Teaching GuideTerm
Higher Technical University College of Nautical Science and Naval Engines
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Grao en Tecnoloxías Mariñas
 Subjects
  Matemáticas II
   Contents
Topic Sub-topic
Lesson 1.- Bilinear forms. Quadratic forms.
1.1.- Bilinear forms. Associated Matrix
1.2.- Symmetrical bilinear forms
1.3.- Quadratic forms
1.4.- Canonical Quadratic form. Reduction to the Canonical Form
1.5.- Classification of the Quadratic Forms
Lesson 2.- Loci in the Plane. Conic sections

2.1.- Loci in the plane
2.2-. Circumference
2.3.- Elipse
2.4.- Hyperbola. Equilateral hyperbola.
2.5.- Parabola
2.6.- Conic sections.

Lesson 3.- General Equation of a Conic Section. Canonical Form 3.1.- General equation
3.2.- Invariants
3.3.- Classification
3.4.- Reduction to the Canonical Form
3.5.- Obtention of Relevant Elements: Centre, Axes, Asymptotes, Focus, Vertices
3.6.- Graphic representation

Lesson 4.- Loci in the space. Quadric surfaces

4.1.- Loci in the Space
4.2.- Ruled surfaces. Surfaces of Revolution
4.3.- Spherical surface
4.4.- Ellipsoid
4.5.- Hyperboloids
4.6.- Paraboloids
4.7.- Cylindrical surfaces
4.8- Conical Surfaces

Lesson 5.- Functions of several real variables. Limits and Continuity. 10.1.- General definitions

5.1.- General definitions
5.2.- Limits
5.3.- Continuity
Lesson 6.- Partial and Directional Derivatives

6.1.- Partial Derivatives. Tangent Plane
6.2.- Directional Derivatives
6.3.- On Partial Derivatives, Directional Derivatives and Continuity
6.4.- Higher Order Partial derivatives.

Lesson 7.- Differentiation


7.1.- General definitions
7.2.- Differentiability, Continuity and Partial Derivatives
7.3.- Chain Rules. Implicit Differentiation
7.4.- Higher order Differentiation

Lesson 8. Taylor's Theorem. Optimization

8.1.- Taylor’s polinomyal and theorem
8.2.- Relative extrema
8.3.- Conditioned extrema. Lagrange Multipliers.

Lesson 9.- Multiple Integrals. Applications 9.1.- General definitions and Properties
9.2.- Iterated Integrals. Fubini's Theorem.
9.3.- Change of Variables
9.4.- Applications
Lesson 10.- Line Integral and Surface Integral 10.1.- Introduction
10.2.- Line Integral
10.3.- Green's Theorem
10.4.- Surface Integral
10.5.- Surface Integral in Curvilinear Coordinates
10.6.- Stoke's Theorem. Gauss-Ostrogradski's Theorem
Lesson 11.- Ordinary Differential Equations of First Order
11.1.- General definitions
11.2.- Ordinary Differential Equations of First Order
11.3.- Main Types of ODE of First Order
Lesson 12.- Higher Order Ordinary Differential Equations 12.1.- Homogeneous and Nonhomogeneous Second Order ODE’s
12.2.- Second Order Linear ODE with constant coefficients
12.3.- Higher order Nonhomogeneous ODE of n-th Order
Lesson 13.- Systems of Ordinary Differential Equations 13.1.- Systems of Ordinary Differential Equations
13.2.- Systems of Linear Differential Equations with Constant Coefficients
Lesson 14.- Laplace Transform. Integraton by Series
14.1.- Laplace Transform
14.2.- Applications of the Laplace Transform
14.3.- Integration of Ordinary Differential Equations by Series
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