85162 - Differential Calculus

Course Unit Page

Academic Year 2021/2022

Learning outcomes

By the end of the course the student learns the basic concepts and techniques of multivariate differential calculus. More specifically, the student is able to calculate partial derivatives, to find maxima and minima and to use the Lagrange multiplicator method.

Course contents

R^n as a (Euclidean) metric space, normed space, and inner product space.

Partial derivatives and differentiable functions. The total differential theorem.

Multivariate Taylor expansion. Hessian matrices, local extrema.

Implicit functions: Dini's theorem.

Simple, open and regular curves in R^n and tangent vectors. Differentiable manifolds in R^n. Jacobian matrices. Tangent spaces and normal spaces. The method of Lagrange multipliers for optimality in constrained problems.

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