2024-2025 Course Catalog
Welcome to Virginia Tech! We are excited that you are here planning your time as a Hokie.
Welcome to Virginia Tech! We are excited that you are here planning your time as a Hokie.
Introduction to the scope and applicability of mathematics and its many sub-disciplines. Introduction to the process of thinking, learning, and writing as a mathematician through topics such as logic systems, recreational mathematics, LaTeX programming, history, ethics, open problems, and research in mathematics. Also includes advising topics such as planning a Virginia Tech course of study. P/F only. Math majors.
Precalculus college algebra, basic functions (algebraic, exponential, logarithmic, and trigonometric), conic sections, graphing techniques, basic probability. Usage of mathematical models, analytical calculations, and graphical or numerical representations of data to analyze problems from multiple disciplines that address intercultural and global challenges in areas such as chemistry, environmental science, the life sciences, finance, and statistics. Use of spreadsheet software. Two units of high school algebra and one of plane geometry are required.
Quantitative and computational thinking to address relevant global issues. Unified calculus course covering techniques and applications of differential and integral calculus for functions of one variable. Constitutes the standard first-year mathematics courses for the life sciences. 1025: Differential calculus, graphing, applications for the life sciences, use of spreadsheet software. Assumes 2 units of high school algebra, 1 unit of geometry, 1/2 unit of trigonometry and precalculus. 1026: Integral calculus, numerical techniques, elementary differential equations, applications for the life sciences, use of spreadsheet and scientific software. A student can earn credit for at most one of 1025 and 1225. A student can earn credit for at most one of 1026 and 1226.
Quantitative and computational thinking to address relevant global issues. Unified calculus course covering techniques and applications of differential and integral calculus for functions of one variable. Constitutes the standard first-year mathematics courses for the life sciences. 1025: Differential calculus, graphing, applications for the life sciences, use of spreadsheet software. Assumes 2 units of high school algebra, 1 unit of geometry, 1/2 unit of trigonometry and precalculus. 1026: Integral calculus, numerical techniques, elementary differential equations, applications for the life sciences, use of spreadsheet and scientific software. A student can earn credit for at most one of 1025 and 1225. A student can earn credit for at most one of 1026 and 1226.
Introduction to the scope and applicability of mathematics and its many sub-disciplines. Introduction to the process of thinking, learning, and writing as a mathematician through topics in pure and applied mathematics and a brief experience with mathematical research. Also includes advising topics such as planning a Virginia Tech course of study. Math majors.
Linear equations, polynomials, relations and functions, rational functions, quadratic equations, radicals and functions with rational exponents, exponentials, logarithms, trigonometric functions, trigonometric identities. Designed as preparation for MATH 1225: Calculus of a Single Variable.
1225-1226: CALCULUS OF A SINGLE VARIABLE Quantitative and computational thinking to address relevant intercultural and global issues. Unified calculus course covering techniques of differential and integral calculus for functions of one variable. Constitutes the standard first-year mathematics courses for science and engineering. 1225: limits, continuity, differentiation, transcendental functions, applications of differentiation, introduction to integration. Assumes 2 units of high school algebra, 1 unit of geometry, 1/2 unit each of trigonometry and precalculus, and placement by Math Dept. 1226: techniques and applications of integration, trapezoidal and Simpson’s rules, improper integrals, sequences and series, power series, parametric curves and polar coordinates, software-based techniques. A student can earn credit for at most one of 1025 and 1225. A student can earn credit for at most one of 1525 and 1225. A student can earn credit for at most one of 1026 and 1226. Pre: Grade of at least C- in 1225 for 1226.
1225-1226: CALCULUS OF A SINGLE VARIABLE Quantitative and computational thinking to address relevant intercultural and global issues. Unified calculus course covering techniques of differential and integral calculus for functions of one variable. Constitutes the standard first-year mathematics courses for science and engineering. 1225: limits, continuity, differentiation, transcendental functions, applications of differentiation, introduction to integration. Assumes 2 units of high school algebra, 1 unit of geometry, 1/2 unit each of trigonometry and precalculus, and placement by Math Dept. 1226: techniques and applications of integration, trapezoidal and Simpson’s rules, improper integrals, sequences and series, power series, parametric curves and polar coordinates, software-based techniques. A student can earn credit for at most one of 1025 and 1225. A student can earn credit for at most one of 1525 and 1225. A student can earn credit for at most one of 1026 and 1226. Pre: Grade of at least C- in 1225 for 1226.
Introduction to programming for mathematical problem solving. Programming software interfaces. Logic and conditional computations. Iterative computations and recursion. Data arrays. Compartmentalized computations using functions. Data visualization. Data input/output. Programming applications such as Monte Carlo simulation, random walks, computational geometry, and graph theory.
