Textbooks:

[1] Zorich V. A. - Mathematical Analysis I

[2] Zorich V. A. - Mathematical Analysis II

Exercise Sheets:

[Sheet 1] [Sheet 2] [Sheet 3] [Sheet 4]

Classroom diary:

1 - 19/03 - Review of normed multilinear maps. The differential of a function between normed vector spaces. Uniqueness of the differential. Differentiability implies continuity. General properties of the differential. Partial derivatives. Ref. [2, Chapters 10.2, 10.3]

2 - 26/03 - Examples of differentials and the geometric meaning of the gradient. Regular curves and their arc-length. Equivalent curves. An example of curve with infinite length. Curvilinear integrals (of the first kind). Ref. [1, Chapters 6.4.2]

3 - 09/04 - The finite increment theorem (opposed to the one-dimensional mean value theorem). Some consequences: sufficient condition for the differentiability, Lipschitz property. Directional derivatives and their relation with the differential. Higher order derivatives and how to calculate them. Ref. [2, Chapters 10.4, 10.5.1, 10.5.2]

4- 11/04 - Symmetry of higher order derivatives. Brief review of bilinear forms and the Hessian matrix. Differential of multilinear functions. Ref. [2, Chapters 10.5.3, 10.5.4]

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5- 16/04 - Taylor polynomial. Study of the Extrema of functions. Some examples. The fundamental lemma of the calculus of variations. Ref. [2, Chapter 10.6]

6- 23/04 - Euler-Lagrange equation. Newton's second law of motion as solution of a variational principle.... Ref. [2, Chapters 10.6]