Take me to the Fall Time Schedule
100-299 |
300-499 |
500-599 |
316. Differential Equations.
Math. 215 and 217. Credit can be received for only
one of Math. 216 or Math. 316, and credit can be received for
only one of Math. 316 or Math. 404. (3). (Excl). (BS).
This is an introduction to differential equations for students
who have studied linear algebra (Math 217). It treats techniques
of solution (exact and approximate), existence and uniqueness theorems, some qualitative theory, and many applications. Proofs
are given in class; homework problems include both computational
and more conceptually oriented problems. First-order equations:
solutions, existence and uniqueness, and numerical techniques;
linear systems: eigenvector-eigenvalue solutions of constant coefficient
systems, fundamental matrix solutions, nonhomogeneous systems;
higher-order equations, reduction of order, variation of parameters, series solutions; qualitative behavior of systems, equilibrium
points, stability. Applications to physical problems are considered throughout. Math 216 covers somewhat less material without the
use of linear algebra and with less emphasis on theory. Math 286
is the Honors version of Math 316. Math 471 and/or 572 are natural
sequels in the area of differential equations, but Math 316 is
also preparation for more theoretical courses such as Math 451.
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333. Directed Tutoring.
Math. 385 and enrollment in the Elementary Program
in the School of Education. (1-3). (Excl). (EXPERIENTIAL). May
be repeated for a total of three credits.
An experiential mathematics course for exceptional upper-level
students in the elementary teacher certification program. Students
tutor needy beginners enrolled in the introductory courses (Math
385 and Math 489) required of all elementary teachers. Permission
of instructor. WL:2
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350/Aero. 350. Aerospace
Engineering Analysis. Math. 216, 256, 286, or 316.
(3). (Excl). (BS).
This is a three-hour lecture course in engineering mathematics
which continues the development and application of ideas introduced
in Math 215 and 216. The course is required in the Aerospace Engineering
curriculum, and covers subjects needed for subsequent departmental
courses. The major topics discussed include vector analysis, Fourier
series, and an introduction to partial differential equations, with emphasis on separation of variables. Some review and extension
of ideas relating to convergence, partial differentiation, and integration are also given. The methods developed are used in the formulation and solution of elementary initial- and boundary-value
problems involving, e.g., forced oscillations, wave motion, diffusion, elasticity, and perfect-fluid theory. There are two
or three one-hour exams and a two-hour final, plus about ten homework
assignments, or approximately one per week, consisting largely
of problems from the text. The text is Mathematical Methods
in the Physical Sciences by M.L. Boas. WL:2
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371/Engin. 371. Numerical
Methods for Engineers and Scientists. Engineering
101, and Math. 216. (3). (Excl). (BS).
This is a survey course of the basic numerical methods which are
used to solve practical scientific problems. Important concepts
such as accuracy, stability, and efficiency are discussed. The
course provides an introduction to MATLAB, an interactive
program for numerical linear algebra, and may provide practice
in FORTRAN programming and the use of a software library subroutine.
Convergence theorems are discussed and applied, but the proofs
are not emphasized. Floating point arithmetic, Gaussian elimination, polynomial interpolation, spline approximations, numerical integration
and differentiation, solutions to non-linear equations, ordinary
differential equations, polynomial approximations. Other topics
may include discrete Fourier transforms, two-point boundary-value
problems, and Monte-Carlo methods. Math 471 is a similar course
which expects one more year of maturity and is somewhat more theoretical
and less practical. The sequence Math 571-572 is a beginning graduate
level sequence which covers both numerical algebra and differential
equations and is much more theoretical. This course is basic for
many later courses in science and engineering. It is good background
for 571-572. WL:2
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385. Mathematics for
Elementary School Teachers. One year each of high
school algebra and geometry. No credit granted to those who have
completed or are enrolled in 485. (3). (Excl).
