Double Majoring in Math and Other Disciplines
In many fields, a strong mathematical background is
useful or indispensable
for deeper understanding. Roughly one half of the math majors at UMCP
also major in another discipline. In addition to the deeper
understanding
attained, they achieve a significantly broader education and gain an
important
credential for employment or graduate school.
Below on this page, we have information in the following categories:
For further information please drop by the
Department
to see the Mathematics Coordinator of Undergraduate Advising, Ida Chan,
Room: 1115, Phone:
(301) 4057582, Email:
ugadvisor@math.umd.edu.
Mathematics Double Major & Courses for Students in
Double Major vs. Double Degree
A "Double Major'' is a student who plans to receive a
single degree,
but who will complete the requirements for two majors. In this case the
last line of the student's transcript will state that a single
Bachelor's
Degree was awarded and that the student had two majors. Alternately, a
student may complete a "Double Degree'' program. This requires the
student
to complete the requirements for both majors and also to obtain a total
of 150 credits. In this case the student may actually receive two
separate
Bachelor's Degrees. For more information, consult the Undergraduate
Catalog
or contact an advisor.
How to Add a Second Major
If you are considering double majoring, you should set
up appointments
with advisors in both disciplines. For an appointment with a
mathematics
advisor, send email to ugadvisor@math.umd.edu.
The advisors will help you to complete the "Courses for
a Double Major''
Form, which establishes a proposed course of study for satisfying the
requirements
of both programs. Once this form is complete, you may apply for the
addition
of a second major by visiting the main office of the college affiliated
with the second major. (Mathematics falls under the CMPS college; their
office is Room 3400 A.V. Williams Building.)
Graduate and Professional
Degree Programs for which
double majoring
would serve as a particularly good preparation:
Program Name & Indicated Double Major
Actuarial Science MATH & BMGT
Artificial Intelligence MATH & CMSC
Biometry/Epidemiology MATH & BIOL
Computer Science MATH & CMSC
Demography MATH & SOCY
Econometrics MATH & ECON
Educational Statistics MATH & EDMS
Engineering Reliability MATH & ENME
Financial Mathematics MATH & BMGT or ECON
Image Processing MATH & CMSC
Network Optimization MATH & BMGT
Network Performance MATH & ENEE
Neural Computing MATH & CMSC
Operations Research MATH & BMGT
Psychological Statistics MATH & PSYC
Scientific Computing MATH & PHYS or CHEM or CMSC
Signal Processing MATH & ENEE
Survey Methodology MATH & SOCY
Mathematics Education MATH & EDUC
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MATHEMATICS DOUBLE MAJOR FOR CHEMISTRY STUDENTS
Chemistry is a fundamental science underlying
biomedical, materials,
environmental and earth sciences. Research in Chemistry involves
mathematical
and statistical tools in several essential ways, which are
indispensable
both to theoreticians and to applied investigators through scientific
computing.
Physical chemistry, like physics, is intrinsically mathematical; and
the
description of chemical reaction kinetics and thermodynamics have
always
been formulated in terms of differential equations. Molecular
simulation
can involve Monte Carlo methods to model interaction potentials to
determine
behaviors of individual molecules or ensembles of molecules.
Computational
Chemistry employs quantum mechanics in particular for the calculation
of
"wavefunctions" to model molecular structure and energetics. Even very
applied chemistry research nowadays involves sophisticated imaging
equipment
and computer datareduction, interpretation, and simulation.
Development
of new chemometric techniques requires some understanding of
mathematical
algorithms and statistical and scientific computing, and a double major
with Mathematics is a natural way to develop such understanding.
Calculus (MATH 140141) is a requirement for UMCP
biochemistry and chemistry
majors. Differential equations are fundamental to chemical
kinetics
and thermodynamics, and all chemistry majors interested in graduate
study
or research would benefit from a basic differential equations course
(MATH
246).