Differential calculus techniques for functions of one and two variables. Emphasis on graphs, rates of change, and optimization of linear, quadratic, exponential, and logistic functions. Terminology and applications for business, including spreadsheet software. Mathematical models of real-world business problems, including discrete and continuous models, that address intercultural and global challenges in such areas as finance, marketing, and accounting. Assumes 2 units of high school algebra and 1 unit of geometry.
A standard first-year mathematics sequence for architecture majors. Mathematical models of real-world problems, including discrete and continuous models, that address relevant global challenges in such areas as urban planning, building construction, and home design. 1535: Euclidean geometry, trigonometry, sequences and the golden ratio, graph theory, tilings, polygons and polyhedra, applications for 2- and 3-dimensional design and construction, use of geometric software. 1536: vectors in the plane and space, descriptive and projective geometry, differential and integral calculus, applications for 2- and 3-dimensional design and construction, including areas, volumes, centroids, and optimization. Assumes 2 unites of high school algebra and 1 unit of high school geometry.
A standard first-year mathematics sequence for architecture majors. Mathematical models of real-world problems, including discrete and continuous models, that address relevant global challenges in such areas as urban planning, building construction, and home design. 1535: Euclidean geometry, trigonometry, sequences and the golden ratio, graph theory, tilings, polygons and polyhedra, applications for 2- and 3-dimensional design and construction, use of geometric software. 1536: vectors in the plane and space, descriptive and projective geometry, differential and integral calculus, applications for 2- and 3-dimensional design and construction, including areas, volumes, centroids, and optimization. Assumes 2 unites of high school algebra and 1 unit of high school geometry.
Study of the nature and structure of numbers for prospective elementary and middle school teachers; number theory, number systems, operations and algebraic thinking, problem solving, and mathematical modeling. 1614 may not be taken by math majors for credit.
Study of key geometry concepts for prospective elementary and middle school teachers; multiple perspectives including transformational, coordinate, Euclidean and analytical geometry; geometric modeling; geometric and spatial reasoning. 1624 may not be taken by math majors for credit.
Vector and matrix algebra systems of linear equations, linear equations, linear independence, bases, orthonormal bases, rank, linear transformations, diagonalization, implementation with contemporary software. Math 1226 or a grade of at least B in VT MATH 1225. A student can earn credit for at most one of 2114 and 2405H.
Vector and matrix algebra systems of linear equations, linear equations, linear independence, bases, orthonormal bases, rank, linear transformations, diagonalization, implementation with contemporary software. Math 1226 or a grade of at least B in VT MATH 1225. A student can earn credit for at most one of 2114H and 2405H.
Calculus for functions for several variables. Planes and surfaces, continuity, differentiation, chain rule, extreme values, Lagrange multipliers, double and triple integrals and applications, software-based techniques. A student can earn credit for at most one of 2204 and 2406H. A student can earn credit for at most one of 2024 and 2204. A student can earn credit for at most one of 2204 and CMDA 2005.
Calculus for functions of several variables. Planes and surfaces, continuity, differentiation, chain rule, extreme values, Lagrange multipliers, double and triple integrals and applications, software-based techniques. A student can earn credit for at most one of 2204H and 2406H. A student can earn credit for at most one of 2024 and 2204H. A student can earn credit for at most one of 2204H and CMDA 2005.
Unified course in ordinary differential equations. First-order equations, second-and-higher-order constant coefficient linear equations, systems of first-order linear equations, and numerical methods. Mathematical models describing motion and cooling, predator-prey population models, SIR-models, mechanical vibrations, electric circuits, rates of chemical reactions, radioactive decay. Quantitative and computational thinking to address relevant intercultural and global issues. A student can earn credit for at most one of 2214 and 2406H. A student can earn credit for at most one of 2214 and CMDA 2006.
Unified course in ordinary differential equations. First-order equations, second-and-higher-order constant coefficient linear equations, systems of first-order linear equations, and numerical methods. Mathematical models describing motion and cooling, predator-prey population models, SIR-models, mechanical vibrations, electric circuits, rates of chemical reactions, radioactive decay. Quantitative and computational thinking to address relevant intercultural and global issues. A student can earn credit for at most one of 2214H and 2406H. A student can earn credit for at most one of 2214H and CMDA 2006
Unified course covering topics from linear algebra, differential equations, and calculus for functions of several variables. Comprises the standard second year mathematics courses for science and engineering. 2405H: Vector and matrix algebra, systems of linear equations, linear independence, bases, orthonormal bases, rank, linear transformations and diagonalization. Ordinary linear homogeneous differential equations, implementation with contemporary software. 2406H: Ordinary nonhomogeneous differential equations, calculus for functions of several variables, planes and surfaces, continuity, differentiation, chain rule, extreme values, Lagrange multipliers, double and triple integrals and applications, with software-based techniques. A student can earn credit for at most one of 2114, 2114H, and 2405H. A student can earn credit for at most one of 2204, 2204H, and 2406H. A student can earn credit for at most one of 2214, 2214H, and 2406H.
