Topic Sub-topic
The space R^n The vector space R^n.
Scalar product: norms and distances.
Classification of points and sets.
Topology of R^n: bounded set, extrema.
Coordinates systems: polar, cylindrical and spherical coordinates.
Functions of several variables Scalar and vector functions.
Level sets.
Continuity in compact sets.
Differenciation of funcions of several variables and applications Directional derivative.
Partial derivatives: properties and practical computing.
Differential map of a function.
Gradient, relation with partial derivatives.
Relation between the differential map and partial derivatives: jacobian matrix.
Higher order partial derivatives.
Introduction to vector calculus.
Taylor theorem for scalar functions.
Critical points. Classification.
Hessian matrix.
Conditioned extremes: dimension reduction, Lagrange multipliers method.
Implicit function theorem and inverse function theorem.

Integration of funcions of one and several variables Riemann sums.
Integrable functions.
Integral Calculus Theorems: Mean Value Theorem, Fundamental Theorem and Barrow's rule.
Primitive Calculus.
Polinomial interpolation.
Numerical integration. Compound Simpson's Rule.
Application of integral calculus to computing arc lengths, volumes of revolution and surface areas of revolution.
Double integrals.
Triple integrals.
Variable change in double and triple integrals.
Application of integrals: calculation of areas and volumes.
Complex numbers The field of complex numbers.
Operations: sum, produt.
Module and argument.
Polar form.
Operating in polar form.