Smooth Spaces 3: Map Spaces, Calculus, and Dynamics
In this video I introduce map spaces and functionals, and describe their applications in physics. I also describe how differentiation, integration, and vector fields can be described using infinitesimals, a new view of the continuum based on infinitesimals, and how we can connect infinitesimals, map spaces, and vector fields with the standard study of continuous dynamical systems via differential equations. This approach is illustrated by looking at how to solve the harmonic oscillator system from physics.
A lot of this content was inspired by Lawvere's paper
https://github.com/mattearnshaw/lawve...
And by Bell's book `A primer of infinitesimal Analysis'
A rigorous approach to this subject, emphasizing the category theoretic foundations, can be found here:
Category Theory For Beginners: Synthetic Differential Geometry
• Category Theory For Beginners: Synthetic D...
More videos about various things to do with synthetic differential geometry can be found within my playlist:
• Synthetic Differential Geometry
A video about intuitionistic logic and type theory can be found here:
• Foundations 5: Intuitionistic Logic and Ty...
A video about the internal language of a topos (the Mitchell-Benabou language) can be found here:
• Category Theory For Beginners: Internal La...