## Mathematics Courses

**Math 130. Basic Math – Algebra and Trigonometry.** 3(1). This course sis designed to help reinforce algebraic and trigonometric skills necessary for success in the technical core. Basic graphing, algebraic manipulation, and trigonometric calculations are covered. This course may be used as an Academy option to fulfill graduation requirements. This course does not fulfill any major’s requirements. Final Exam. Prereq: Can only be enrolled in the course by recommendation of the Department of Mathematical Sciences. Sem hrs: 3 fall.

**Math 141.**** Calculus I.** 3 (1). The study of differential calculus. Topics include functions and their applications to physical systems; limits and continuity; vectors and vector arithmetic; a formal treatment of derivatives; numeric estimation of derivatives at a point; basic differentiation formulas for elementary functions; product, quotient, and chain rules; implicit differentiation; and mathematical and physical applications of the derivative, to include extrema, concavity, and optimization. Significant emphasis is placed on using technology to solve and investigate mathematical problems. Final exam. Sem hrs: 3 fall or spring

**Math 142. Calculus II. **3(1). A study of integral calculus with a focus on the Fundamental Theorems and their application. Topics include: estimating area under a curve, antiderivatives, numeric integration methods, antiderivative formulas for the elementary functions, integration by substitution, parts and tables, improper integrals, differential equations, exponential growth and decay, an introduction to Taylor Series, and mathematical and physical applications of the Fundamental Theorems. Physical applications include area and volume problems and the concept of work. Final exam. Prereq: Math 141. Sem hrs: 3 fall or spring.

**Math 152****. Advanced Placed Calculus II.** 3(1). A more intense study of integral calculus for advanced-placed fourth-class cadets. Content is similar to Math 142, with the addition of an introduction to polar coordinates, vector arithmetic, and complex arithmetic. Additional emphasis is placed on the mathematical and physical applications in preparation for cadets interested in pursuing a technical major or minor. Final exam. Prereq: For fourth-class cadets--qualifying performance on DFMS placement exams; for third-class cadets, Department Head approval. Sem hrs: 3 fall.

**Math 243. Calculus III.** 3(1). Multivariate calculus, including vector functions, partial differentiation, directional derivatives, line integrals, and multiple integration. Maxima and minima in multiple dimensions and the method of Lagrange Multipliers. Solid analytical geometry to include lines, planes, and surfaces in 3-space. Designed for cadets who indicate an interest in a technical major. Final exam. Prereq: C or better in Math 142 or advanced-placement through DFMS exams. Waiver authority is Deputy Head for Academics. Sem hrs: 3 fall or spring.

**Math 245. Differential Equations. **3(1). Modeling with and analysis of linear ordinary differential equations. Includes matrix algebra and matrix inverses, first-order ordinary differential equations (numerical methods, separation of variables, integrating factors, and method of undetermined coefficients), and second-order linear differential equations/first-order linear systems (Laplace transforms, determinants, eigenvalues, eigenvectors, and stability). Applications may include population growth, predator/prey, and mass-spring system modeling. Final exam. Prereq: Prior completion of Math 243 (or Math 253) is strongly recommended. Co better in Math 142 (or Math 152) or advanced-placement through DFMS exams. Waiver authority is the DFMS Deputy Head for Academics. Sem Hrs: 3 fall or spring.

**Math 253. Advanced Placed Calculus III. **3(1). A more intense study of multivariate calculus for advanced-placed fourth-class cadets. Content is similar to Math 243. Additional emphasis is placed on mathematical and physical applications in preparation for cadets interested in pursuing a technical major. Final exam. Prereq: Advanced placement through DFMS exams. Sem hrs: 3 fall.

**Math 300. Introduction to Statistics.** 3(1). Topics include descriptive statistics, emphasizing graphical displays; basic probability and probability distributions; sampling distribution of the mean and the Central Limit Theorem; statistical inference including confidence intervals and hypothesis testing; correlation; and regression. Math 300 is designed primarily for majors in the Social Sciences and Humanities. It emphasizes the elements of statistical thinking, focuses on concepts, automates most computations, and has less mathematical rigor than Math 356. Final exam. Prereq: Math 142/152 or department permission. Sem hrs: 3 fall or spring.

