Differential Geometry I

This is your first real encounter with the geometry of smooth curves and surfaces in Euclidean space, where calculus stops being about graphs and starts being about shape, how a curve bends and twists, how a surface curves in different directions at a point, and which of those properties survive bending versus stretching. You'll spend the semester computing Frenet frames, first and second fundamental forms, Gaussian and mean curvatures, and working through proof-style problem sets on regular surfaces, geodesics, and the Gauss-Bonnet theorem. Multivariable calculus and linear algebra are the working tools throughout, and the payoff is the language you need for manifolds, general relativity, and any future course where "curvature" means something intrinsic rather than visual.