Beyond this level, the mathematical tools of interest
vary for specialties
within chemistry, but a mathematically strong student who aspires to a
solid foundation for graduate work should take multivariable calculus
(MATH
241) and linear algebra (MATH 240 or 461), and should seriously
consider
advanced calculus (MATH 410411) and other 400level math courses (see
the discussion for PHYSICS majors). Partial
differential
equations (MATH 462) are the mathematical tools which a chemist would
use
to describe chemical reaction kinetics, mass transport and
thermodynamics.
More advanced mathematical analysis (Fourier analysis in MATH 464, plus
Hilbert space theory) is crucial to quantum mechanics and to theory
within
physical chemistry. Group theory (treated with other abstractalgebra
topics
in MATH 402 or 403) enters in the study of symmetries in
crystallography.
Geometry enters in the description and visualization of spatial
molecular
structure through projective imaging techniques (so Chemistry students
may be interested in MATH 431, Geometry for Computer Graphics).
Fourier
analysis (MATH 464) is a crucial tool in unraveling the information
contained
in chemometric imaging technologies. Probability theory (STAT 400 or
410)
arises in chemistry in several ways, specifically in describing and
simulating
relative frequencies of sequence patterns in DNA and other biological
molecules,
in modeling
structure of DNAprotein complexes and polymer sequence configurations,
and more generally in Monte Carlo simulation methodology. Statistical
tools
(STAT 401,430) often enter applied chemistry research, in roles ranging
from patternrecognition techniques of multivariate data analysis
(e.g.,
in screening large collections of compounds for specific types of
reactive
properties), to validation of compound or sequence identifications from
sequences of titrations which are not individually conclusive (as in
DNA
sequencing and chemical forensics), to statistical quality control of
chemical
processes. A basic understanding of elementary probability and
statistics
can be obtained from STAT 400.
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MATHEMATICS DOUBLE MAJOR FOR COMPUTER SCIENCE STUDENTS
Computer Science interacts closely with mathematics in
several ways,
which go far beyond the topics in discrete mathematics which all
Computer
Science majors learn. The interactions are sketched briefly below by
reference
to computer science specializations within which mathematical topics
arise.
For some of these interactions, the relationship with mathematics is
primarily
conceptual and serves as theoretical background; for others, such as
the
study of algorithms and error estimates in parallel computing and the
statistical
methods used in artificial intelligence and image processing,
mathematical
thinking is essential in practice.
Modeling of computer network performance and
demand/traffic streams
is a central and growing part of the design and testing of computer
architectures,
priority schemes, and resourceallocation algorithms. Probability,
statistics,
and stochastic processes are important mathematical tools for these
tasks.
STAT 410 (Probability) and STAT 405 (Stochastic Processes and Queueing)
are the relevant MATH courses in this direction.
Formal logic appears early and often in theoretical
computer science;
and as is emphasized in computer science courses on Discrete
Mathematics,
many computerscience problems related to datastructures are
essentially
combinatorial. These connections explain why certain types of
mathematics
necessarily arise in theoretical computer science. Issues concerning
the
types of questions which can be decided computationally using languages
and dataprocessing of various types are also research topics in Logic.
The Mathematics Department's logic courses are MATH 450 at an
elementary
level, and MATH 446447 at a deeper level.
There is a close connection between Mathematics and the design
and
performance of algorithms. For example, questions of efficient
largescale
parallel scientific computations are studied via theoretical numerical
analysis. Since the performance of algorithms is often assessed in
terms
of their use of time and resources in larger and larger problems, they
require the study of asymptotic rates of growth of combinatorial
structures;
of the worst or most costly configurations of data to which the
algorithms
could be applied; and also, of average behavior when the algorithm is
applied
to randomly configured inputs. Courses of particular relevance to these
topics are MATH 475 (Combinatorics) and STAT 410 (Probability).