**Math 310. Mathematical Modeling.** 3(1). An introductory course in mathematical modeling. Students model various aspects of real-world situations chosen from Air Force applications and from across academic disciplines, including military sciences, operations research, economics, management, and life sciences. Topics include: the modeling process, graphical models, proportionality, model fitting, optimization, and dynamical systems. Several class periods are devoted to in-class work on small projects. Math 310 is not appropriate for Math or OR majors. Final exam. Prereq: Completion of core math. Sem hrs: 3 spring.

**Math 320. Foundations of Mathematics.** 3(1). Course emphasizes exploration, conjecture, methods of proof, ability to read, write, speak, and think in mathematical terms. Includes an introduction to the theory of sets, relations, and functions. Topics from algebra, analysis, or discrete mathematics may be introduced. A cadet cannot receive credit for both Math 320 and Math 340. Final exam or final project. Prereq: Completed Math 142/152 with a ‘C’ or better. Wavier authority is the Deputy Head for Academics. Sem hrs: 3 fall or spring.

**Math 340. Discrete Mathematics. **3(1). Useful for cadets interested in applications of mathematics to computer science and electrical engineering. Propositions and logic; sets and operations on sets; functions, recursion, and induction; graphs, trees, and their applications; discrete counting and combinatorics. Final exam. Prereq: Comp Sci 110 and C3C standing or department permission. Sem hrs: 3 spring.

**Math 342. Numerical Analysis with Differential Equations. **3(1). An introductory numerical analysis course with an emphasis on solving differential equations. Specific topics include finite precision arithmetic, root finding, fixed point iteration, interpolating polynomials, solving initial value problems via Runge-Kutta and multi-step methods including systems of equations, and solutions of linear systems of equations. The approach is a balance between the theoretical and applied perspectives with some computer programming required. Final exam. Prereq: Math 245 and Comp Sci 110. Sem hrs: 3 spring.

**Math 344. Matrices and Differential Equations. **3(1). Properties, types, and operations of matrices; solutions of linear systems; Euclidean vector spaces, linear independence, and bases; eigenvalues and eigenvectors. Computational aspects. Applications to differential equations. First- and second-order differential equations and systems. Models may include population growth, warfare, and economics. A cadet cannot receive credit for both Math 344 and Math 360. Final exam. Prereq: Math 142/152 or department permission. Sem hrs: 3 spring.

**Math 346. Engineering Math.** 3(1). Provides advanced mathematical concepts and skills necessary for technical disciplines. Topics include differential and integral vector calculus (gradient, directional derivative, divergence, curl, Divergence Theorem, Stokes’ Theorem), Fourier series, orthogonal functions, and partial differential equations (separation of variables, transform methods, numerical techniques). Final exam. Prereq: C or better in both Math 243 (or Math 253) and Math 245. Wavier authority: Deputy Head for Academics. Sem hrs: 3 fall or spring.

**Math 356. Probability and Statistics for Engineers and Scientists.**3(1). Topics include classical discrete and continuous probability distributions; generalized univariate and bivariate distributions with associated joint, conditional, and marginal distributions; expectations of random variables; Central Limit Theorem with applications in confidence intervals and hypothesis testing; regression; and analysis of variance. This course is a core substitute for Math 300. Credit will not be given for both Math 300 and Math 356, not for both Math 356 and Math 377. Designed for cadets in engineering, science, or other technical disciplines. Math majors and Operations Research majors should take the Math 377/378 sequence. Final Exam. Prereq: Math 142/152. Sem hrs: 3 fall or spring.

