Smooth Spaces 1: Introduction To Synthetic Differential Geometry

In this introductory video we outline the coming course in synthetic differential geometry. We explain the advantages to the synthetic approach to differential geometry, which include being able to set up calculus with greater ease, and being able to think about a large variety of spaces more easily. The remainder of the course will be somewhat informal, although we will make efforts to cover the underlying axioms in an exhaustive way, and we will put a lot of effort into describing numerous applications (which include solving physics problems, computing lengths, areas, volumes and curvatures etc.). In this introductory video we introduce important novel ideas, like the notions of function spaces, nilsquare infinitesimals, and the synthetic approach to defining the operations of calculus. A rigorous approach to this subject, emphasizing the category theoretic foundations, can be found here: Category Theory For Beginners: Synthetic Differential Geometry https://youtu.be/y9ieJ1Gpl6Y More videos about various things to do with synthetic differential geometry can be found within my playlist: https://youtube.com/playlist?list=PLCTMeyjMKRkqy3NUHnMeY-PGgsZ963F58