Videos

Exercises on norms and limits of sequences. Infinite limit. Definition of limit for a vector valued function of vector variable. 

Test for non existence of limit of a function. Examples and exercises. 

Exercises on calculus of limits using polar and spherical coordinates. 

Exercises on calculus of limits. Definition of continuous function. Examples.

Basic tolopogical definitions: open and closed ball, interior and boundary points, open and closed sets. Examples and exercises. Cantor's characterization of closed sets.

Topological properties of sets defined by inequalities and equalities. Bounded and unbounded sets. Examples. Compact sets. Weierstrass' theorem. Exercises. 

Exercises on open, closed, bounded, compact sets of R^d. Existence of min/max for continuous functions on closed and unbounded domains of R^d. Definition of directional derivative. Examples and exercises. 

Partial derivatives: definition and examples. Differentiability, jacobian matrix and connection with directional differentiability. 

Remarks on differentiability: directional differentiability and continuity. Gradient vector. Exercises: checking differentiability. Test of differentiability. 

Exercises on differentiability. Local min/max points. Stationary points. Examples. 

Exercises on stationary points. Fermat's theorem: remarks and proof. Optimization problems: exercises.

Exercises on free optimization. 

Convexity and concavity, sufficiency of the first order condition. Second derivative: Hessian matrix, examples, Schwarz theorem. Positive/negative definite matrices.

(second part of the class missing because of a technical failure).

Taylor's formula and classification of stationary points. Exercises.

Constrained optimization. Lagrange multiplier theorem. Examples and exercises. 

Exercises on Lagrange's multiplier theorem. Submersion maps. 

Exercises on submersion maps and multiple constraints optimization. 

Vector fields: definition and examples. Conservative fields and potentials. Examples. Irrotational fields, examples and exercises. 

Exercises on vector fields and potentials. Line integral.

Circulations and characterization of conservative fields through null circulations. Examples and exercises. 

Introduction to Multiple Integrals: motivations, and definition of the integral for positive and bounded functions.

Definition of integral for a function of variable sign. Reduction formula: Fubini's theorem, examples. 

Exercises on reduction formula. Test of integrability: Tonelli's theorem. Examples and exercises.

Exercises on reduction formula for double integrals. Reduction formula for triple integrals. Examples. 

Exercises reduction formula. Definition of area and volume, slicing formula. Volume of the sphere. Change of variable: recap from one variable calculus, change of variable formula for multiple integrals. Integration in polar coordinates. 

Exercises: computing integrals in polar coordinates. Spherical and cylindrical coordinates, exercises. 

Exercises on integration in spherical/cylindrical coordinates and general change of variables. 

Exercises on change of variable. Green's formula and irrotational fields. 

Exercises on Green's formula, area formula. Introduction to functions of complex variable. 

Series of complex numbers: definition of convergence, Geometric series, test for convergence, absolute convergence. Power series, radius of convergence, disk of convergence. Exponential function: definition and main properties. 

Chauchy-Riemann equations. Exercises. 

Differential Equations: definition of solution, Cauchy problem. Existence and uniqueness: examples, example of non uniqueness.

Local existence and uniqueness. Qualitative study of solutions of differential equations. Exercises. 

Exercises on qualitative study of differential equations.