**Math 359. Design and Analysis of Experiments. **3(1). An introduction to the philosophy of experimentation and the study of statistical designs. The course requires a knowledge of statistics at the Math 300 level. Topics include analysis of variance for K treatments, various two- and three-level designs, interactions, unbalanced designs, and regression analysis. A valuable course for all science and engineering majors. Final project. Prereq: Math 300, Math 356 or Math 378. Sem hrs: 3 spring.

**Math 360. Linear Algebra.** 3(1). A first course in linear algebra focusing on Euclidean vector spaces and their bases. Using matrices to represent linear transformation, and to solve systems of equations, is a central theme. Emphasizes theoretical foundations (computational aspects are covered in Math 344). A cadet cannot receive credit for both Math 344 and Math 360. Final exam or final project. Prereq/Coreq: Math 320 or department permission. Sem hrs: 3 fall.

**Math 366. Real Analysis I.**3(1). A theoretical study of functions of one variable focused on proving results related to concepts first introduced in differential and integral calculus. This course is an essential prerequisite for graduate work in mathematical analysis, differential equations, optimization, and numerical analysis. Final exam or final project. Prereq: Math 360 or department permission. Sem hrs: 3 spring.

**Math 377. Advanced Probability.** 3(1). Topics include probability fundamentals, discrete and continuous random variables, single and multivariate probability distributions, functions of random variables, sampling distributions and the Central Limit Theorem. This course is designed for Mathematics and Operations Research majors. Final exam. Prereq: Math 243 (or Math 253). Sem hrs: 3 fall.

**Math 378. Advanced Statistics.** 3(1). Topics include point and interval estimation, properties of point estimators, sample inferential statistics with confidence intervals, hypothesis testing, ANOVA, linear regression, design and analysis of experiments, and nonparametric statistics. This course is a core substitute for Math 300 but has much more rigor and depth. Credit will not be given for both Math 300 and Math 378. Final exam. Prereq: Math 377. Sem hrs: 3 spring.

**Math 420. Mathematics Capstone I. **1.5(1). The first semester of the mathematics capstone experience. Students will decide on a topic of independent research in, or related to, the mathematical sciences and begin work with a faculty advisor. Significant progress toward a thesis will be made during the semester. Final project. Prereq: C1C standing in the Mathematics major. Sem hrs: 1.5 fall.

**Math 421. Mathematics Capstone II. **1.5(1). The second semester of the mathematics capstone experience. Students will compete work on their independent research project and produce a thesis to present their findings. Final project. Prereq: C1C standing in the Mathematics major. Sem hrs: 1.5 spring.

**Math 443. Numerical Analysis of Differential Equations. **3(1). An intermediate numerical analysis course with an emphasis on solving differential equations. Specific topics include solving linear and nonlinear systems; solutions of initial value problems via Runge-Kutta, Taylor, and multistep methods; convergence and stability; and solutions of boundary value problems. Other topics typically include approximating eigenvalues and eigenvectors and numerically solving partial differential equations. The approach is a balance between the theoretical and applied perspectives with some computer programming required. Final exam or final report. Prereq: Math 346 or Math 469 and one of Math 342 or Physics 356, or department permission. Sem hrs: 3 spring of even-numbered years.

**Math 451. Complex Variables.** 3(1). A valuable course for cadets intending to pursue graduate work in mathematics or its applications, particularly in areas involving partial differential equations. Analytic functions; integration; the Cauchy Integral Theorem and applications; power and Laurent series, residues and poles; conformal mapping with applications to potential theory and fluid flows. Final exam. Prereq: Math 243/253 and Math 245 or department permission. Sem hrs: 3 spring.

**Math 465. Modern Algebra. **3(1). A valuable course for cadets intending to pursue graduate work in mathematics or its applications. Focuses on the study of algebraic structures and functions between these structures. Topics include: cyclic groups, permutation groups, normal subgroups and quotient groups; rings, ideals, polynomial rings and fields. Depending on instructor and student preferences, applications to coding theory, crystallography, or combinatorics are explored. Final exam. Prereq: Math 320 or department permission. Sem hrs: 3 fall.