This course, together with its sequel Math 489, provides a coherent
overview of the mathematics underlying the elementary and middle
school curriculum. It is required of all students intending to
earn an elementary teaching certificate and is taken almost exclusively
by such students. Concepts are heavily emphasized with some attention
given to calculation and proof. The course is conducted using
a discussion format. Class participation is expected and constitutes
a significant part of the course grade. Enrollment is limited
to 30 students per section. Although only two years of high school
mathematics are required, a more complete background including
pre-calculus or calculus is desirable. Topics covered include
problem solving, sets and functions, numeration systems, whole
numbers (including some number theory) and integers. Each number
system is examined in terms of its algorithms, its applications, and its mathematical structure. There is no alternative course.
Math 489 is the required sequel. WL:2
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395(295). Honors Analysis
I. Math. 296 or permission of the Honors advisor.
(4). (Excl). (BS).
This course is a continuation of the sequence Math 295-296 and has the same theoretical emphasis. Students are expected to understand and construct proofs. This course studies functions of several
real variables. Topics are chosen from elementary linear algebra:
vector spaces, subspaces, bases, dimension, solutions of linear
systems by Gaussian elimination; elementary topology: open, closed, compact, and connected sets, continuous and uniformly continuous
functions; differential and integral calculus of vector-valued
functions of a scalar; differential and integral calculus of scalar-valued
functions on Euclidean spaces; linear transformations: null space, range, matrices, calculations, linear systems, norms; differential
calculus of vector-valued mappings on Euclidean spaces: derivative, chain rule, implicit and inverse function theorems. WL:2
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399. Independent Reading.
(1-6). (Excl). (INDEPENDENT). May be repeated for
credit.
Designed especially for Honors students. WL:2
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412. Introduction to
Modern Algebra. Math. 215, 255, or 285; and 217.
No credit granted to those who have completed or are enrolled
in 512. Students with credit for 312 should take 512 rather than
412. One credit granted to those who have completed 312. (3).
(Excl). (BS).
This course is designed to serve as an introduction to the methods
and concepts of abstract mathematics. A typical student entering this course has substantial experience in using complex mathematical
(calculus) calculations to solve physical or geometrical problems, but is unused to analyzing carefully the content of definitions
of the logical flow of ideas which underlie and justify these
calculations. Although the topics discussed here are quite distinct
from those of calculus, an important goal of the course is to
introduce the student to this type of analysis. Much of the reading, homework exercises, and exams consists of theorems (propositions, lemmas, etc.) and their proofs. Math 217 or equivalent
required as background. The initial topics include ones common
to every branch of mathematics: sets, functions (mappings), relations, and the common number systems (integers, rational numbers, real
numbers, complex numbers). These are then applied to the study
of particular types of mathematical structures such as groups, rings, and fields. These structures are presented as abstractions
from many examples such as the common number systems together
with the operations of addition or multiplication, permutations
of finite and infinite sets with function composition, sets of
motions of geometric figures, and polynomials. Notions such as
generator, subgroup, direct product, isomorphism, and homomorphism
are defined and studied. Math 312 is a somewhat less abstract
course which substitutes material on finite automata and other
topics for some of the material on rings and fields of Math 412.
Math 512 is an Honors version of Math 412 which treats more material
in a deeper way. A student who successfully completes this course
will be prepared to take a number of other courses in abstract
mathematics: Math 416, 451, 475, 575, 481, 513, and 582. All of these courses will extend and deepen the student's grasp of modern
abstract mathematics. WL:2
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413. Calculus for Social
Scientists. Not open to freshmen, sophomores or mathematics
concentrators. (3). (Excl). (BS).
A one-term course designed for students who require an introduction
to the ideas and methods of the calculus. The course begins with
a review of algebra and then surveys analytic geometry, derivatives, maximum and minimum problems, integrals, integration, and partial
derivatives. Applications to business and economics are given
whenever possible, and the level is always intuitive rather than
highly technical. This course should not be taken by those who
have had a previous calculus course or plan to take more than
one or two further courses in mathematics. The course is specially
designed for graduate students in the social sciences. WL:2
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416. Theory of Algorithms.