Within the areas of Network Design and System
Security, the mathematics
of Cryptology (MATH 456)  closely related to Number Theory, which is
introduced in MATH 406  is especially useful. Computer Scientists
encounter
this material also in studying coding for datacompression.
Several directions in Computer Science are explicitly
statistical: Artificial
Intelligence, Pattern Recognition, and Neural Networks all address
aspects of adaptivity and automated learning based upon data. By their
nature, these approaches to automated learning involve highly
parameterized
models, and the estimation of the necessary parameters relies on the
mathematics
of optimization and numerical analysis. Moreover, although formal
probabilistic
models of data are not always relevant to practice in these fields, a
conceptual
grounding in probability and statistics is very useful. The
courses
offered in these areas are MATH 477 (Optimization), MATH 466 (Numerical
Analysis), and in Statistics: STAT 400401 at an introductory level,
and
STAT 410420 (which can be taken with only multivariablecalculus
prerequisites)
at a deeper level.
Another important and highly mathematical set of topics
is Computer
Vision and Image Processing. The relevant mathematics here includes
geometry (MATH 431, and at a deeper level Projective and Differential
Geometry
in MATH 436); differential equations (MATH 246, MATH 414) for the
description
of motion; and also Statistics (to make formal sense of blurring,
distortion,
and superpositions of signals). Other topics in mathematical analysis,
such as Fourier Analysis (MATH 417) and Wavelets, also play a role in
image
analysis.
Software Engineering employs statistical tools in
several ways.
Either in management of the development cycle, or in modeling and
testing
of software reliability, designed datacollection
of
software "metrics' is used to characterize difficulty and cost of
(portions of) software projects. Basic statistical tools and concepts
are
covered in STAT 400401. But Software Engineers should also be educated
consumers of statistical methods such as regression and analysis of
variance
(covered in STAT 450 and, with a more computational focus, in
STAT
430), and experimental design.
For Scientific Computation, the study of
numerical analysis and
differential equations is essential. Numerical analysis is introduced
in
MATH/AMSC 460 and 466; 466 is a more theoretical and rigorous version
of
460. The introductory differential equations course is MATH 246
and
there are several upperlevel courses (MATH 414, 415, 462).
Differential equations are also important in the study of image
processing,
as mentioned above.
Many new directions in Computer Science can be
seen to rely heavily
on calculusbased mathematical analysis. The rigorous foundation for
all
of the Analysis topics mentioned above (including differential
equations,
numerical analysis, Fourier analysis, probability and statistics) is
taught
in MATH 410 and 411 respectively in one and several dimensions. The
more
deeply one needs to understand any of the mathematical topics listed
(for
example, the more clearly one needs to understand the performance of
algorithms
in scientific computation), the more important a grasp of this basic
theory
becomes. A student inclined toward advanced work (or graduate school)
in
a related area should consider taking these courses.
MATH courses connected with computer science
specializations:
Course & Title/Topic; CMSC Specialization
MATH 401 Applic. of Linear Algebra; Systems, AI
MATH 406 Intro. to Number Theory; Systems
MATH 417 Fourier Analysis; Signal Processing
MATH 420 Mathematical Modeling; Systems, Software
Eng'g, AI
MATH 431 Math. of Computer Graphics; AI, Interfaces
MATH 446 Axiomatic Set Theory; CS Theory
MATH 447 Math. Logic; CS Theory
MATH 450 Logic for Computer Sci.; all areas
MATH 456 Cryptology; Systems
MATH 475 Combinatorics & Graph Theory; all
areas
MATH 477 Optimization & AI, Scientific
Computing
AMSC 466 Numerical Analysis; Scientific Computing, AI
STAT 400 Intro to Prob/Stat, I; all areas
STAT 401 Intro to Prob/Stat, II; all areas
STAT 405 Stochastic Processes/Queueing; Systems,
Algorithms
STAT 410 Probability Theory; Theory, Algorithms
STAT 430 SAS & Introductory Regression; Systems,
Software
Eng'g
STAT 450 Regression & ANOVA; Systems, Software
Eng'g
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MATHEMATICS DOUBLE MAJOR FOR PHILOSOPHY STUDENTS
Philosophers study various aspects of the relation
between formal or
axiomatic structures and human rationality or understanding.
