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Course Criteria
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5.00 Credits
Covers much of the same material as 18.03 with more emphasis on theory. The point of view is rigorous and results are proven. Local existence and uniqueness of solutions.
Prerequisite:
Prereq: None. Coreq: Calculus II (GIR)
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5.00 Credits
Study of ordinary differential equations (ODEs), including modeling physical systems. Solution of first-order ODEs by analytical, graphical, and numerical methods. Linear ODEs, primarily second order with constant coefficients. Complex numbers and exponentials. Inhomogeneous equations: polynomial, sinusoidal, and exponential inputs. Oscillations, damping, resonance. Fourier series inputs; resonant terms. Matrix methods: eigenvalues and eigenvectors, matrix powers and exponentials. Applications to linear systems. Nonlinear autonomous systems: critical point analysis, phase plane diagrams, applications to modeling. Enrollment limited.
Prerequisite:
Prereq: None. Coreq: Calculus II (GIR)
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4.00 Credits
Complex algebra and functions; analyticity; contour integration, Cauchy's theorem; singularities, Taylor and Laurent series; residues, evaluation of integrals; multivalued functions, potential theory in two dimensions; Fourier analysis, Laplace transforms, and partial differential equations.
Prerequisite:
Prereq: Calculus II (GIR); 18.03 or 18.034
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4.00 Credits
Elementary introduction with applications. Basic probability models. Combinatorics. Random variables. Discrete and continuous probability distributions. Statistical estimation and testing. Confidence intervals. Introduction to linear regression.
Prerequisite:
Prereq: Calculus I (GIR)
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4.00 Credits
Basic subject on matrix theory and linear algebra, emphasizing topics useful in other disciplines, including systems of equations, vector spaces, determinants, eigenvalues, singular value decomposition, and positive definite matrices. Applications to least-squares approximations, stability of differential equations, networks, Fourier transforms, and Markov processes. Uses MATLAB. Compared with 18.700, more emphasis on matrix algorithms and many applications.
Prerequisite:
Prereq: Calculus II (GIR)
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5.00 Credits
Elementary discrete mathematics for computer science and engineering. Emphasis on mathematical definitions and proofs as well as on applicable methods. Topics: formal logic notation, proof methods; induction, well-ordering; sets, relations; elementary graph theory; integer congruences; asymptotic notation and growth of functions; permutations and combinations, counting principles; discrete probability. Further selected topics such as: recursive definition and structural induction; state machines and invariants; recurrences; generating functions.
Prerequisite:
Prereq: Calculus I (GIR)
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3.00 Credits
Covers functions of a complex variable; calculus of residues. Includes ordinary differential equations; Bessel and Legendre functions; Sturm-Liouville theory; partial differential equations; heat equation; and wave equations.
Prerequisite:
Prereq: Calculus II (GIR); 18.03
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3.00 Credits
Review of linear algebra, applications to networks, structures, and estimation, finite difference and finite element solution of differential equations, Laplace's equation and potential flow, boundary-value problems, Fourier series, discrete Fourier transform, convolution. Frequent use of MATLAB in a wide range of scientific and engineering applications.
Prerequisite:
Prereq: Calculus II (GIR); 18.03 or 18.034
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3.00 Credits
Initial value problems: finite difference methods, accuracy and stability, heat equation, wave equations, conservation laws and shocks, level sets, Navier-Stokes. Solving large systems: elimination with reordering, iterative methods, preconditioning, multigrid, Krylov subspaces, conjugate gradients. Optimization and minimum principles: weighted least squares, constraints, inverse problems, calculus of variations, saddle point problems, linear programming, duality, adjoint methods.
Prerequisite:
Prereq: Calculus II (GIR); 18.03 or 18.034
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0.00 - 6.00 Credits
One-week review of one-variable calculus (18.01), followed by concentrated study covering multivariable calculus (18.02), two hours per day for five weeks. Primarily for graduate students in Course 2N. Degree credit allowed only in special circumstances.
Prerequisite:
Prereq: Permission of instructor
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