MATHEMATICAL ANALYSIS B
- Overview
- Assessment methods
- Learning objectives
- Contents
- Full programme
- Bibliography
- Teaching methods
- Contacts/Info
Analysis A
Final exam with written (4 problems and 2 questions in 2 hours, open questions) and oral examination.
The final score will be made of the score obtained in the written examination out of 30 eventually averaged with the score of the oral examination always out of 30.
AIM OF THE COURSE
Learning basic tools in Linear Algebra and multivariable Calculus. In particular, we introduce linear complex spaces, the algebra of matrices, representations of linear functions between linear spaces, connections with solvability of linear systems and change of variables. This language is then applied to study multivariable functions, local and global properties, polynomial approximation. Optimization problems also with constraint. We introduce the Riemann integral for multivariable functions and develop calculus methods and applications. We finally introduce vector fields with applications to the Gauss principle.
LEARNING OUTCOMES
At the end of the course, the student will be able to:
1. mastering basic tools of Analysis of multivariable functions;
2. applying theoretical results to more applied topics in the Physics and Engineering within the modelling as well calculus environment;
3. studying problems with both qualitative and quantitative approach;
4. adapting and implementing basic approximation tools in applications.
The complex field: Gauss' plane, operations, algebraic and trigonometric form of complex numbers, De Moire formula, theorem on roots of complex numbers, the fundamental theorem of Algebra, subsets of the Gauss plane. Linear spaces, basis and dimension: scale product, triangular inequality, Cauchy-Schwarz inequality, orthonormal basis and Gram-Schmidt procedure, projections theorem. Linearity: approximation and the superposition principle, the vector space Mat(m,n), row-column product, transpost, diagonal, triangular, and symmetric matrices, rotations in the plane, representation theorem, linear maps between finite dimensional vector spaces, composition of linear maps. Determinants, Laplace theorem and properties. Binet formula, rank of a matrix, kernel and rank of linear maps, nullstellensatz theorem. Inverse of a matrix and applications to linear systems of equations, Touche-Capelli theorem. Eigenvalues and eigenvectors, diagonalization. Real functions of several variables, level curves, limits and continuity, properties. Directional derivatives, linear approximation and the tangent plane, differentiability and the gradient formula. Optimization and the method of restrictions, the increment method. Fermat's theorem, higher order derivatives, Schwarz theorem, the chain rule, taylor expansion. The Hessian matrix. Constrained optimization, the Lagrange multiplier theorem. Surfaces, vector fields and the Jacobian matrix, coordinates transformations in the plane and in the space. Curve integrals and vector fields, conservative and local conservatives fields. Divergence, Curl, Laplacian and differential identities. Multiple integrals, Riemann sums and integrability of continuous functions, simple domains and iterated integrals. Properties of the Riemann integral, change of variables in double and triple integrals, polar coordinates, spherical and elliptic coordinates. Surface integrals, the divergence theorem, the Stoke theorem.
The complex field: Gauss' plane, operations, algebraic and trigonometric form of complex numbers, De Moire formula, theorem on roots of complex numbers, the fundamental theorem of Algebra, subsets of the Gauss plane (4 h.). Linear spaces, basis and dimension: scale product, triangular inequality, Cauchy-Schwarz inequality, orthonormal basis and Gram-Schmidt procedure, projections theorem (4h.). Linearity: approximation and the superposition principle, the vector space Mat(m,n), row-column product, transpost, diagonal, triangular, and symmetric matrices, rotations in the plane, representation theorem, linear maps between finite dimensional vector spaces, composition of linear maps (4h.). Determinants, Laplace theorem and properties. Binet formula, rank of a matrix, kernel and rank of linear maps, nullstellensatz theorem. Inverse of a matrix and applications to linear systems of equations, Rouche-Capelli theorem (4h.). Eigenvalues and eigenvectors, diagonalization (4 h.). Real functions of several variables, level curves, limits and continuity, properties (4h.). Directional derivatives, linear approximation and the tangent plane, differentiability and the gradient formula (4h.). Optimization and the method of restrictions, the increment method (4h.). Fermat's theorem, higher order derivatives, Schwarz theorem, the chain rule, taylor expansion. The Hessian matrix (4h.). Constrained optimization, the Lagrange multiplier theorem (2h.). Surfaces, vector fields and the Jacobian matrix, coordinates transformations in the plane and in the space (2h.). Curve integrals and vector fields, conservative and local conservatives fields (4h.). Divergence, Curl, Laplacian and differential identities (2h.). Multiple integrals, Riemann sums and integrability of continuous functions, simple domains and iterated integrals (4h.). Properties of the Riemann integral, change of variables in double and triple integrals, polar coordinates, spherical and elliptic coordinates (2h.). Surface integrals, the divergence theorem, the Stoke theorem (4h.). (* means including proofs.)
Book: R. Adams, Calculus: A Complete Course (7th Edition), Pearson.
Classical lectures in presence and on-line through Microsoft Teams. Slides and further teaching files available through e-learning.
Professor is available to meeting students before and after lectures and in his office upon email appointment: daniele.cassani@uninsubria.it