**Math 467. Real Analysis II.** 3(1). A theoretical study of functions of several variables to include topology of cartesian spaces, compact and connected sets, convergence of sequences and functions, continous functions, fixed point theorems, contractions, Stone-Weierstrass approximation theorems, differentiation, partial differentiation, mapping theorems, and Implicit Function Theorem. Final exam. Prereq: Math 366 or department permission. Sem hrs: 3 fall of odd-numbered years.

**Math 468. Dynamical Systems.**3(1). The study and application of linear and nonlinear differential equations to physical systems from both computational and analytical points of view. Topics vary. Typical choices include systems of differential equations, stability analysis, bifurcations, maps, and chaos. Final exam or final report. Prereq: Math 243/253 and Math 245 or department permission. Sem hrs: 3 fall of even numbered-years.

**Math 469. Partial Differential Equations. **3(1). Solutions of boundary value problems with applications to heat flow, wave motion, and potential theory. Methods of solution include separation of variables and eigenfunction expansion, including Fourier series. Topics typically include the method of characteristics, generalizations to higher dimensions, and the use of non-Cartesian coordinate systems. Additional topics may include numerical methods, nonlinear equations, and transform methods. Final exam or final report. Prereq: Math 243/253 and Math 245 or department approval. Sem hrs: 3 fall.

**Math 470. Mathematical Physics.** 3(1). An introduction to various mathematical topics needed in graduate-level physics and applied mathematics courses. Topics vary; typical choices include special functions (Legendre polynomials, Bessel functions, etc.), calculus of variations, complex functions (Laurent series, contour integration, and the Residue Theorem), Fourier series and their convergence properties, integral transform concepts (Fourier and Laplace transforms, Green's functions), dynamical systems. Final exam. Prereq: Math 346 or Math 469 or department approval. Sem hrs: 3 spring of odd-numbered years.

**Math 472. Introduction to Number Theory.** 3(1). Basic facts about integers, the Euclidean algorithm, prime numbers, congruencies and modular arithmetic, perfect numbers and the Legendre symbol will be covered and used as tools for the proof of quadratic reciprocity. Special topics such as public key cryptography and the Riemann Zeta function will be covered as time allows. Final exam. Prereq: Math 320 Sem hrs: 3 spring of odd years.

**Math 473. Introduction to Point-Set Topology.** 3(1). Review of set theory; topology on the real line and on the real plane; metric spaces; abstract topological spaces with emphasis on bases; connectedness and compactness. Other topics such as quotient spaces and the separation axioms may be included. A valuable course for all math majors in the graduate school option. Final exam. Prereq: Math 320. Sem hrs: 3 fall of odd years.

**Math 474. Combinatorics and Graph Theory.** 3(1). Combinatorics and Graph Theory. 3(1). Permutations, combinations, recurrence relations, inclusion-exclusion, connectedness in graphs, colorings, and planarity. Theory and proofs, as well as applications to areas such as logistics, transportation, scheduling, communication, biology, circuit design, and theoretical computer science. Final exam or final project. Prereq: Math 340; or Math 320 and one of Math 245 or Math 344 or Math 360; or department permission. Sem hrs: 3 spring.

**Math 495. Special Topics.** 3(1). Selected advanced topics in mathematics. Final exam or final report. Prereq: Department permission. Sem hrs: 3 fall or spring.

**Math 499. Independent Study and Research. **3(0). Individual study and/or research under the direction of a faculty member. Oral midterm and final; final paper. Prereq: Department permission. Sem hrs: 3 fall or spring.

**Math 499A. Independent Study and Research. **2(0). Individual study and/or research under the direction of a faculty member. Oral midterm and final; final paper. Prereq: Department permission. Sem hrs: 2 fall or spring.

**Math 499B. ***Independent Study and Research.*** **1.5(0). Individual study and/or research under the direction of a faculty member. Oral midterm or final; final paper. Prereq: Department permission. Sem hrs: 1.5 fall or spring.

