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• Department of Mathematics
1845 E Northgate Dr
Irving, TX 75062
Phone: 972-721-5175
Fax: 972-721-4148

math@udallas.edu

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## Mathematics Concentration

### I. Applied Mathematics Concentration

Philosophy: Much of the history and philosophy of Applied Mathematics can be summarized by a quote from the preface to The Functions of Mathematical Physics by Harry Hochstadt, "The topics covered... were first studied by the outstanding mathematicians of the eighteenth and nineteenth centuries. Among the many who devoted themselves to these studies are Gauss, Euler Fourier, Legendre, and Bessel. These men did not recognize the modern and somewhat artificial distinction between pure and applied mathematics. Much of their work was stimulated by physical problems that led to the studies of differential equations. Frequently they developed generalizations to obtain results having no immediate or obvious applications. As a consequence mathematics was often ahead of its time having tools ready before physicists and engineers felt the need for them." The concentration reflects this historic interplay by presenting topics of obvious interest to applied scientists as well as being of purely mathematical interest.

The concept of transformations plays a central role in applied mathematics. Partial differential equations are transformed into ordinary differential equations. Ordinary differential equations are transformed in algebraic equations. Algebraic systems are transformed into simple algebraic systems. Thus, one can understand why linear algebra plays a fundamental role in the concentration.

Content: The concentration consists of five courses. The core of the Applied Mathematics Concetration is made up of the three courses: Calculus III (Math 2412), Linear Algebra (Math 3310), and Applied Mathematics (Math 4315). Fundamental to modern applied mathematics is the study of structures known as vector spaces and the linear operations on those spaces. The student is introduced to these concepts in linear algebra. These ideas are expanded in Calculus III where the linearity and multidimensionality introduced in linear algebra are combined with the infinite processes of calculus. These concepts continue to be drawn together in Applied Mathematics, where the analogy is completed between discrete problems, continuous one-dimensional problems, and continuous multi-dimensional problems.

The fourth course is an applied mathematics elective such as Differential Equations (Math 3324), Probability (Math 3326), Statistics (Math 3327), Numerical Analysis(Math 3338), or a Computer Science course approved by director.

The fifth course is an elective from a field other than mathematics. This allows the student to tailor the concentration to his or her own interests and reinforces the concentration's interdisciplinary nature. Possible choices include:

CHE 3331  Physical Chemistry I
ECO 3327  Statistical Theory and Methods
ECO 3328  Business and Economic Forecasting
PHI 4333  Philosophy of Science
PHY 3341  Optics
PHY 3363  Computational Physics
PHY 4327  Electromagnetic Theory
PHY 4423  Theoretical Mechanics
PHY 4424  Quantum Mechanics
PSY 3337  Statistical Methods
Other electives as approved by the department.

### II. Pure Mathematics Concentration

The concentration provides a coherent set of courses for students interested in mathematics, short of a major, in areas distinct from those of applied mathematics.

The concentration consists of five mathematics classes (fifteen credits): Math 3310 (Linear Algebra),  Math 3321 (Linear Point Set Theory); two of Math 4332 (Abstract Algebra I), Math 4334 (Topology), or Math 4341 (Analysis I); and a fifth class, selected from the following list:

MAT 3320  Foundations of Geometry
MAT 3322  History and Philosophy of Mathematics
MAT 3331  Number Theory
MAT 4332 or 4333  Abstract Algebra I or II
MAT 4334  Topology
MAT 4341 or 4342  Analysis I or II
MAT 4V43/4V44  Research Hours
Courses as approved by the department.