MIT Single Variable Calculus

Course description:

This introductory calculus course covers differentiation and integration of functions of one variable, with applications.

Course topics:

Derivatives, slope, velocity, rate of change. Limits, continuity - Trigonometric limits. Derivatives of products, quotients, sine, cosine. Chain rule - Higher derivatives. Implicit differentiation, inverses. Exponential and log - Logarithmic differentiation; hyperbolic functions. Hyperbolic functions. Linear and quadratic approximations. Curve sketching. Max-min problems. Related rates. Newton's method. Mean value theorem. Inequalities. Differentials, antiderivatives. Differential equations, separation of variables. Definite integrals. First fundamental theorem of calculus. Second fundamental theorem. Applications to logarithms and geometry. Volumes by disks and shells. Work, average value, probability. Numerical integration. Trigonometric integrals and substitution. Integration by inverse substitution; completing the square. Partial fractions. Integration by parts, reduction formulae. Parametric equations, arclength, surface area. Polar coordinates; area in polar coordinates. Indeterminate forms - L'Hôspital's rule. Improper integrals. Infinite series and convergence tests. Taylor's series.

MIT Multivariable Calculus

Course description:

This course covers vector and multi-variable calculus. It is the second semester in the freshman calculus sequence. Topics include vectors and matrices, partial derivatives, double and triple integrals, and vector calculus in 2 and 3-space.

Course topics:

Dot product. Determinants; cross product. Matrices; inverse matrices. Square systems; equations of planes. Parametric equations for lines and curves. Velocity, acceleration - Kepler's second law. Level curves; partial derivatives; tangent plane approximation. Max-min problems; least squares. Second derivative test; boundaries and infinity. Differentials; chain rule. Gradient; directional derivative; tangent plane. Lagrange multipliers. Non-independent variables. Partial differential equations. Double integrals. Double integrals in polar coordinates. Change of variables. Vector fields and line integrals in the plane. Path independence and conservative fields. Gradient fields and potential functions. Green's theorem. Flux; normal form of Green's theorem. Simply connected regions. Triple integrals in rectangular and cylindrical coordinates. Spherical coordinates; surface area. Vector fields in 3D; surface integrals and flux. Divergence theorem. Line integrals in space, curl, exactness and potentials. Stokes' theorem. Topological considerations - Maxwell's equations.

MIT Linear Algebra

Course description:

This is a basic subject on matrix theory and linear algebra. Emphasis is given to topics that will be useful in other disciplines, including systems of equations, vector spaces, determinants, eigenvalues, similarity, and positive definite matrices.

Course topics:

The geometry of linear equations. Elimination with matrices. Multiplication and inverse matrices. Factorization into A = LU. Transposes, permutations, spaces R^n. Column space and nullspace. Solving Ax = 0: pivot variables, special solutions. Solving Ax = b: row reduced form R. Independence, basis, and dimension. The four fundamental subspaces. Matrix spaces; rank 1; small world graphs. Graphs, networks, incidence matrices. Orthogonal vectors and subspaces. Projections onto subspaces. Projection matrices and least squares. Orthogonal matrices and Gram-Schmidt. Properties of determinants. Determinant formulas and cofactors. Cramer's rule, inverse matrix, and volume. Eigenvalues and eigenvectors. Diagonalization and powers of A. Differential equations and exp(At). Markov matrices; fourier series. Symmetric matrices and positive definiteness. Complex matrices; fast fourier transform. Positive definite matrices and minima. Similar matrices and jordan form. Singular value decomposition. Linear transformations and their matrices. Change of basis; image compression. Left and right inverses; pseudoinverse.

MIT Differential Equations

Course description:

Differential Equations are the language in which the laws of nature are expressed. Understanding properties of solutions of differential equations is fundamental to much of contemporary science and engineering. Ordinary differential equations (ODE's) deal with functions of one variable, which can often be thought of as time. Topics include: Solution of first-order ODE's by analytical, graphical and numerical methods; Linear ODE's, especially second order with constant coefficients; Undetermined coefficients and variation of parameters; Sinusoidal and exponential signals: oscillations, damping, resonance; Complex numbers and exponentials; Fourier series, periodic solutions; Delta functions, convolution, and Laplace transform methods; Matrix and first order linear systems: eigenvalues and eigenvectors; and Non-linear autonomous systems: critical point analysis and phase plane diagrams.

