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.- Espazos Vectoriais
 1.1.- Vector space. Definition. Examples and Properties
1.2.- Vector subspace.
1.3.- System of Generators of a Subspace
1.4.- Linear Independence
1.5.- Basis of a Vector Space. Finite Dimensional Spaces.
1.6.- Change of Basis in a Vector Space
1.7.- Union and Intersection of Subspaces
1.8.- Sum of Subspaces. Direct sum. Supplementary Subspaces.
1.9.- Product of Vectorial Spaces
Lesson 2.- Linear Functions. Matrices. 2.1.- Linear Function: Definition, Examples, Properties and Types of Linear Functions.
2.2.- Kernel and Image of a Linear Function.
2.3.- Existence and obtention of an Associated Matrix to a Linear Function.
2.4.- Addition of Linear Functions. Product by a Scalar. Associated Matrices.
2.5.- Vector Spaces of Matrices
2.6.- Composition of Linear Functions. Associated Matrix.
2.7.- Product of Matrices. Ring of Square Matrices
2.8.- Some Particular Types of Matrices
2.9.- Transpose Matrix. Symmetric, Antisymmetric and Orthogonal Matrices.
2.10.- Matrices of Complex Elements.
Lesson 3.- Determinants.
 3.0.- Permutations. Class of a Permutation.
3.1.- Determinant of a Square Matrix. Sarrus Rule.
3.2.- Properties of Determinants.
3.3.- Methods for Calculation of Determinants. Cofactor Matrix.
3.4.- Product of Determinants.
3.5.- Some Particular Examples of Determinants.
3.6.- Reverse Matrix.
3.7.- Rank of a Matrix.
3.8.- Rank of a System of Vectors
3.9.- Expression of the Change of Base of a Vectorial Space in shape Matrix
Lesson 4.- Systems of Linear Equations.
 4.1.- Definitions. Classification. Matrix notation.
4.2.- Equivalent systems.
4.3.- System of Cramer. Rule of Cramer
4.4.- General System of Linear Equations. Theorem of Rouché-Frobenius
4.5.- Homogeneous Systems.
4.6.- Methods of Resolution by Reduction. Gauss' 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.
7.11.- Combined Products.
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. Applications.


 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.
Lesson 12.- Graphic representation of Real Valued Functions.
12.1.- Domain and Continuity
12.2.- Symmetries
12.3.- Periodicity.
12.4.- Intersection with the coordinates axis.
12.5.- Use of successive derivatives and applications: Monotonicity. Extremal values. Concavity and Convexity. Inflection points.
12.6.- Asymptotes and Parabolic Branches

Lesson 13.- Sequences and Series.
 13.1.- General definitions. Types of Sequences.
13.2.- Practical calculation of Limits
13.3.- General definitions. Main Types of Numerical Series.
13.4.- Properties of the Numerical Series. Criteria of Convergence for Series of Positive Terms.
13.5.- Series of Positive and Negative Terms. Alternated Series.

Lesson 14.- Functional Sequences and Series. Series of powers. 14.1.- General definitions.
14.2.- Series of Powers. Convergence.
14.3.- Series expansions.
14.4.- Series of Taylor and Mac Laurin.
14.5.- Binomial Series.
14.6.- Method of the Undetermined Coefficients.

Lesson 15.- Indefinite integration of Functions of a Real Variable 15.1.- General definitions. Table of Primitives.
15.2.- Immediate integration
15.3.- Integration by Parts
15.4.- Integration of Rational Functions
15.5.- Integration by Replacement or Change of Variable
Lesson 16.- Definite Integration. Applications. 16.1.- General definitions
16.2.- Properties
16.3.- Mean Value Theorem. Barrow's Rule.
16.4.- Evaluation of Definite Integrals.
16.5.- Improper Integral.
16.6.- Applications of the Definite Integral

Lesson 17.- Complex Numbers 17.1.- General definitions
17.2.- Fundamental operations
17.3.- Powers and Roots
17.4.- Exponential form of a Complex
17.5.- Logarithms And Complex Powers.
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