One model for a damped harmonic oscillator is described by a linear autonomous ordinary differential equation (ODE). One way to solve this system is to diagonalize the matrix of coefficients (or put into Jordan form if not diagonalizable) and compute the matrix exponential. The meaning of complex eigenvalues is given in this simple example in terms of oscillations.
This is part of a series of lectures on Mathematical Analysis II. Topics covered include continuous and differentiable multi-variable functions on Euclidean space, the chain rule, the implicit function theorem, manifolds, tangent spaces, vector fields, the degree and index of a smooth map, the Euler characteristic, metric spaces, the contraction mapping theorem, existence and uniqueness of solutions to ordinary differential equations, and integral equations.
I speak rather slowly, so you may wish to increase the speed of this video.
These videos were created during the 2017 Spring semester at the UConn CETL Lightboard Room.