Mathematics
can play a fundamental role in elucidating understanding through the
study
of Logic (MATH 450, 446, and 447), for example through the study of
decidability
of propositions, types of knowledge accessible to a computer, etc.
Problems
of Epistemology and the Philosophy of Science can be addressed through
the study of Inference (STAT 400, 401, 410, 420, 450), mathematical
modelbuilding
(MATH 420), and in particular the theory of dynamical models (MATH 246,
414, and 452) elucidating what can and cannot be predicted from such
models.
In addition, Philosophers are interested in formal models of learning
and
consciousness through neural networks, artifical intelligence, and the
like, many of which concern problems of Optimization (studied in MATH
477).
Other formal approaches to the study of human decisions and rationality
make formal use of game theory (addressed in MATH 475).
Many of the interfaces between Philosophy and
Mathematics also concern
topics in Computer Science, but to the extent that these questions are
going to be accessible to theory, a strong background in mathematics is
likely to be needed. See the page on Computer
Science and Math double majors for further discussion and
information.
The upperlevel mathematics courses likely to be of
greatest interest
to Philosophy students are:
Course Number & Title/Topic
MATH 246 Differential Equations
MATH 414 Differential Equations
MATH 420 Mathematical Modeling
MATH 446 Axiomatic Set Theory
MATH 447 Mathematical Logic
MATH 450 Logic for Computer Scientists
MATH 452 Dynamics & Chaos
MATH 475 Graph Theory & Combinatorics
MATH 477 Optimization
STAT 400 Intro. to Prob/Stat I
STAT 401 Intro. to Prob/Stat II
STAT 410 Probability Theory (adv.calc. level)
STAT 420 Statistical Theory
STAT 450 Regression
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MATHEMATICS DOUBLE MAJOR FOR ECONOMICS STUDENTS
Modern economic theory has become more and more
quantitative over the
last two generations, and in recent years the mathematical demands and
expectations of many graduate schools in economics have escalated
significantly.
An undergraduate interested in pursuing graduate studies either in
either
economics or econometrics will be well advised to take upperlevel
courses
in mathematical analysis, differential equations, and statistics.
Moreover,
many employment opportunities in economics involve analysis and
forecasting
of economic data, often in the form of time series.
The behavior over time of formally defined economic
systems is often
modeled in terms of ordinary differential equations (ODE's). Therefore
economists are likely to benefit from courses on solution techniques
(MATH
246), qualitative behavior of solutions (such as chaotic behavior,
blowup,
stability, convergence, etc., in courses like MATH 414 and 452), and
numerical
analysis of methods for computing solutions (AMSC 460 or 466).
Other theoretical approaches to economics involve
characterizing the
equilibria of economic systems. "General Equilibrium Theory" in
Economics
rests on a mathematical basis of rigorous analysis (such as would
be introduced in the courses MATH 410411) including properties of
convex
function and some introductory topology of metric spaces. These topics
would be elaborated in a form suitable for economists in a course on
Optimization
(MATH 477).
Finally, all branches of economics nowadays rest on the
extraction of
estimates of theoretical parameters from empirical data, and
Econometrics
makes this a specialization. Useful introductory courses on concepts of
Probability and Statistics include STAT 400 at univariatecalculus
level,
continuing in STAT 401; or STAT 410420 as the advancedcalculuslevel
introductory sequence which would be particularly suitable as a
preparation
for graduate study. A more advanced course on Regression (STAT 450), or
a new course (STAT 430) emphasizing Regression topics using the SAS
statistical
software package could serve economics students as a mathematical
capstone.
Students with some interest in Insurance (e.g. the Actuarial
profession)
would take the Actuarial Mathematics course, STAT 470.