Course topics:

Direction Fields, Integral Curves. Euler's Numerical Method. Solving First-order Linear ODE's. Steady-state and Transient Solutions. First-order Substitution Methods: Bernouilli and Homogeneous ODE's. First-order Autonomous ODE's: Qualitative Methods, Applications. Complex Numbers and Complex Exponentials. First-order Linear with Constant Coefficients: Behavior of Solutions, Use of Complex Methods. Continuation; Applications to Temperature, Mixing, RC-circuit, Decay, and Growth Models. Solving Second-order Linear ODE's with Constant Coefficients: The Three Cases. Continuation: Complex Characteristic Roots; Undamped and Damped Oscillations. Theory of General Second-order Linear Homogeneous ODE's: Superposition, Uniqueness, Wronskians. Continuation: General Theory for Inhomogeneous ODE's. Stability Criteria for the Constant-coefficient ODE's. Finding Particular Sto Inhomogeneous ODE's: Operator and Solution Formulas Involving Ixponentials. Interpretation of the Exceptional Case: Resonance. Introduction to Fourier Series; Basic Formulas for Period 2(pi). Continuation: More General Periods; Even and Odd Functions; Periodic Extension. Finding Particular Solutions via Fourier Series; Resonant Terms; Hearing Musical Sounds. Introduction to the Laplace Transform; Basic Formulas. Derivative Formulas; Using the Laplace Transform to Solve Linear ODE's. Convolution Formula: Proof, Connection with Laplace Transform, Application to Physical Problems. Using Laplace Transform to Solve ODE's with Discontinuous Inputs. Use with Impulse Inputs; Dirac Delta Function, Weight and Transfer Functions. Introduction to First-order Systems of ODE's; Solution by Elimination, Geometric Interpretation of a System. Homogeneous Linear Systems with Constant Coefficients: Solution via Matrix Eigenvalues (Real and Distinct Case). Continuation: Repeated Real Eigenvalues, Complex Eigenvalues. Sketching Solutions of 2x2 Homogeneous Linear System with Constant Coefficients. Matrix Methods for Inhomogeneous Systems: Theory, Fundamental Matrix, Variation of Parameters. Matrix Exponentials; Application to Solving Systems. Decoupling Linear Systems with Constant Coefficients. Non-linear Autonomous Systems: Finding the Critical Points and Sketching Trajectories; the Non-linear Pendulum. Limit Cycles: Existence and Non-existence Criteria. Relation Between Non-linear Systems and First-order ODE's; Structural Stability of a System, Borderline Sketching Cases; Illustrations Using Volterra's Equation and Principle.

MIT Computational Science and Engineering I

Course description:

This course provides a review of linear algebra, including applications to networks, structures, and estimation, Lagrange multipliers. Also covered are: differential equations of equilibrium; Laplace's equation and potential flow; boundary-value problems; minimum principles and calculus of variations; Fourier series; discrete Fourier transform; convolution; and applications.

Course topics:

Four special matrices. Differential eqns and Difference eqns. Solving a linear system. Delta function day! Eigenvalues. Positive definite. Positive definite day! Springs and masses; the main framework. Oscillation. Finite differences in time; least squares. Least square. Graphs and networks. Kirchhoff's Current Law. Trusses and ATCA. Finite elements in 1D. Quadratic/cubic elements. Element matrices; 4th order bending equations. Boundary conditions, splines, gradient and divergence. Laplace's equation. Fast Poisson solver. Finite elements in 2D. Fourier series. Discrete Fourier series. Examples of discrete Fourier transform; fast Fourier transform; convolution. Filtering. Filters. Fourier integral transform. Convolution equations: deconvolution; convolution in 2D. Sampling Theorem.

MIT Computational Science and Engineering II

Course description:

This graduate-level course is a continuation of MIT Computational Science and Engineering I. Topics include numerical methods; initial-value problems; network flows; and optimization.

Course topics:

Difference Methods for Ordinary Differential Equations. Finite Differences, Accuracy, Stability, Convergence. The One-way Wave Equation and CFL, von Neumann Stability. Comparison of Methods for the Wave Equation. Second-order Wave Equation (including leapfrog). Wave Profiles, Heat Equation, point source. Finite Differences for the Heat Equation. Convection-Diffusion, Conservation Laws. Conservation Laws, Analysis, Shocks. Shocks and Fans from Point Source. Level Set Method. Matrices in Difference Equations (1D, 2D, 3D). Elimination with Reordering: Sparse Matrices. Financial Mathematics, Black-Scholes Equation. Iterative Methods and Preconditioners. General Methods for Sparse Systems. Multigrid Methods. Krylov Methods, Multigrid Continued. Conjugate Gradient Method. Fast Poisson Solver. Optimization with constraints. Weighted Least Squares. Calculus of Variations, Weak Form. Error Estimates, Projections. Saddle Points, Inf-sup condition. Two Squares, Equality Constraint Bu = d. Regularization by Penalty Term. Linear Programming and Duality. Duality Puzzle, Inverse Problem, Integral Equations.

