MATH202: Linear Algebra (Fall 2019)
A first course in linear algebra, for undergraduate mathematics majors.
Linear algebra, at the level we’ll be examining it, is the study of systems of linear equations. Linear equations end up giving us an extremely good description (at least to the lowest order approximation) of all sorts of phenomena, from physics to economics and finance to the social sciences. As a result, on a practical level, linear algebra touches on pretty much any field with quantitative aspects to it. It’s also a deeply rewarding field of study in its own right: the “coincidence” that linear equations end up describing so much of the world is due to some deep and fascinating properties of the underlying structure of mathematics itself. (This is still a topic of active research!) We’ll explore a bit of this structure, and see how a lot of the practical techniques and concepts across all different disciplines emerge from the properties of the structure itself.
Solutions to linear systems of equations; fields, vector spaces, and subspaces; rank and nullity; the four fundamental subspaces; determinants and inverse matrices; applications of Gauß-Jordan elimination; change of basis; linear transformations; norms, Lp norms, and inner product; orthonormal basis and Gram-Schmidt, eigenvectors and eigenvalues, diagonalization; symmetric, orthogonal, Hermitian, and unitary matrices; spectral decomposition; orthogonal decomposition; singular value decomposition; generalized eigenvectors and Jordan normal form.
MATH206: Complex Analysis (Fall 2019)
A first course in the analysis of functions of a single complex variable, for undergraduate mathematics majors.
Complex differentiability; differentiability properties of elementary functions and harmonic functions; the reflection principle; integration along contours; the Cauchy-Goursat theorem; simple and multiple connectedness of domains; the Cauchy integral formula; the maximum modulus principle; Taylor and Laurent series; analytic continuation; absolute and uniform series convergence; classification of isolated singularities; branch cuts; zeroes and poles; residues; the Cauchy residue theorem and Jordan's lemma; principal values; computation of definite and improper integrals involving polynomials and trigonometric functions; contour deformation around branch points and branch cuts; Rouché's theorem.
MATH303: Partial Differential Equations (Spring 2018)
A course in the linear partial differential equations encountered in physics and engineering, for undergraduate applied mathematics majors.
Classification of second-order PDEs with constant coefficients; boundary value problems on an interval; Robin, Dirichlet, and von Neumann conditions; existence and uniqueness; separation of variables; eigenvalue problems and eigenfunction expansion; Fourier series and applications; inhomogenous problems; integral transform methods; the fundamental solution; Green functions; maximum-minimum principles; the method of characteristics; parabolic equations and the heat equation; elliptic equations and the Laplace equation; the steady-state and equilibrium problems for elliptic equations; the energy integral for the Laplace equation; the Dirichlet problem for the Laplace equation; wave propagation; the d'Alembert solution for hyperbolic equations; the Cauchy problem for hyperbolic equations; the method of characteristics.
PHYS101: Introduction to Physics (Fall 2018, Term 2)
A first course in classical mechanics, for first-year undergraduate students. Previous exposure to differentiation and integration in one dimension is expected.
Units and dimensional analysis; coordinate systems and vectors; unit vectors, orthogonality, inner products, and cross products; displacement, velocity, and acceleration; free fall, projectile motion, and circular motion; Newton's laws; free body diagrams; force balance; friction; centripetal force; partial derivatives; gradient, divergence, and curl; line integrals of scalar and vector quantities; work; kinetic energy; potential energy; conservative and nonconservative forces; conversion of energy; gravitational and elastic potential energy; momentum and impulse; conservation of momentum; elastic and inelastic collisions; center of mass; angular velocity and acceleration; rotational kinetic energy; moment of inertia; torque; work and power for angular motion; angular momentum; conservation of angular momentum; center of gravity.