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Operations Research Courses

These courses are offered in one of the following departments: Department of Computer Science (DFCS), Department of Economics and Geography (DFEG), Department of Management (DFM), and Department of Mathematical Sciences (DFMS)

**Ops Rsch 310. Systems Analysis.**3(1). This course provides an introduction to rigorous quantitative modeling methods that have broad application. The course focuses on the mathematics of the models, the computer implementation of the models, and the application of these models to practical decision-making scenarios. By demonstrating the application of these techniques to problems in a wide range of disciplines, the course is relevant to technical and non-technical majors at USAFA. The course consists of six distinct blocks: decision analysis and utility theory, linear and nonlinear optimization, project management, queuing theory, simulation, and the systems approach to engineering and decision-making. Administered by the Department of Management. Instruction provided by inter-departmental Operations Research faculty. Final exam. Prereq: Comp Sci 110, Math 142. Sem hrs: 3 Fall and Spring.

**Ops Rsch 311. ***Deterministic Models*** .** 3(1). Topics include linear programming (with sensitivity analysis and applications) and non-linear programming. Both the theory and the computer implementation of these techniques are addressed. Administered by the Department of Mathematical Sciences. Final exam. Prereq: Math 343 or Math 360; and either Ops Rsch 310 or department permission. Sem hrs: 3 fall.

**Ops Rsch 312. Probabilistic Models. **3(1). Selected probabilistic models (such as random walks, Markov Chains, queues, inventories and reliability models) are analyzed as stochastic processes. Administered by the Department of Mathematical Sciences. Final exam. Prereq: Math 377 or department permission. Sem hrs: 3 spring.

**Ops Rsch 405. First-Class Seminar.** 0(1). A course for First-Class Operations Research majors that provides for presentation of cadet and faculty research; guest lectures; field trips; seminars on career and graduate school opportunities for scientific analysts in the Air Force; goal setting exercises; and applications of Operations Research. The class meets once each week. Open only to First-Class Operations Research majors. Pass/Fail. No final exam. Prereq: C1C standing. Sem hrs: 0 fall.

**Ops Rsch 421. ***Case Studies in Operations Research.*** **1.5(1) . The study of methodologies associated with business and operations management. A case-based course intended to provide the proper foundation needed to conduct effective analyses supporting a variety of scenarios. Students will evaluate various cases, develop plans for and conduct analyses, and create effective written and oral presentations. Final Project or Exam. Sem hrs: 1.5 fall.

**Ops Rsch 422. Capstone in Operations Research. **3(2). Capstone course in Operations Research. Case studies in advanced Operations Research techniques with emphasis on problem recognition, model formulation and Air Force applications. Final project. No final exam. Administered by the Department of Management. Prereq: Ops Rsch 310, Ops Rsch 321, Ops Rsch 411, and C1C standing. Sem hrs: 3 spring.

**Ops Rsch 495. Special Topics.** 1-3(1). Selected advanced topics in Operations Research. Final exam or final report. Offered by DFCS, DFEG, DFM or DFMS. Prereq: department permission. Sem hrs and offering time determined by department (NTE 3 sem hrs).

**Ops Rsch 499. Independent Study.**3(0). Individual study and/or research in Operations Research, under the supervision of a faculty member. Final exam or final report. Offered by DFCS, DFEG, DFM, or DFMS. Prereq: department permission. Sem hrs: 3 fall or spring.

**Ops Rsch 499A. Independent Study. **2(0). Individual study and/or research in Operations Research, under the supervision of a faculty member. Final exam or final report. Offered by DFCS, DFEG, DFM, or DFMS. Prereq: department permission. Sem hrs: 2 fall or spring.

**Ops Rsch 499B. ***Independent Study**.*** **1.5(0). Individual study and/or research in Operations Research, under the supervision of a faculty member. Final exam or final report. Offered by DFCS, DFEG, DFM, or DFMS. Prereq: department permission. Sem hrs: 1.5 fall or spring.