Math. 312 or 412 or CS 303, and CS 380. (3). (Excl).
(BS).
Many common problems from mathematics and computer science may
be solved by applying one or more algorithms
- well-defined procedures that accept input data specifying a
particular instance of the problem and produce a solution. Students
entering Math 416 typically have encountered some of these problems
and their algorithmic solutions in a programming course. The goal
here is to develop the mathematical tools necessary to analyze
such algorithms with respect to their efficiency (running time)
and correctness. Different instructors will put varying degrees
of emphasis on mathematical proofs and computer implementation
of these ideas. Typical problems considered are: sorting, searching, matrix multiplication, graph problems (flows, travelling salesman), and primality and pseudo-primality testing (in connection with
coding questions). Algorithm types such as divide-and-conquer, backtracking, greedy, and dynamic programming are analyzed using
mathematical tools such as generating functions, recurrence relations, induction and recursion, graphs and trees, and permutations. The
course often includes a section on abstract complexity theory
including NP completeness. This course has substantial overlap
with EECS 586 – more or less depending on the instructors. In
general, Math 416 will put more emphasis on the analysis
aspect in contrast to design of algorithms. Math 516
(given infrequently) and EECS 574 and 575 (Theoretical Computer
Science I and II) include some topics which follow those of this
course. WL:2
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417. Matrix Algebra I.
Three courses beyond Math. 110. No credit granted
to those who have completed or are enrolled in 217, 419, or 513.
(3). (Excl). (BS).
Many problems in science, engineering, and mathematics are best
formulated in terms of matrices – rectangular arrays of numbers.
This course is an introduction to the properties of and operations
on matrices with a wide variety of applications. The main emphasis
is on concepts and problem-solving, but students are responsible
for some of the underlying theory. Diversity rather than depth
of applications is stressed. This course is not intended for mathematics
concentrators, who should elect Math 217 or 513 (Honors). Topics
include matrix operations, echelon form, general solutions of
systems of linear equations, vector spaces and subspaces, linear
independence and bases, linear transformations, determinants, orthogonality, characteristic polynomials, eigenvalues and eigenvectors, and similarity theory. Applications include linear networks, least
squares method (regression), discrete Markov processes, linear
programming, and differential equations. Math 419 is an enriched
version of Math 417 with a somewhat more theoretical emphasis.
Math 217 (despite its lower number) is also a more theoretical
course which covers much of the material of 417 at a deeper level.
Math 513 is an Honors version of this course, which is also taken
by some mathematics graduate students. Math 420 is the natural
sequel but this course serves as prerequisite to several courses:
Math 452, 462, 561, and 571. WL:2
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419/EECS 400/CS 400.
Linear Spaces and Matrix Theory. Four terms of college
mathematics beyond Math 110. No credit granted to those who have
completed or are enrolled in 217 or 513. One credit granted to those who have completed Math. 417. (3). (Excl). (BS).
Math 419 covers much of the same ground as Math 417 but presents the material in a somewhat more abstract way in terms of vector
spaces and linear transformations instead of matrices. There is
a mix of proofs, calculations, and applications with the emphasis
depending somewhat on the instructor. A previous proof-oriented
course is helpful but by no means necessary. Basic notions of
vector spaces and linear transformations: spanning, linear independence, bases, dimension, matrix representation of linear transformations;
determinants; eigenvalues, eigenvectors, Jordan canonical form, inner-product spaces; unitary, self-adjoint, and orthogonal operators
and matrices, applications to differential and difference equations.
Math 417 is less rigorous and theoretical and more oriented to
applications. Math 217 is similar to Math 419 but slightly more
proof-oriented. Math 513 is much more abstract and sophisticated.
