Course Notes of Ashwani Kapila
Includes Lectures, Exams, and HW (with Solutions)
Rensselaer Polytechnic Institute
2016-2023


Courses

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.