Columbia University in the City of New York · Courses
APMA E
26 courses with the subject APMA E, each shown exactly as we captured it from the college's catalog, with every element we hold. Where the wording looks broken, that is our reading of the catalog, not the college's text.
APMA E2000MULTV. CALC. FOR ENGI ＆ APP SCI4.00
Differential and integral calculus of multiple variables. Topics include partial differentiation; optimization of functions of several variables; line, area, volume, and surface integrals; vector functions and vector calculus; theorems of Green, Gauss, and Stokes; applications to selected problems in engineering and applied science
Prerequisites: MATH V1201 A unified, single-semester introduction to differential equations and linear algebra with emphases on (1) elementary analytical and numerical technique and (2) discovering the analogs on the continuous and discrete sides of the mathematics of linear operators: superposition, diagonalization, fundamental solutions. Concepts are illustrated with applications using the language of engineering, the natural sciences, and the social sciences. Students execute scripts in Mathematica and MATLAB (or the like) to illustrate and visualize course concepts (programming not required)
Prerequisites: (MATH UN2030) MATH V2030; or equivalent. Introduction to partial differential equations; integral theorems of vector calculus. Partial differential equations of engineering in rectangular, cylindrical, and spherical coordinates. Separation of the variables. Characteristic-value problems. Bessel functions, Legendre polynomials, other orthogonal functions; their use in boundary value problems. Illustrative examples from the fields of electromagnetic theory, vibrations, heat flow, and fluid mechanics
Prerequisites: Written permission from instructor and approval from adviser. This course may be repeated for credit, but no more than 6 points of this course may be counted toward the satisfaction of the B.S. degree requirements. Candidates for the B.S. degree may conduct an investigation in applied mathematics or carry out a special project under the supervision of the staff. Credit for the course is contingent upon the submission of an acceptable thesis or final report
Subject
APMA E
Credits (min)
0
Credits (max)
4
Credit unit
points
Type
course
Repeatable
may be repeated for credit, but no more than 6 points of this course may be counted toward the satisfaction of the B.
Prerequisites: Cannot be taken together with APMA E3101 Fundamentals of Linear Algebra including vector and Matrix algebra, solution of linear systems, existence and uniqueness, gaussian elimination, gauss-jordan elimination, the matrix inverse, elementary matrices and the LU factorization, computational cost of solutions. Vector spaces and subspaces, linear independence, basis and dimension. The 4 fundamental subspaces of a matrix. Orthogonal projection onto a subspace and solution of Linear Least Squares problems, unitary matrices, inner products, orthogonalization algorithms and the QR factorization, applications. Determinants and applications. Eigen problems including diagonalization, symmetric matrices, positive-definite systems, eigen factorization and applications to dynamical systems and iterative maps. Introduction to the singular value decomposition and its applications
Prerequisites: APMA E3101 AND APMA E4007; After the approval of APMA E4007, it could also serve as a pre-requisite to 4008. Advanced topics in linear algebra with applications to data analysis, algorithms, dynamics and differential equations, and more. (1) General vector spaces, linear transformations, spaces isomorphisms; (2) spectral theory - normal matrices and their spectral properties, Rayleigh quotient, Courant-Fischer Theorem, Jordan forms, eigenvalue perturbations; (3) least squares problem and the Gauss-Markov Theorem; (4) singular value decomposition, its approximation properties, matrix norms, PCA and CCA
Prerequisites: Strong background in multivariate calculus is required. Knowledge in elementary analysis such as the delta-epsilon definition of continuity, convergence of sequences of real numbers, topocontinuity of real-valued functions
Prerequisites: Advanced calculus and course in basic analysis, or instructor's permission. Introduction to modern tools in functional analysis that are used in the analysis of deterministic and stochastic partial differential equations and in the analysis of numerical methods: metric and normed spaces. Banach space of continuous functions, measurable spaces, the contraction mapping theorem, Banach and Hilbert spaces bounded linear operators on Hilbert spaces and their spectral decomposition, and time permitting distributions and Fourier transforms
Prerequisites: Course in ordinary differential equations. Techniques of solution of partial differential equations. Separation of the variables. Orthogonality and characteristic functions, nonhomogeneous boundary value problems. Solutions in orthogonal curvilinear coordinate systems. Applications of Fourier integrals, Fourier and Laplace transforms. Problems from the fields of vibrations, heat conduction, electricity, fluid dynamics, and wave propagation are considered
