Functions of several variables: continuity, partial derivatives, gradient, max adn minimum, Lagrange multipliers and max and min with constrains, derivatives of second and superior order.
Double and triple integrals: solution via iteration, non-rectangular domains, double integrals in polar coordinates, triple integrals in clindrical and spherical coordinates.
Vector valued functions: gradient, curl and divergence.
Stokes and divergence theorems
Knowledge of the main tools related to several variables funcions and vector valued functions.
Prerequisites
The elements of mathematical Analysis in the first part of the course
Teaching Methods
Lectures. Lessons and exercises
Type of Assessment
Written exam with oral check and discussion of the results
Course program
Several Variables Functions
Cartesian coordinates in dimension 3.
Equations and graphs
Linear Equations
Sphere
Cylinder
Orientation
Several variables functions
Partial derivatives
partial derivatives and contour maps
partial derivatives and linear approximation
Optimization
Multiple Integrals
Double Integrals
Integral as a limit
Calculus of integrals by iteration
Integrals on non-rectangular regions
Double integrals in polar coordinates
Cylindrical and spherical coordinates
Triple Integrals.
Derivatives
Stationary points, max and min
Gradient and linear approximation
Derivatives of composition of functions
Differentiable functions
Lagrange multipliers
Gradient and Lagrange conditions
Integration
Change of variables in multiple integrals
Line integrals
Vector fields
Oriented curves
Green theorem
Green theorem in regions with holes
Surfaces and Integration
Curves, surfaces,
Parametriation of a surface. Examples
Surface integrals
Derivatives and Integrals for vector fields
Flow integrala
Flux and circulation
Stokes and divergence theorem