#Math101 Session 43

Time：Nov. 12, 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 (III)

Abstract：This session introduces tangent spaces and the differential of a smooth map. Starting with vectors in Euclidean space, we use coordinate charts to define tangent vectors on a smooth manifold and explain how their coordinate representations transform under a change of charts. We then construct the tangent space at each point and show that its vector space structure is independent of the chosen coordinates. For a smooth map between manifolds, we define its differential, or tangent map, as a linear map between the corresponding tangent spaces. We establish its coordinate representation by the Jacobian matrix and discuss its basic properties through examples.

（文：申爽/审核：郭千桥）