Prerequisites: (MATH UN1202) MATH V1202; or equivalent. Complex numbers, functions of a complex variable, differentiation and integration in the complex plane. Analytic functions, Cauchy integral theorem and formula, Taylor and Laurent series, poles and residues, branch points, evaluation of contour integrals. Conformal mapping, Schwarz-Christoffel transformation. Applications to physical problems
Prerequisites: (APMA E4300) and APMA E4300; and application and knowledge in C, Fortran or similar complied language. Introduction to the key concepts and issues in computational science aimed at getting students to a basic level of understanding where they can run simulations on machines aimed at a range of applications and sizes from a single workstation to modern super-computer hardware. Topics include but are not limited to basic knowledge of UNIX shells, version control systems, reproducibility, Open MP, MPI, and many-core technologies. Applications will be used throughout to demonstrate the various use cases and pitfalls of using the latest computing hardware
Prerequisites: Elementary probability theory IEOR E3658 or above, stochastic process the first part of IEOR E4106 or STAT G4264, knowledge on analysis MATH GU4601 or above, differential equations APMA E4200, numerical methods APMA E4300 & programming skills Provides elementary introduction to fundamental ideas in stochastic analysis for applied mathematics. Core material includes: (i) review of probability theory (including limit theorems), and introduction to discrete Markov chains and Monte Carlo methods; (ii) elementary theory of stochastic process, Ito's stochastic calculus and stochastic differential equations; (iii) introductions to probabilistic representation of elliptic partial differential equations (the Fokker-Planck equation theory); (iv) stochastic approximation algorithms; and (v) asymptotic analysis of SDEs
x An x following the course number means that the course meets in the fall semester. y A y following the course number means that the course meets in the spring semester.
Required for, and can be taken only by, all applied mathematics majors in the junior year. Introductory seminars on problems and techniques in applied mathematics. Typical topics are nonlinear dynamics, scientific computation, economics, operation research, etc
3-4 pts. Lect: 1 Required for all applied mathematics majors in the senior year. Term paper required. Examples of problem areas are nonlinear dynamics, asymptotics, approximation theory, numerical methods, etc. Approximately three problem areas are studied per term. Fall 2026: APMA E4903 Course Number Section/Call Number Times/Location Instructor Points Enrollment APMA 4903 001/14137 M W 11:40am - 12:55pm 501 Schermerhorn Hall Chris Wiggins 3.00-4.00 67/70
Prerequisites: Advanced calculus and junior year applied mathematics, or their equivalents. May be repeated for credit. Topics and instructors from the Applied Mathematics Committee and the staff change from year to year. For advanced undergraduate students and graduate students in engineering, physical sciences, biological sciences, and other fields. Examples of topics include multi-scale analysis and Applied Harmonic Analysis
Lect: 3. An M.S. degree requirement. Students attend at least three Applied Mathematics research seminars within the Department of Applied Physics and Applied Mathematics and submit reports on each. Spring 2026: APMA E6100 Course Number Section/Call Number Times/Location Instructor Points Enrollment APMA 6100 D01/19995 Kui Ren 0.00 2/2 Fall 2026: APMA E6100 Course Number Section/Call Number Times/Location Instructor Points Enrollment APMA 6100 001/14142 T 2:45pm - 3:35pm 1024 Seeley W. Mudd Building Qiang Du 0.00 47/55 APMA 6100 001/14142 T 4:10pm - 5:00pm 1024 Seeley W. Mudd Building Qiang Du 0.00 47/55 APMA 6100 V01/19588 Qiang Du 0.00 10/99
Prerequisites: (APMA E3101) and (APMA E4200) APMA E3101 AND APMA E4200; or their equivalents. Advanced calculus, basic concepts in analysis, or instructor's permission. Introduction to analytic theory of PDEs of fundamental and applied science; wave (hyperbolic), Laplace and Poisson equations (elliptic), heat (parabolic) and Schroedinger (dispersive) equations; fundamental solutions, Greens functions, weak/distribution solutions, maximum principle, energy estimates, variational methods, method of characteristics; elementary functional analysis and applications to PDEs; introduction to nonlinear PDEs, shocks; selected applications
Prerequisites: (APMA E3102) or (APMA E4200) APMA E3102 OR APMA E4200 Numerical analysis of initial and boundary value problems for partial differential equations. Convergence and stability of the finite difference method, the spectral method, the finite element method and applications to elliptic, parabolic, and hyperbolic equations
