Includes Lectures, Exams, and HW (with Solutions)

Rensselaer Polytechnic Institute

2016-2023

**MATH 4600 - Advanced Calculus:** Here

A course emphasizing advanced concepts and methods from calculus. Topics include: multivariable
integral theorems (Green's, divergence, Stokes', Reynolds transport), extrema of multivariable
functions (including Taylor's theorem and Lagrange multipliers), the calculus of variations
(Euler-Lagrange equations, constraints, principle of least action), and Cartesian tensors
(calculus, invariants, representations).

**MATH 6500 - Partial Differential Equations:** Here

A course dealing with the basic theory of partial differential equations. It includes such topics
as properties of solutions of hyperbolic, parabolic, and elliptic equations in two or more
independent variables; linear and nonlinear first order equations; existence and uniqueness
theory for general higher order equations; potential theory and integral equations.

**MATH 6600 - Methods of Applied Mathematics:** Here

Linear vector spaces; eigenvalues and eigenvectors in discrete systems; eigenvalues and eigenvectors
in continuous systems including Sturm-Liouville theory, orthogonal expansions and Fourier series,
Green's functions; elementary theory of nonlinear ODEs including phase plane, stability and
bifurcation; calculus of variations. Applications will be drawn from equilibrium and dynamic
phenomena in science and engineering.

**MATH 6620 - Perturbation Methods:** Here

This course is devoted to advanced methods rather than theory. Content includes such topics
as matched asymptotic expansions, multiple scales, WKB, and homogenization. Applications are
made to ODEs, PDEs, difference equations, and integral equations. The methods are illustrated
using currently interesting scientific and engineering problems that involve such phenomena
as boundary or shock layers, nonlinear wave propagation, bifurcation and stability, and resonance.

**MATH 6640 - Complex Variables and Integral Transforms with Applications:** Here

Review of basic complex variables theory; power series, analytic functions, singularities,
and integration in the complex plane. Integral transforms (Laplace, Fourier, etc.) in the
complex plane, with application to solution of PDEs and integral equations. Asymptotic expansions
of integrals (Laplace method, methods of steepest descent and stationary phase), with emphasis
on extraction of useful information from inversion integrals of transforms. Problems to be drawn
from linear models in science and engineering.