MIT Introduction to Algorithms

Course description:

This course teaches techniques for the design and analysis of efficient algorithms, emphasizing methods useful in practice. Topics covered include: sorting; search trees, heaps, and hashing; divide-and-conquer; dynamic programming; amortized analysis; graph algorithms; shortest paths; network flow; computational geometry; number-theoretic algorithms; polynomial and matrix calculations; caching; and parallel computing.

Course topics:

Analysis of Algorithms, Insertion Sort, Mergesort. Asymptotic Notation - Recurrences - Substitution, Master Method. Divide-and-Conquer: Strassen, Fibonacci, Polynomial Multiplication. Quicksort, Randomized Algorithms. Linear-time Sorting: Lower Bounds, Counting Sort, Radix Sort. Order Statistics, Median. Hashing, Hash Functions. Universal Hashing, Perfect Hashing. Relation of BSTs to Quicksort - Analysis of Random BST. Red-black Trees, Rotations, Insertions, Deletions. Augmenting Data Structures, Dynamic Order Statistics, Interval Trees. Skip Lists. Amortized Algorithms, Table Doubling, Potential Method. Competitive Analysis: Self-organizing Lists. Dynamic Programming, Longest Common Subsequence. Greedy Algorithms, Minimum Spanning Trees. Shortest Paths I: Properties, Dijkstra's Algorithm, Breadth-first Search. Shortest Paths II: Bellman-Ford, Linear Programming, Difference Constraints. Shortest Paths III: All-pairs Shortest Paths, Matrix Multiplication, Floyd-Warshall, Johnson.

UC Berkeley Statistics

Course topics:

Population and variables. Standard measures of location, spread and association. Normal approximation. Regression. Probability and sampling. Binomial distribution. Interval estimation. Some standard significance tests.

Stanford The Fourier Transform and Its Applications

Course description:

The goals for the course are to gain a facility with using the Fourier transform, both specific techniques and general principles, and learning to recognize when, why, and how it is used. Together with a great variety, the subject also has a great coherence, and the hope is students come to appreciate both.

Course topics:

The Fourier transform as a tool for solving physical problems. Fourier series, the Fourier transform of continuous and discrete signals and its properties. The Dirac delta, distributions, and generalized transforms. Convolutions and correlations and applications; probability distributions, sampling theory, filters, and analysis of linear systems. The discrete Fourier transform and the FFT algorithm. Multidimensional Fourier transform and use in imaging. Further applications to optics, crystallography. Emphasis is on relating the theoretical principles to solving practical engineering and science problems.

Stanford Introduction to Linear Dynamical Systems

Course description:

Introduction to applied linear algebra and linear dynamical systems, with applications to circuits, signal processing, communications, and control systems.

Course topics:

Least-squares aproximations of over-determined equations and least-norm solutions of underdetermined equations. Symmetric matrices, matrix norm and singular value decomposition. Eigenvalues, left and right eigenvectors, and dynamical interpretation. Matrix exponential, stability, and asymptotic behavior. Multi-input multi-output systems, impulse and step matrices; convolution and transfer matrix descriptions. Control, reachability, state transfer, and least-norm inputs. Observability and least-squares state estimation.

Stanford Convex Optimization I

Course topics:



Concentrates on recognizing and solving convex optimization problems that arise in engineering. Convex sets, functions, and optimization problems. Basics of convex analysis. Least-squares, linear and quadratic programs, semidefinite programming, minimax, extremal volume, and other problems. Optimality conditions, duality theory, theorems of alternative, and applications. Interiorpoint methods. Applications to signal processing, control, digital and analog circuit design, computational geometry, statistics, and mechanical engineering.

Stanford Convex Optimization II

Course topics:

Continuation of Convex Optimization I. Subgradient, cutting-plane, and ellipsoid methods. Decentralized convex optimization via primal and dual decomposition. Alternating projections. Exploiting problem structure in implementation. Convex relaxations of hard problems, and global optimization via branch & bound. Robust optimization. Selected applications in areas such as control, circuit design, signal processing, and communications. Course requirements include a substantial project.

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