LEARNING OUTCOMES

Students will become acquainted with the most important topics in several (real) variables and how they are used in practice.  We  only present proofs that illustrate fundamental principles and are free of technicalities.
Applications to Physics and Probability are emphasised.

AIMS AND LEARNING OUTCOMES

At the end of the first semester, students will be able to manipulate functions of several variables and solve basic optimization problems. Moreover they will be at their ease with mean and conditional expectation. 

Physics students will be able to apply vector calculus: use double, triple and line integrals in applications, including Green's Theorem, Stokes' Theorem and Divergence Theorem.

PREREQUISITES

First year calculus (derivatives and integrals for functions of a single variable, sequences and numerical series). Vector spaces, eigevalues and eigenvectors are frequently used.

TEACHING METHODS

Both theory and exercises are presented by the teacher. The first semester consists of 12 weeks with four hours of theory and two hours of exercises per week. The second semester consists of 12 weeks with three hours of theory and two hours of exercises per week.  

SYLLABUS/CONTENT

Differential Calculus

1. Vectors, scalar product, norm, distance
2. Elements of Topology: open and closed sets, bounded, compact and connected sets, isolated, cluster and boundary points. Heine-Borel's Theorem
3. Functions: examples and graphs, level sets. Continuous functions and their local properties. Algebra of continuous functions.
4. Limits.
5. Global properties of continuous functions: Weierstrass Theorem and Intermediate value Theorem
6. Differential calculus: partial derivatives, gradient, jacobian matrix. Tangent vector to a curve. Tangent plane to the graph of a 2 variable function. Chain rule. Functions with null gradient. Second derivatives,  Schwarz' Theorem. Hessian and Taylor formula of order 2. Relative maxima and minima. Diffeomorphisms. Inverse Function Theorem. Implicit functions. Lagrange Multipliers.

Integral Calculus

1. 2 dimensional Riemann integral: definition. Fubini's Theorem. Normal domains. Change of variables. 
2. Convergence of improper integrals.
3. Triple integrals. Spherical and cylindrical coordinates.
4. Parametric integrals.
5. Elements of Lebesgue measure. Monotone convergence Theorem and Lebesgue Theorem. 

Sequences and series

1. Function sequences: point and uniform convergence. Relation with continuity, integration and differentiation of the limit function.
2. Function series: Weierstrass' test. 
3. Power series. 
4. Taylor series. 
5. Fourier series

Differential geometry of curves and surfaces.

1. Curves: speed and velocity. Length, arc length, curvature. Line integrals.
2. Parametric surfaces: tangent plane, oriented surfaces. Area. Surface integrals and fluxes.
3. Vector fields in R^2. Bounded regular domains and their boundaries. Green's Theorem. Vector fields in R^3. Curl and divergence. Gauss and Stokes' Theorems.
4. Potential Theory

Differential equations

1. Ordinary differential equations and IVPs (Initial Value Problems).
2. Systems of differential equations.
3. Existence and uniqueness for IVPs (local and global)
4. Linear systems: structure of solutions. Constant coefficient systems.

RECOMMENDED READING/BIBLIOGRAPHY

Serge Lang - Calculus of Several Variables, Third Edition, Undergraduate Texts in Mathematics, Springer, 1987.