Math 420 is the natural sequel, but this course serves as prerequisite
to several courses: Math 452, 462, 561, and 571. WL:2
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422. Topics in Actuarial
Mathematics I. Math. 216, 256, 286, or 316. (3).
(Excl). (BS).
We will explore how much insurance affects the lives of students
(automobile insurance, social security, health insurance, theft
insurance) as well as the lives of other family members (retirements, life insurance, group insurance). While the mathematical models
are important, an ability to articulate why the insurance options
exist and how they satisfy the customer's needs are equally important.
In addition, there are different options available (e.g.
in social insurance programs) that offer the opportunity of discussing
alternative approaches. WL:2
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423. Mathematics of Finance.
Math. 217 and 425; CS 183. (3). (Excl). (BS).
This course is an introduction to the mathematical models used
in finance and economics with particular emphasis on models for
pricing derivative instruments such as options and futures. The
goal is to understand how the models derive from basic principles
of economics, and to provide the necessary mathematical tools
for their analysis. A solid background in basic probability theory
is necessary. Topics include risk and return theory, portfolio theory, capital asset pricing model, random walk model, stochastic
processes, Black-Scholes Analysis, numerical methods, and interest
rate models. WL:2
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425/Stat. 425. Introduction
to Probability. Math. 215, 255, or 285. (3). (MSA).
(BS).
Sections 001, 002, and 004. See Statistics
425.
Sections 003 and 005.
This course introduces students to useful and interesting ideas
of the mathematical theory of probability and to a number of applications
of probability to a variety of fields including genetics, economics, geology, business, and engineering. The theory developed together
with other mathematical tools such as combinatorics and calculus
are applied to everyday problems. Concepts, calculations, and derivations are emphasized. The course will make essential use
of the material of Math 116 and 215. Math concentrators should
be sure to elect sections of the course that are taught by Mathematics
(not Statistics) faculty. Topics include the basic results and methods of both discrete and continuous probability theory: conditional
probability, independent events, random variables, jointly distributed
random variables, expectations, variances, covariances. Different
instructors will vary the emphasis. Math 525 is a similar course
for students with stronger mathematical background and ability.
Stat 426 is a natural sequel for students interested in statistics.
Math 523 includes many applications of probability theory. WL:2
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431. Topics in Geometry
for Teachers. Math. 215, 255, or 285. (3). (Excl).
(BS).
This course is a study of the axiomatic foundations of Euclidean
and non-Euclidean geometry. Concepts and proofs are emphasized;
students must be able to follow as well as construct clear logical
arguments. For most students this is an introduction to proofs.
A subsidiary goal is the development of enrichment and problem
materials suitable for secondary geometry classes. Topics selected
depend heavily on the instructor but may include classification
of isometries of the Euclidean plane; similarities; rosette, frieze, and wallpaper symmetry groups; tessellations; triangle groups;
finite, hyperbolic, and taxicab non-Euclidean geometries. Alternative
geometry courses at this level are 432 and 433. Although it is
not strictly a prerequisite, Math 431 is good preparation for
531. WL:2
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433. Introduction to
Differential Geometry. Math. 215, or 255 or 285, and Math. 217(3). (Excl). (BS).
