathematics 236 or instructor permission*6 credit; Formal or Statistical Reasoning; offered Winter 2018*·

**Stephen F Kennedy**

#### MATH 361: Complex Analysis

The theoretical foundations for the calculus of functions of a complex variable. Not open to students who have taken Mathematics 351 Functions of a Complex Variable.Prerequisites: Mathematics 321 or instructor permission*6 credit; Formal or Statistical Reasoning; offered Spring 2018*·

**Liz Sattler**

#### MATH 365: Stochastic Processes

Introduction to the main discrete and continuous time stochastic processes. Topics include Markov chains, Poisson process, continuous time Markov chains, Brownian motion. Use of R and/or Mathematica.Prerequisites: Mathematics 232 and 265*6 credit; Formal or Statistical Reasoning, Quantitative Reasoning Encounter; not offered 2017–2018*

#### MATH 395: Topics in the Theory of Elliptic Curves

Introduction to the geometry and arithmetic of elliptic curves, with selected advanced topics. Introductory topics include the geometry of cubics, the group law on an elliptic curve, points of finite order, the group of rational points, heights and the Mordell-Weil theorem. Students will have the opportunity to explore advanced topics such as: integral points on elliptic curves; elliptic curves over finite fields; elliptic curves and cryptography; and links between elliptic curves and Fermat’s Last Theorem.

Prerequisites: Mathematics 342, an equivalent Budapest or Moscow Semester in Mathematics course or instructor permission*6 credit; Formal or Statistical Reasoning; not offered 2017–2018*

#### MATH 400: Integrative Exercise

Either a supervised small-group research project or an individual, independent reading. Required of all senior majors.Prerequisites: Mathematics 236 and successful completion of three courses from among: Mathematics courses numbered above 236, Computer Science 252, Computer Science 254*3 credit; S/NC; offered Fall 2017, Winter 2018, Spring 2018*·

**Rafe Jones**,

**Mark Krusemeyer**,

**Gail S Nelson**,

**Liz Sattler**,

**Katie St. Clair**,

**Rob Thompson**,

**Owen D Biesel**,

**Stephen F Kennedy**,

**Adam Loy**

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