Economics majors may also be interested in Mathematical
Finance, the
theoretical underpinning for the methods of pricing Options and complex
derivative securities on Wall Street. Some necessary background for the
deeper study of these topics would include partial differential
equations
(MATH 462), probability (STAT 410), and numerical/computer methods
(AMSC
460).
Upperlevel math courses of general interest to
Economics majors:
Course Number & Title/Topic
MATH 401 Appl. of Linear Algebra
MATH 405 Linear Algebra Theory
MATH 410 Advanced Calculus I
MATH 411 Advanced Calculus II
MATH 414 Differential Equations
MATH 420 Modeling
MATH 452 Dynamics & Chaos
AMSC 460 Computational Methods
MATH 462 Partial Differential Equations
AMSC 466 Numerical Analysis
AMSC 477 Optimization
STAT 400401 Prob/Stat
STAT 410 Probability Theory (advancedcalculus level)
STAT 420 Intro. to Statistical Theory
STAT 450 Regression
STAT 470 Actuarial Math.
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MATHEMATICS DOUBLE MAJOR FOR BUSINESS STUDENTS
Quantitative topics of immediate importance to business
majors include:
optimal resourceallocation, routing and scheduling, inventory or
portfolio
management; modeling of risk and uncertainty for business decisions;
forecasting
of macroeconomic and market conditions; statistical methods of quality
control; and statistical methods of sampling and auditing. The primary
mathematical topics which lend insight in these areas are (Network and
Combinatorial) Optimization (treated in MATH 477), Combinatorics and
graph
theory (MATH 475), and Probability and Statistics (STAT 400401,
410420,
430, and 450). Probabilistic network models and stochastic processes
are
covered in STAT 405.
Business majors may have interests in Actuarial
Science, the mathematical
subject describing the calculations of fair premiums and probabilities
of loss in Insurance. The Mathematics Department offers courses
relevant to actuarial science
(in particular, STAT 470),
as well as advising related
to preparation for the Actuarial Examinations.
BMGT majors may also be interested in Mathematical
Finance, the theoretical
underpinning for the methods of pricing Options and complex derivative
securities on Wall Street. Some necessary background for the deeper
study
of these topics would include partial differential equations (MATH
462),
probability (STAT 410), and numerical/computer methods (AMSC 460).
Upperlevel math courses of general interest to Business
majors:
Course Number & Title/Topic
MATH 401 Appl. of Linear Algebra
MATH 420 Modeling
MATH 460 Computational Methods
MATH 462 Partial Differential Equations
MATH 475 Combinatorics & Graph Theory
MATH 475 Graph Theory & Combinatorics
AMSC 477 Optimization
STAT 400401 Prob/Stat
STAT 405 Stochastic Processes & Queueing
STAT 410 Probability Theory (advancedcalculus level)
STAT 420 Statistical Theory
STAT 430 SAS & Introductory Regression
STAT 440 Survey Sampling
STAT 450 Regression
STAT 470 Actuarial Math.
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MATHEMATICS DOUBLE MAJOR FOR ENGINEERING STUDENTS
Virtually all engineering undergraduates at UMCP move
through the freshmansophomore
level of the mathematics major  MATH 140, 141, 241, 246, 240 (often
with
a comparable linear algebra course, such as MATH 461, in place of 240).
From there the great range of engineering activities is reflected by a
great variation in further mathematics coursework. Generally
speaking,
the most compelling reason for engineering undergraduates to attain a
strong
mathematical background, such as a double major, is the range of
advanced
techniques of engineering analysis and algorithm design which require
advanced
mathematics, such as: the numerical analysis of differential equations
for the strength of materials, automatic tracking and control of system
trajectories, signal analysis and filtering, reliability of engineering
systems, analysis of the behavior of engineering systems under
stochastic
fluctuations of demand or loading, and stochastic simulation.