Unified course covering topics from linear algebra, differential equations, and calculus for functions of several variables. Comprises the standard second year mathematics courses for science and engineering. 2405H: Vector and matrix algebra, systems of linear equations, linear independence, bases, orthonormal bases, rank, linear transformations and diagonalization. Ordinary linear homogeneous differential equations, implementation with contemporary software. 2406H: Ordinary nonhomogeneous differential equations, calculus for functions of several variables, planes and surfaces, continuity, differentiation, chain rule, extreme values, Lagrange multipliers, double and triple integrals and applications, with software-based techniques. A student can earn credit for at most one of 2114, 2114H, and 2405H. A student can earn credit for at most one of 2204, 2204H, and 2406H. A student can earn credit for at most one of 2214, 2214H, and 2406H.
Emphasis on topics relevant to computer science. Topics include logic, propositional calculus, set theory, relations, functions, mathematical induction, elementary number theory and Boolean algebra. Does not carry credit for mathematics majors, but may be used as though it were a 3000-level elective course for the mathematics minor. Two units of high school algebra, one unit of geometry, one-half unit each of trigonometry and precalculus mathematics required. 2534 may not be taken by math majors for credit without special permission. A student can earn credit for at most one of 2534 and 3034.
Introduction to professional, culturally respectful mathematics tutoring. Development of listening and questioning skills, assessment of students’ mathematical difficulties. Exploration of teaching and learning processes, effectively utilizing technology, and adjusting instruction to diversity in students’ mathematical reasoning. Concurrent mathematics tutoring experience required. May be repeated twice with different leadership expectations for a maximum of 3 credits.
Honors section.
Practice in writing mathematical proofs. Exercises from set theory, number theory, and functions. Propositional logic, set operations, equivalence relations, methods of proof, mathematical induction, the division algorithm and images and pre-images of sets. A student can earn credit for at most one of 2534 and 3034.
Introduction to abstract algebraic structures (groups, rings, and fields) and structure-preserving maps (homomorphisms) for these structures. Proof-intensive course illustrating the rigorous development of a mathematical theory from initial axioms.
Emphasis on concepts related to computational theory and formal languages. Includes topics in graph theory such as paths, circuits, and trees. Topics from combinatorics such as permutations, generating functions, and recurrence relations.
Introductory course in linear algebra. Abstract vector spaces, linear transformations, algorithms for solving systems of linear equations, matrix analysis. This course involves mathematical proofs.
Fundamental calculus of functions of two or more variables. Implicit function theorem, Taylor expansion, line integrals, Greens theorem, surface integrals.
Theory of limits, continuity, differentiation, integration, series. 3224 duplicates 4525.
Computational methods for numerical solution of non-linear equations, differential equations, approximations, iterations, methods of least squares, and other topics. A grade of C or better required in CS prerequisite 1044 or 1705. A student can earn credit for at most one of 3414 and 4404.
Arithmetic of complex numbers. Geometry of the complex plane. Geometry of exponentiation and roots. Complex exponential, trigonometric and hyperbolic functions. Continuity and differentiability. Analytic and harmonic functions.
Historical development of mathematics from antiquity to modern times. Senior standing in mathematics or mathematics education required.
An introduction to the theory of groups and rings. Topics include normal subgroups, permutation groups, Sylows Theorem, Abelian groups, Integral Domains, Ideals, and Polynomial Rings.
Introduction to elementary number theory. Topics covered may include divisibility, greatest common divisors, unique prime factorization, congruences, Fermat's Little Theorem, Chinese Remainder Theorem, multiplicative number-theoretic functions, Diophantine equations, primitive roots, and the Quadratic Reciprocity Law.
Second course in linear algebra. Similarity invariants, Jordan canonical form, inner product spaces, self-adjoint operators, selected applications.
4175: Introduction to classical and modern symmetric-key cryptography; alphabetic ciphers, block ciphers and stream ciphers; background in modular arithmetic and probability; perfect secrecy; linear and differential cryptanalysis; Advanced Encryption Standard; hashing. 4176: Introduction to modern public-key cryptography and cryptanalysis; RSA algorithm, ElGamal algorithm, Diffie-Hellman algorithm; digital signatures; background in group theory and number theory; algorithms for primality testing, factoring, and discrete logarithms; elliptic curves.
