DIFFERENTIAL CALCULUS

Content:

1. The concept of a derivative

2. Mean-value theorems

3. The concept of diffeomorphism, and the solution of equations

4. Higher-order derivatives

5. The exponential function, and linear differential equations with constant coefficients

6. The integral product, an d linear differential equations

7. Vector fields and differential equations

8. Conjugacy and local coordinates

9. Differentiable submanifolds

10. Calculus of variations

Appendix A: Banach spaces and multilinear maps

Appendix B: The Banach fixed-point theorem

Appendix C: Newton’s method

Appendix D: Global inverse function theorems

Appendix E: Reduction of linear endomorphisms

Appendix F: Linear differential equations with periods coefficients

Appendix G: Existence of solutions of differential equations and their dependence on initial data

Appendix H: Simplicity of SO(3)