Certainly, any engineering student who intends to go on
to
graduate engineering coursework in mathematically demanding
specialties should
consider a double major with mathematics.
ANALYSIS
The mathematics used in engineering (differential
equations, probability,
statistics, Fourier analysis, etc.) belongs largely to Analysis,
the branch of mathematics growing out of calculus. We begin with some
perspective
on analysis.
Roughly speaking, there are three levels of command in
analysis.
The first level is what you have after your freshmansophomore
coursework.
Then there is a big jump to the second level, in prooforiented courses
like MATH 410411, where the curtain is pulled back and you have the
opportunity
to understand clearly what calculus in one and several dimensions is
about.
The jump to the third level of Math 630631 (the graduate analog of
MATH
410411, covering measure theory and Lebesgue integration) is not the
end
of analysis, but arguably there are no jumps to follow which are as
dramatic
as the jumps to the first three levels.
For many engineering activities, a level 1 foundation is
adequate: techniques
are developed in solving specific types of problems in engineering
courses.
Mathematics courses in the "460" series broaden the range of problem
solution
techniques available at this level, in areas ranging from numerical
computing
(MATH 460), linear algebra (MATH 461), partial differential equations
(MATH
462), complex analysis and power series (MATH 463), through MATH 464
(Fourier
series and transform methods). Level 2, in which the rigorous
foundation
goes beyond techniques for problem solving, is the gateway and
prerequisite
for more advanced mathematical topics, but is initiated in MATH
410411.
(By the way, the Math Department is initiating a new course, MATH 412,
which may be more appropriate for some engineers than MATH 411.) A
large
body of immensely useful material for engineering, from the more
advanced
study of differential equations and numerical analysis, to probability
and stochastic processes, to control theory, cannot be properly
understood
without level 2 and (in graduate work) level 3.
Mathematical Topics for Engineering Specialties
What follows is an indication of the main upperdivision
mathematical
topics used in different engineering specialties. In addition to MATH
410411
(or 412), which supplies a rigorous foundation for the
upperlevel mathematics used in all areas. In addition, every
field of engineering nowadays makes heavy use of Optimization
(treated
in MATH 477), Simulation (with relevant courses STAT 410
and
405 in the probabilistic context), and statistical data analysis
(introduced
in STAT 400401 and STAT 430). All engineers confront issues of
Mathematical
Modeling, and so may benefit from MATH 420; and almost all must
understand
something about Reliability, which relies on Probability (STAT 400 or
410).
AEROSPACE ENGINEERING (ENAE)
Vehicle design requires analysis of stress, material
strength and elasticity.
The mathematical models involve partial differential equations (MATH
462
or 415). Fourier analysis (MATH 464 or 417) is used to
approximate
solutions to equations. Computer simulations involve numerical
analysis
(AMSC 460 or 466). Navigation, tracking, and control also heavily
involve differential equations and numerical analysis.
Automaticcontrol
methods involving filtering rely on probability (STAT 410) and
stochastic
processes.
BIOLOGICAL RESOURCES ENGINEERING (ENBE)
Beyond sophomore coursework, statistical methods (STAT
401, STAT 430)
are probably most useful.
CHEMICAL ENGINEERING (ENCH)
Masstransport models involve partial differential
equations (MATH 462
or 415). Solutions of these involve Fourier analysis (MATH 464 or
417). Numerical analysis (AMSC 460 or 466) is used for computer
simulations.
Optimization techniques are fundamental. Probability and statistics 
particularly quality control and experimental design (introduced in
STAT
430)  are used to design, track and evaluate
systems.
CIVIL ENGINEERING (ENCE)
Civil engineers also analyze stress on materials. The
models involve
partial differential equations (MATH 462 or 415), their solution
involves
Fourier analysis (MATH 464 or 417), and the computer simulations
involve
finite element methods and eigenvalue problems in numerical analysis
(AMSC
460 or 466). The calculus of variations is used in advanced
structural
analysis. This is not a subject in the undergraduate curriculum, but
MATH
410411 provides a good preparation for this material.