4175: Introduction to classical and modern symmetric-key cryptography; alphabetic ciphers, block ciphers and stream ciphers; background in modular arithmetic and probability; perfect secrecy; linear and differential cryptanalysis; Advanced Encryption Standard; hashing. 4176: Introduction to modern public-key cryptography and cryptanalysis; RSA algorithm, ElGamal algorithm, Diffie-Hellman algorithm; digital signatures; background in group theory and number theory; algorithms for primality testing, factoring, and discrete logarithms; elliptic curves.
Real number system, point set theory, limits, continuity, differentiation, integration, infinite series, sequences and series of functions.
Real number system, point set theory, limits, continuity, differentiation, integration, infinite series, sequences and series of functions.
Analytic functions, complex integration, series representation of analytic functions, residues, conformal mapping, applications
Solution techniques, linear systems, the matrix exponential, existence theorems, stability, non-linear systems, eigenvalue problems.
Survey of basic concepts in chaotic dynamical systems. Includes material on bifurcation theory, conjugacy, stability, and symbolic dynamics.
Basic concepts of topological spaces, continuous functions, connected spaces, compact spaces, and metric spaces.
Transformational approach to Euclidean geometry including an in-depth study of isometries and their application to symmetry, geometric constructions, congruence, coordinate geometry, and non-Euclidean geometries.
Interpolation and approximation, numerical integration, solution of equations, matrices and eigenvalues, systems of equations, approximate solution of ordinary and partial differential equations. Applications to physical problems. A student can earn credit for at most one of 3414 and 4404.
Theory and techniques of modern computational mathematics, computing environments, computational linear algebra, optimization, approximation, parameter identification, finite difference and finite element methods and symbolic computation. Project-oriented course; modeling and analysis of physical systems using state-of-the-art software and packaged subroutines.
Separation of variables for heat, wave, and potential equations. Fourier expressions. Application to boundary value problems. Bessel functions. Integral transforms and problems on unbounded domains.
Separation of variables for heat, wave, and potential equations. Fourier expressions. Application to boundary value problems. Bessel functions. Integral transforms and problems on unbounded domains.
4445: Vector spaces and review of linear algebra, direct and iterative solutions of linear systems of equations, numerical solutions to the algebraic eigenvalue problem, solutions of general non-linear equations and systems of equations. 4446: Interpolation and approximation, numerical integration and differentiation, numerical solutions of ordinary differential equations. Computer programming skills required.
4445: Vector spaces and review of linear algebra, direct and iterative solutions of linear systems of equations, numerical solutions to the algebraic eigenvalue problem, solutions of general non-linear equations and systems of equations. 4446: Interpolation and approximation, numerical integration and differentiation, numerical solutions of ordinary differential equations. Computer programming skills required.
Analysis of classical and modern applications of mathematics in the physical, biological and social sciences. Emphasis on problem formulating, modeling, solving, simulating, and analyzing results. Programming language required.
Laplace transformations, Fourier series, partial differential equations and separation of variables, boundary value problems, and Sturm-Liouville theory.
Vector Analysis: Greens theorem, potential theory, divergence, and Stokes theorem. Complex Analysis: Analyticity, complex integration, Taylor series, residues, conformal mapping, applications. 4574 may not be taken by math majors for credit.
Course activities will emphasize the curricular themes of problem solving, reasoning and proof, communication, connections, and representation. 4625: Topics in discrete mathematics and algebra from a secondary teaching perspective. 4626: Topics in trigonometry, geometry, measurement, statistics, and probability from a secondary teaching perspective.
Course activities will emphasize the curricular themes of problem solving, reasoning and proof, communication, connections, and representation. 4625: Topics in discrete methematics and algebra from a secondary teaching perspective. 4626: Topics in trigonometry, geometry, measurement, statistics, and probability from a secondary teaching perspective.
Use and impact of technology in secondary mathematics curriculum. Various technologies including graphing calculators, calculator based laboratory and probes (CBLs), computer algebra systems, spreadsheets, dynamic geometry software and the Internet will be used to explore secondary mathematical concepts from an advanced viewpoint.
A review of basic principles and problem-solving techniques in the eleven topics covered by the Praxis II (Mathematics Content Knowledge) examination. Passing the Praxis II examination prior to student teaching is a state requirement for all students seeking secondary licensure. Passing Praxis I required.
May be repeated for a maximum of 12 credits.
Honors section.
Honors section.
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