#Math101 Session 41

Time：Oct. 29, 2026, 13:30-15:30

Location：Online via Tencent Meeting (ID: 927-712-396)

Speaker：Mr. Chen, Jiahou (Class of 2023)

Title：An invitation to differentiable manifolds (II)

Abstract：This session continues the exploration of differentiable manifolds by formalizing the transition from topological spaces to differential calculus on manifolds. We begin by reviewing topological manifolds, local coordinate charts, atlases, and compatible atlases, clarifying how coordinate charts allow us to pull back local neighborhoods into Euclidean space to perform standard calculus. Building on this foundation, we examine smooth maps between manifolds, their local coordinate representations, and the independence of smoothness from the choice of charts. Furthermore, we define partial derivatives, directional derivatives, and local linear approximations via coordinate charts, verifying their consistency across compatible atlases.

（文：申爽/审核：郭千桥）