Probability
and statistics (STAT 400401 or 410) are important for modeling and
evaluation
and for the reliability aspects of ENCE. Traffic flow is one of
the
more mathematical areas of ENCE, involving partial differential
equations
(MATH 462), queueing theory (STAT 405), and computer simulations (AMSC
460, STAT 405).
ELECTRICAL ENGINEERING (ENEE)
Overall, ENEE may be the most mathematical area of
engineering. Looking
over just UMCP's core graduate courses in ENEE,
one sees for example information theory, random processes,
control
theory and signal processing. A student in ENEE needs a strong
undergraduate
background in mathematics.
In particular, a strong ENEE undergraduate is well
advised to take MATH
410411. It is also advisable to take probability (STAT 410), needed
for
information theory, random processes, queueing theory in STAT 405,
etc.),
statistics (STAT 401 or 420, the essential mathematics for
understanding
real data probabilistically), Fourier analysis (the sine qua non of
signal
processing and an essential mathematical tool), complex analysis ("i's"
are everywhere in EENE, you might as well take the course which gives
you
a basic understanding of complex vs. real analysis), MATH 464 (Laplace
and Fourier transform methods; the course may contain basic "wavelet
theory";
check with the Math advisor) and numerical
analysis
(AMSC 460 or 466). At this point a math doublemajor could be
completed
with a useful algebra course, MATH 403 (abstract algebra) or MATH 405
(advanced
linear algebra).
FIRE PROTECTION ENGINEERING (ENFP)
Probability and statistics (e.g. STAT 400) are used in
fire risk assessment
and computer models. Queueing theory (STAT 405) is used for
resource
allocation which is relevant to prioritizing fire protection
investments.
Burning rate and heat transfer are modeled via partial differential
equations
(MATH 462) and computer simulations and numerical analysis (AMSC 460 or
466, MATH 420).
MATERIALS ENGINEERING (ENMA)
Some basic group theory (covered in greater depth as
part of MATH 403)
is essential in the diffraction analysis of materials. Thermodynamics
and
kinetics involve differential equations (MATH 246 or 414), partial
differential
equations (MATH 462) and their solutions with Fourier analysis (MATH
464).
Numerical analysis (AMSC 460 or 466) is used in computer simulations.
Stastistical
experimentaldesign techniques (STAT 430) are important in optimal
design,
as is Probability (STAT 400 or 410) in Reliability.
MECHANICAL ENGINEERING (ME)
The importance of mathematics varies in ME with the
particular track
followed, but differential equations, numerical analysis, and
stability/control
theory appear everywhere: the relevant undergraduate math courses are
MATH
246, 462 and 464, AMSC 460 or 466, MATH 414 or 417. In addition,
engineering
Reliability involves probability (STAT 400 or 410), and statistical
experimental
design (STAT 430) is also important.
NUCLEAR ENGINEERING (ENNU)
Heat transfer and computer simulations are of particular
importance
in this branch of engineering. The mathematics involved includes
probability
(STAT 400 etc.), multi linear algebra (tensors), perturbation theory
and
variational methods. A good foundation in analysis (MATH
410,
411, 462, 464) can be helpful.
RELIABILITY ENGINEERING (ENRE)
Probability and statistics (STAT 400401, 410, and 430)
are of particular
importance in ENRE.
SYSTEMS ENGINEERING (ENSE)
ENSE uses optimization (MATH 477), probability and
statistics (STAT
400 etc.) and numerical analysis (AMSC 460 or 466) in computer
simulations.
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MATHEMATICS DOUBLE MAJOR FOR PHYSICS AND ASTRONOMY
STUDENTS
No science is more deeply mathematical than
physics.
Nature speaks to us in the language of
mathematics.
Richard Feynmann, Physics Nobel
Laureate
Mathematics courses not only support undergraduate work
in physics.