This course is about the analysis of curves and surfaces in 2-
and 3-space using the tools of calculus and linear algebra. There
will be many examples discussed, including some which arise in
engineering and physics applications. Emphasis will be placed
on developing intuitions and learning to use calculations to verify
and prove theorems. Students need a good background in multivariable
calculus (215) and linear algebra (preferably 217). Some exposure
to differential equations (216 or 316) is helpful but not absolutely
necessary. Topics covered include (1) curves: curvature, torsion, rigid motions, existence and uniqueness theorems; (2) global properties
of curves: rotation index, global index theorem, convex curves, 4-vertex theorem; (3) local theory of surfaces: local parameters, metric coefficients, curves on surfaces, geodesic and normal curvature, second fundamental form, Christoffel symbols, Gaussian and mean
curvature, minimal surfaces, classification of minimal surfaces
of revolution. 537 is a substantially more advanced course which
requires a strong background in topology (590), linear algebra
(513) and advanced multivariable calculus (551). It treats some
of the same material from a more abstract and topological perspective
and introduces more general notions of curvature and covariant
derivative for spaces of any dimension. Math 635 and Math 636
(Topics in Differential Geometry) further study Riemannian manifolds
and their topological and analytic properties. Physics courses
in general relativity and gauge theory will use some of the material
of this course. WL:2
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450. Advanced Mathematics
for Engineers I. Math. 216, 256, 286, or 316. (4).
(Excl). (BS).
Although this course is designed principally to develop mathematics
for application to problems of science and engineering, it also
serves as an important bridge for students between the calculus
courses and the more demanding advanced courses. Students are
expected to learn to read and write mathematics at a more sophisticated
level and to combine several techniques to solve problems. Some
proofs are given, and students are responsible for a thorough
understanding of definitions and theorems. Students should have
a good command of the material from Math 215, and 216 or 316, which is used throughout the course. A background in linear algebra, e.g., Math 217, is highly desirable, as is familiarity
with Maple software. Topics include a review of curves
and surfaces in implicit, parametric, and explicit forms; differentiability
and affine approximations; implicit and inverse function theorems;
chain rule for 3- space; multiple integrals; scalar and vector
fields; line and surface integrals; computations of planetary
motion, work, circulation, and flux over surfaces; Gauss' and Stokes' Theorems, derivation of continuity and heat equation.
Some instructors include more material on higher dimensional spaces
and an introduction to Fourier series. Math 450 is an alternative
to Math 451 as a prerequisite for several more advanced courses.
Math 454 and 555 are the natural sequels for students with primary
interest in engineering applications. WL:2
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451. Advanced Calculus
I. Math. 215 and one course beyond Math. 215; or
Math. 255 or 285. Intended for concentrators; other students should
elect Math. 450. (3). (Excl). (BS).
This course has two complementary goals: (1) a rigorous development
of the fundamental ideas of calculus, and (2) a further development
of the student's ability to deal with abstract mathematics and mathematical proofs. The key words here are "rigor"
and "proof"; almost all of the material of the course
consists in understanding and constructing definitions, theorems
(propositions, lemmas, etc.), and proofs. This is considered
one of the more difficult among the undergraduate mathematics
courses, and students should be prepared to make a strong commitment
to the course. In particular, it is strongly recommended that
some course which requires proofs (such as Math 412) be taken
before Math 451. Topics include: logic and techniques of proof;
sets, functions, and relations; cardinality; the real number system
and its topology; infinite sequences, limits and continuity; differentiation;
integration, the Fundamental Theorem of Calculus; infinite series;
sequences and series of functions. There is really no other course
which covers the material of Math 451. Although Math 450 is an
alternative prerequisite for some later courses, the emphasis
of the two courses is quite distinct. The natural sequel to Math
451 is 452, which extends the ideas considered here to functions
of several variables. In a sense, Math 451 treats the theory behind
Math 115-116, while Math 452 does the same for Math 215 and a
part of Math 216. Math 551 is a more advanced version of Math
452. Math 451 is also a prerequisite for several other courses:
Math 575, 590, 596, and 597. WL:2
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454. Boundary Value Problems
for Partial Differential Equations. Math. 216, 256, 286, or 316. Students with credit for Math. 354 can elect Math.
454 for one credit. (3). (Excl). (BS).
