I am currently in my $2$nd Year, $3$rd Semester of Undergraduate Statistics Major Course. I have Analysis III Course which should mainly revolve around Multivariable Calculus(Differentiation & Integration). But I am facing difficulty in following the course mainly because my Prof. is teaching some really advanced stuffs. The things he have covered is mentioned below:
- Inverse Function Theorem
- Contraction principle
- Inverse Function Theorem for Lipschitz maps
- Local Immersion and Submersion Theorem
- Local Diffeomorphism
- Diffeomorphism
- Rank Theorem
- Implicit Function Theorem
- Trace of a Parametrization
- Manifolds er definitions in different ways and the thing that they imply each other
- Milnor definition implies the original definition of manifold
- Coordinate chart
- Tangent space
- Unconstrained optimization
- Constrained optimization
- Lagrange Multiplier Theorem
- Regular point, critical point
- Local maximum
- Ordinary Differential equations
- Picard Lindelof Theorem
- Initial value problem
- Local flow
- Theorem regarding existence of unique flow
- Local uniqueness of solution of first order ODE
- Patching Lemma theorem
- Liouville Theorem
- Gradient field gradient flow
- Trajectory of a gradient flow
- Equality inequality constraints
- Star convex
- Smoothness
- Poincare Lemma for $\mathbb{R}^2$
- Connected topological spaces
- Alternating linear maps
- Exterior differential
- pth DrRham Cohomology group
- Curl, ker curl, image grad
I need some suggestion for materials or books or lecture videos which cover these topics.
