Teaching GuideTerm Higher Technical University College of Nautical Science and Naval Engines |
Grao en Náutica e Transporte Marítimo |
Subjects |
Mathematics I |
Contents |
Identifying Data | 2016/17 | |||||||||||||
Subject | Mathematics I | Code | 631G01101 | |||||||||||
Study programme |
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Descriptors | Cycle | Period | Year | Type | Credits | |||||||||
Graduate | 1st four-month period |
First | FB | 6 | ||||||||||
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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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