Teaching GuideTerm
Higher Technical University College of Nautical Science and Naval Engines
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Grao en Náutica e Transporte Marítimo
 Subjects
  Mathematics I
   Contents
Topic Sub-topic
Lesson 1.- Matrices and Determinants. 1.1.- Matrices. Operations with matrices.
1.2.- Determinants.
Lesson 2.- Vector spaces.
 2.1.- Introduction.
2.2.- Definition, examples and properties.
2.3.- Linear subspace.
2.4.- Linear dependence and linear independence.
2.5.- Generator systems.
2.6.- Bases. Dimension.
2.7.- Equations of a linear subspace.
2.8.- Range of a system of vectors.
Lesson 3.- Linear Mappings 3.1.- Introduction.
3.2.- Linear mappings.
3.3.- Matrix associated to a linear mapping.
3.4.- Change of basis matrix.
Lesson 4.- Systems of Linear Equations.
 4.1.- Introduction.
4.2.- Definition, examples.
4.3.- Existence and uniqueness of solution. Rouche-Frobenius theorem.
4.4.- Cramer's rule.
4.5.- Gauss and Gauss-Jordan method.
Lesson 5.- Matrix Diagonalization. 5.1.- Eigenvectors and Eigenvalues. Properties.
5.2.- Characteristic polynomial. Properties.
5.3.- Diagonalizable Matrices. Diagonalization.
5.4.- Diagonalization Of Symmetric Matrices.

Lesson 6.- Affine Space E3. Problems of Incidence and Parallelism.
 6.1.- Affine Space Associated to a Vector Space. System of Reference. Coordinates.
6.2.- Equations of Straight Lines.
6.3.- Relative positions of Straight Lines.
6.4.- Equations of a Plane.
6.5.- Relative positions of Planes. Bundles of Planes.
6.6.- Relative positions of Straight Lines and Planes.

Lesson 7.- Euclidean Vector Spaces. Scalar product, Vector product. Mixed Product. 7.1.- Scalar product
7.2.- Determination of a Scalar Product. Gram Matrix.
7.3.- Euclidean Vector Space. Definition.
7.4.- Norm of a Vector. Relevant Equalities and Inequalities.
7.5.- Angle of two Vectors. Orthogonality.
7.6.- Orthonormal Basis. Expression of the Scalar Product in an Orthonormal Basis.
7.7.- Euclidean Space E3.
7.8.- Orientation in E3.
7.9.- Vector product in R3 . Properties. Analytical expression.
7.10.- Mixed product. Analytical expression. Geometrical interpretation.
Lesson 8.- Metric Problems in Euclidean Spaces.
 8.1.- Normal equation of a Plane.
8.2.- Angles between Linear Manifolds in R3: Angle of Two Planes, Angle of Two Straight Lines, Angle of Straight Line and Plane.
8.3.- Distance between Linear Manifolds in R3: Distance of a Point to a Plane, Distance of a Point to a Straight Line. Distance between two Planes, Distance between Straight Line and Plane. Distance between two Straight Lines. Common Perpendicular to two Straight Lines.
8.4.- Cylindrical coordinates and Spherical coordinates in R3.

Lesson 9.-Real valued functions of a Real Variable. Continuity.
 9.1.- Basic definitions.
9.2.- Functional limits.
9.3.- Continuity. Types of Discontinuity.
9.4.- Properties and Theorems on Continuous Functions.

Lesson 10.- Differentiability and Applications of the Derivatives.
 10.1.- Derivative and Differential of a Function in a Point. Geometrical meaning.
10.2.- Properties and Calculation of Derivatives.
10.3.- Derivative function. Successive derivatives.
10.4.- Applications of the Derivatives to the Local Study of a Function: Growth and Decreasing. Maxima and Minima. Concavity and Convexity. Inflection points.
10.5.- Theorems of Rolle and Mean Value Theorem.
10.6.- Rules of L´Hôpital

Lesson 11.- Theorem of Taylor and Applications. Plot of a real function.


 11.1.- Expression of a Polynomial by means of his Derivatives in a Point.
11.2.- Polynomial and Theorem of Taylor. Formulae of Taylor and Mac Laurin.
11.3.- Expression of Lagrange for the Residual. Bounds for the residual.
11.4.- Applications to the Local Study of a Function: Monotonicity. Extremal values. Concavity and Convexity. Inflection points. Plot of a real function.
Lesson 12.- Indefinite integration of Functions of a Real Variable 12.1.- General definitions. Table of Primitives.
12.2.- Immediate integration
12.3.- Integration by Parts
12.4.- Integration of Rational Functions
12.5.- Integration by Replacement or Change of Variable
Lesson 13.- Definite Integration. Applications. 13.1.- General definitions
13.2.- Properties
13.3.- Mean Value Theorem. Barrow's Rule.
13.4.- Evaluation of Definite Integrals.
13.5.- Improper Integral.
13.6.- Applications of the Definite Integral

Lesson 14.- Complex Numbers 14.1.- General definitions
14.2.- Fundamental operations
14.3.- Powers and Roots
14.4.- Exponential form of a Complex
14.5.- Logarithms And Complex Powers.
The development and overcoming of these contents, together with those corresponding to other subjects that include the acquisition of specific competencies of the degree, guarantees the knowledge, comprehension and sufficiency of the competencies contained in Table AII / 2, of the STCW Convention, related to the level of management of chief mates of the Merchant Navy, on ships without gross tonnage limitation and Master up to a maximum of 500 GT. Table A-II / 2 of the STCW Convention.

Mandatory minimum requirements for certification of masters and chief mates on chief on ships of 500 gross tonnage or more.
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