Tangent Vectors: Definition and Applications

This lecture covers the definition and applications of tangent vectors, focusing on their role in determining the direction of a curve on a surface. Starting with the concept of tangent vectors and their composition with curves, the instructor explains how these vectors are anchored in the tangent plane. Special cases and examples, such as cylinders and spheres, are used to illustrate the properties and behaviors of tangent vectors. The lecture also delves into the regularity of points in parameter settings and the implications for surfaces. The importance of understanding tangent vectors in various contexts is emphasized throughout the presentation.