The tangent space of Euclidean space (or of an open set in Euclidean space) at a point is a vector space of the same dimension as that Euclidean space. One can use a chart to define the tangent space of a differentiable manifold at a point by taking the image of the differential of that chart.
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.