They provide a foundation for understanding the diverse higher
mathematics
with which a good physicist becomes comfortable.
For physics, most areas of mathematics are
important. Differential
Equations are pervasive since Newton, with Ordinary Differential
Equations
arising in Classical Mechanics and Partial Differential Equations in
Electromagnetic
Theory, Continuum Mechanics and Field Theories. Fourier Analysis
is a key tool for solution of differential equations and analyzes of
spectra,
and in problems where explicit solution is not possible, Topology and
Differential
Geometry aid in understanding the qualitative behavior of solution
trajectories.
The rapidly developing field of Nonlinear Dynamics  of great
interest
both to mathematicians and physicists  concerns the search for
general
qualitative structure of complicated trajectories of dynamical systems.
Probability is fundamental in dynamics, statistical mechanics and
quantum
mechanics. Probability is also the basis of Statistics, which is
essential to deciphering the messiness of real data. Lie Groups
provide
the setting for exploiting symmetry in problems of mechanics.
Differential
Geometry is essential for general relativity. Complex Analysis is
fundamental
from quantum mechanics through string theory. Functional Analysis
provides
the framework for quantum mechanics.
One may surmise that a physicist needs to know more math
than a mathematician.
This might be true! The mathematician studies what can be proved
rigorously. The physicist does too, but may often proceed by
approximation, analogy, and numerical conjecture: all of these
activities
require a strong mathematical foundation.
The undergraduate math requirements in physics and
astronomy (MATH 140,
141, 240, 241, 246) provide some initial ideas and vocabulary, and of
course
more mathematics is taught in the physics courses themselves. We
discuss next some of the other mathematics courses of particular
relevance
for physics (including astronomy). Appropriate preparation and
timing
for some of these courses (especially MATH 410411) varies with the
student
and should be discussed with an advisor.
MATH 463 is a course about calculus, power series with
complex numbers
and potential theory.
MATH 452 is an undergraduate introduction to nonlinear
dynamics and
chaos. College Park is one of the world's major research centers in
this
area.
MATH 410411 is the advanced calculus sequence. This
sequence is heavily
oriented to proof and theory and it is a watershed in the student's
mathematical
development. The material is prerequisite to further mathematics in
differential
equations, functional analysis, dynamics, geometry, etc. This is
the key optional sequence a mathematically strong student aiming at
physics
graduate school should consider. It is one of the most difficult
undergraduate mathematics courses, but not bad in comparison to
graduate
school in mathematics or physics.
MATH 403 introduces the basic abstract algebraic
structures: groups,
modules, rings, fields. This course is less crucial for a physicist
than
410411, but these fundamental structures (especially groups) arise in
physics and a course like this is a good place to get clear about
them.
MATH 414 and MATH 415 are introductions (requiring MATH
410 and 411)
to ordinary and partial differential equations at a deeper mathematical
level, respectively going beyond the solutiontechniques
in MATH 246 (ODE) and MATH 462 (PDE) to study rigorously qualitative
theory of solutions.
MATH 417 is a mathematical introduction to Fourier
analysis. MATH 464
(Transform Methods) addresses at a lower mathematical level some
practical
computational uses of the Fourier and Laplace transforms.
MATH 436 is a rigorous course in Differential Geometry.
STAT 410 is an introduction to Probability at
advancedcalculus level,
proving limit theorems like the Central Limit Theorem and Law of Large
Numbers.
There are several undergraduate courses in applied and
computational
mathematics : AMSC 460 (Computational Methods) (emphasizing
computation);
AMSC 466 (like 460, but at a more rigorous mathematical level); and
MATH
472 and MATH 473 (Methods and Models in Applied Mathematics I and II)
(still
more challenging).
Finally: some of the very best undergraduates take
graduate courses
in mathematics, especially MATH 630631, which is to graduate
mathematical
study what MATH 410411 is to undergraduate.
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