| Topic |
Sub-topic |
| Subjet 1. The fields of real and complex numbers |
Numerical sets. The prinicple of induction. The real numbers. Characteristic properties. The axiom of the suprem. Intervals in R. Absolute value. Extended real line.
The field of complex numbers. Representation of complex numbers. Module and argument. Euler's Formula. Operations with complex numbers in polar form: powers (De Moivre's formula), roots, exponentials, logarithms. |
| Subject 2. Vector spaces |
The vector space Rn. Vector Subspaces. Linear combination. Linear dependency and independence. Finite-dimensional vector spaces. Basis and dimension. Rank. |
Subject 3. Linear applications
|
Linear applications. Properties. Kernel and image of a linear application. Characterization of injectives and surjectives linear applications. Rank of a linear application. Associated matrix to a linear application. |
Subject 4. Matrices
|
Definition and types of matrices. Vectorial space of the matrices mxn. Product of matrices. Regular matrix. Rank of a matrix. Calculation of the rank of a matrix and of the reverse matrix by means of elementary operations.
|
Subject 5. Determinants
|
Determinants of a square matrix of order 2 and of order 3. Properties. Development by adjoints. Calculation of the reverse matrix. Calculation of the rank of a matrix.
|
Subject 6. Systems of linear equations
|
Systems of linear equations: matrix form. Equivalent systems. Existence of solutions: theorem of Rouché-Frobenius. Homogeneous systems. Resolution of systems of linear equations: resolution by means of the methods of elimination of Gauss and Gauss-Jordan. Resolution of a system of Cramer. Resolution of a general system using the Cramer's rule.
|
Subject 7. Euclidean vector space
|
Scalar product. Norm. Distance. Orthogonality. Scalar product with respect to a basis. Orthogonal and orthonormal systems. Vector product. Mixed product. Areas and volumes.
|
Subject 8. Geometry
|
Three-dimensional affine space. The straight lines in the affine space. Equations of the straight line. The plane in the affine space. Equations of the plane. Relations of incidence between straight lines and planes. Angles: of two straight lines, of two planes and of a straight line and a plane. Distances: of a point to a plane, of a straight line to a plane and of two croseed straight lines. Metric study of the conical.
|
Subject 9. Diagonalization of endomorphisms and matrices
|
Eigenvectors and eigenvalues. Eigensubspaces. Characteristic polynomial. Diagonalization: Conditions. Annulator polynomial. Theorem of Cayley-Hamilton. Applications.
|
Subject 10. Convergence in R.
|
Topology of the real line: distinguished points, compact sets. Convergent sequences in R. Operations with limits. Calculation of limits: indeterminations, rules of Stolz, of the arithmetical and geometrical means and of the root.
Numerical series. Geometrical and telescopic series. Series of positive terms. Criteria of convergence. Alternating series. Abel's Criterion. Absolute convergence. Sumation of some elementary series.
|
Subject 11. Limit and continuity of functions of a real variable
|
Limit of a function in a point. Sequential limit. Properties of the limits. Calculation of limits. Continuity of real functions. Discontinuity: Types. Operations with continuous functions. Theorems relative to the global continuity: continuous image of a compact, Bolzano-Weierstrass' theorem, Bolzano's theorem: consequences. Continuity of the reverse function and of the composition of functions. |
| Subject 12. Differential calculus of a variable |
Derivative of a function in a point. Geometric interpretation of the concept of derivative. The differential. Derived function. Successive derivatives. Relationship between continuity and derivability. Calculation of derivatives: derivative of the composition of functions and of the inverse function. Theorems relative to derivable functions: Rolle's theorem, consequences; The mean value theorem, consequences; The rule of L'Hôpital, calculation of indeterminate limits. Taylor polynomials of a function. Taylor's theorem. Maximum and minimum Problems. Study of concavity and convexity. Inflection points. Graphical representation of functions |
Subject 13. Integration of functions of a variable
|
The Riemann integral: partitions, upper and lower sums, upper and lower integral, integral functions, the integral as sum limit. Properties. Theorem of the mean value. The fundamental theorem of integral calculus. Barrow's rule. Primitives. General methods for the calculation of primitives. Improper integrals. Geometric applications of the integral. |
Subject 14. Informatics
|
Operating systems: classification, components, examples. Programming fundamentals. Organization of archives. Methods of sorting and searching. Concept and types of databases. |
| LABORATORY PRACTICE AGENDA |
|
| Practice 1. Introduction to the syntax of a computer algebra system. |
Basic commands of a computer algebra system. |
| Practice 2. Complex Numbers |
Complex arithmetic in binomial form. Polar form. Arithmetic in polar form |
Practice 3. Vector Spaces
|
Operations with vectors. Linear independence of vectors and calculation of bases. Generator systems. Range of a vector system. |
| Practice 4. Linear Applications |
Calculation of the associated matrix. Calculation of the kernel, image and rank |
| Practice 5. Matrices and determinants |
Operations with matrices. Calculation of the determinant of a square matrix. Calculation of the rank of a matrix and the inverse matrix |
| Practice 6. Systems of linear equations |
Resolution of linear systems. Cramer's Rule and Gauss and Gauss-Jordan Elimination Methods. Applications. |
| Practice 7. Euclidean Vector Space and Geometry |
Calculation of the scalar product, vector product and mixed product. Calculation of areas, volumes, angles and distances. Conical curves |
| Practice 8. Diagonalization |
Calculation of the eigenvalues and eigenvectors of a square matrix. Diagonalization of matrices. Applications |
Practice 9. Convergence and Series
|
Limit of numerical sequences. Application of the convergence criteria of series. Sum of series. |
Practice 10. Functions
|
Calculation of the limit of a function at a point. Graphical representation of functions. Study of continuity. |
Practice 11. Derivatives.
|
Derivative of functions. Calculation of tangent and normal lines. Problems of relative extremes. Developments in Taylor series. Local study of functions.
|
Practice 12. Integration
|
Calculation of primitives. Applications: calculation of areas, volumes, arc lengths, moments of inertia, etc |
Subject 13. Informatics
|
Programming Fundamentals. Development and management of databases |
