Tangent Vector and the Arc Length

introduce an apparatus that completely describes all the relevant properties of a curve. After that, we move on to surfaces and their properties. We shall study the first and second fundamental forms, geodesics, and curvature – an apparatus that completely describes a surface. We shall formulate Gauss’s Theorema Egregium (Remarkable Theorem) that allows the concept of curvature to be generalized to curvature of higher dimensional manifolds and enables you to understand the language used in special and general relativity. and to generalize the content of this course to higher dimensions.