Differential Calculus and Differential Equations

In the first part: a more in-depth look at the basic concepts of differential calculus covered in the second year.

In the second part: introduce the qualitative analysis of differential equations.

This lesson will cover the following topics:

Differential Calculus

    - definition of the differential, inequality of finite increments, Ck-class functions, Taylor's formula.

    - The concept of a diffeomorphism, the local inversion theorem, global inversion, and the implicit function theorem.

 Differential Equations

    - Maximum solution, global solution, Cauchy–Lipschitz theorem, finite-time blow-up, global Cauchy–Lipschitz theorem, exit theorem for any compact set.

    - Linear differential equations: the linear Cauchy-Lipschitz theorem, linear systems with a constant operator, general linear systems

    - Autonomous equations and vector fields: flow and integral curve of a vector field; stable and unstable equilibrium. Autonomous differential systems of order 2.

The analysis courses in the first and second years, specifically:

- HAX302X Analysis III: Integration and Elementary Differential Equations

- HAX404X Topology of^{R^n} and Functions of Several Variables

Recommended prerequisites: L2 Math

Hourly volumes:

            CM: 27

            TD: 27

            Practical Work: -

            Land: -