55544 - Differential Calculus

Learning outcomes

By the end of the course the student should know the basics of the mathematical analysis of multivariable functions. In particular the student should be able:- to perform multivariable differential calculus and compute partial derivatives - to identify maxima and minima for functions of several variables- to use constrained optimization: method of Lagrange multipliers- to compute multiple integrals

Course contents

I) METRIC SPACES AND NORMED SPACES

Metric Spaces and Topology. Continuous functions. Metric spaces and sequences.

II) R ^ n EUCLIDEAN.

The Euclidean metric in R^n. First topological concepts in R^n. Continuous functions on domains in R ^ N euclidean. Sequences inthe  Euclidean space R^n; convergent sequences and Cauchy sequences. R ^ n euclidean as a normed space. Normed and metrics. R ^ n euclidean as a space with inner product. Geometric interpretation: Inner Products and Angles. Inner products, norms and metrics.

III) DIFFERENTIABLE FUNCTIONS WITH REAL VALUES ON DOMAIN IN R ^ n

Directional derivatives and partial derivatives. Differentiable functions. "Geometric" Interpretation of Differentiation. Differentiation and directional derivative. Differentials, gradient vector and inner products.  Directional derivatives  and differentiatials. Differentiability and continuity. The Total Differential Theorem. "Legitimate" operations between functions that are differentiable at a point. How do you "write intrinsically" the differentials? "Canonical basis" of the dual space (R ^ n)^ *. Subsequent (or mixed) derivatives. Deriving Composite Functions.

IV) FREE OPTIMIZATION: MAXIMUM AND MINIMUM RELATED

Maximum and minimum points for more multivariate functions. Hessian matrices. Necessary conditions. Sufficient conditions.

 

V) OPTIMIZATION WITH CONSTRAINTS

Vector Valued Functions. Differentiable vector valued functions. Curves in R ^ n. Tangent vectors. Regular varieties in R ^ n. Jacobian Matrices.

Regular varieties and Regular points. Curves, regular varieties, tangent spaces and normal spaces. Critical Points with respect to a Constraint. The Lagrange Multipliers Theorem.

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Readings/Bibliography

Teacher Notes on Pdf Files downloadable from the site

Teaching methods

We will introduce general concept and methods pertaining to the Differentil Calculus for functions in several variables.

We also analyze some concrete problems, in order to stimulate the student to find solutions in an autonomous way.

Assessment methods

The examination consists of an oral examination lasting 45 minutes. Will occur 'the student's competency both in terms of acquisition of concepts and methods, with application to concrete cases.

The student will carefully study five proofs at his own choice
(among the proofs explained in the couse). One of them might be
discussed during the exam.

Office hours

See the website of Andrea Brini