This course is devoted to the use of Fourier series and other
orthogonal expansions in the solution of boundary-value problems
for second-order linear partial differential equations. Emphasis
is on concepts and calculation. The official prerequisite is ample
preparation. Classical representation and convergence theorems
for Fourier series; method of separation of variables for the
solution of the one-dimensional heat and wave equation; the heat
and wave equations in higher dimensions; spherical and cylindrical
Bessel functions; Legendre polynomials; methods for evaluating
asymptotic integrals (Laplace's method, steepest descent); Fourier
and Laplace transforms; applications to linear input-output systems, analysis of data smoothing and filtering, signal processing, time-series
analysis, and spectral analysis. Both Math 455 and 554 cover many
of the same topics but are very seldom offered. Math 454 is prerequisite
to Math 571 and 572, although it is not a formal prerequisite, it is good background for Math 556. WL:2
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471. Introduction to
Numerical Methods. Math. 216, 256, 286, or 316; and 217, 417, or 419; and a working knowledge of one high-level computer
language. (3). (Excl). (BS).
This is a survey of the basic numerical methods which are used
to solve scientific problems. The emphasis is evenly divided between the analysis of the methods and their practical applications.
Some convergence theorems and error bounds are proved. The course
also provides an introduction to MATLAB, an interactive
program for numerical linear algebra, as well as practice in computer
programming. One goal of the course is to show how calculus and linear algebra are used in numerical analysis. Topics may include
computer arithmetic, Newton's method for non-linear equations, polynomial interpolation, numerical integration, systems of linear
equations, initial value problems for ordinary differential equations, quadrature, partial pivoting, spline approximations, partial differential
equations, Monte Carlo methods, 2-point boundary value problems, Dirichlet problem for the Laplace equation. Math 371 is a less
sophisticated version intended principally for sophomore and junior
engineering students; the sequence Math 571-572 is mainly taken
by graduate students, but should be considered by strong undergraduates.
Math 471 is good preparation for Math 571 and 572, although it
is not prerequisite to these courses. WL:2
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481. Introduction to
Mathematical Logic. Math. 412 or 451 or equivalent
experience with abstract mathematics. (3). (Excl). (BS).
All of modern mathematics involves logical relationships among
mathematical concepts. In this course we focus on these relationships themselves rather than the ideas they relate. Inevitably this
leads to a study of the (formal) languages suitable for expressing
mathematical ideas. The explicit goal of the course is the study
of propositional and first-order logic; the implicit goal is an
improved understanding of the logical structure of mathematics.
Students should have some previous experience with abstract mathematics
and proofs, both because the course is largely concerned with theorems and proofs and because the formal logical concepts will
be much more meaningful to a student who has already encountered these concepts informally. No previous course in logic is prerequisite.
In the first third of the course the notion of a formal language
is introduced and propositional connectives (and, or, not, implies), tautologies, and tautological consequence are studied.
The heart of the course is the study of first-order predicate
languages and their models. The new elements here are quantifiers
('there exists' and 'for all'). The study of the notions of truth, logical consequence, and provability lead to the completeness
and compactness theorems. The final topics include some applications
of these theorems, usually including non-standard analysis. Math
681, the graduate introductory logic course, also has no specific
logic prerequisite but does presuppose a much higher general level
of mathematical sophistication. Philosophy 414 may cover much
of the same material with a less mathematical orientation. Math
481 is not explicitly prerequisite for any later course, but the
ideas developed have application to every branch of mathematics.
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497. Topics in Elementary
Mathematics. Math. 489. (3). (Excl). (BS). May be
repeated for a total of six credits.
This is an elective course for elementary teaching certificate
candidates that extends and deepens the coverage of mathematics
begun in the required two-course sequence Math 385-489. Topics
are chosen from geometry, algebra, computer programming, logic, and combinatorics. Applications and problem-solving are emphasized.
The class meets three times per week in recitation sections. Grades
are based on class participation, two one-hour exams, and a final
exam. Selected topics in geometry, algebra, computer programming, logic, and combinatorics for prospective and in-service elementary, middle, or junior-high school teachers. Content will vary from
term to term. WL:2
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100-299 |
300-499 |
500-599 |
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