Explore the Frenet Serret equations in this comprehensive differential geometry lecture. Delve into the mathematical description of unit speed space curves twisting and rotating in three-dimensional space using vector-valued derivatives. Learn about the triple of unit vectors T, N, and B (tangent, normal, and bi-normal) that form a mutually perpendicular frame of basis vectors at each point of the curve. Discover how curvature and torsion play crucial roles in describing the curve's behavior as it moves through space. Gain insights into topics such as normalization, vector quantities, osculating circles, and the Frenet frame. This in-depth exploration of space curves and their properties provides a solid foundation for understanding complex geometric concepts in three-dimensional space.