Teaching GuideTerm
Higher Technical University College of Civil Engineering
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Grao en Tecnoloxía da Enxeñaría Civil
 Subjects
  Cálculo infinitesimal II
   Contents
Topic Sub-topic
I. INTEGRATION. 1. Primitive of a function: definition and necessary condition of existence.
2. Riemann Integral: Darboux Sums; integrability conditions; properties.
3. Theorem of the mean.
4. First Fundamental Theorem of Calculus. Barrow's rule.
5. Second Fundamental Theorem of Calculus.
6. Improper integrals.
7. Applications of the definite integral: areas, volumes, arcs and surfaces of revolution.
II. VECTOR FUNCTIONS. 1. Types of functions.
2. Euclidean space: ordinary scalar product; euclidean norm and distance.
3. Vector functions of real variable: limit; continuity; differentiability.
4. Real functions of vector variable: functional and directional limit; continuity; differentiability; directional derivative and partial derivative; differential; gradient.
5. Vector functions of a vector variable: limit; continuity; differentiability.
6. Composition of functions: continuity and differentiability of the composite function; chain rule.
7. Higher order derivatives and differentials.
8. Taylor expansion: general expression; matrix expression.
9. Relative extremes: extreme conditions; determination of the type of quadratic form.
10 Implicit function: definition; existence and differentiability theorem for two variables; generalization.
11. Extremes of functions with constraints: method of Lagrange multipliers.
III. NUMERICAL SERIES. 1. Definition.
2. Geometric series.
3. Necessary condition of convergence.
4. Properties of the series.
5. Cauchy convergence criterion.
6. Series of positive terms. Convergence criteria: majorant and minorant; Riemann series; series comparison; root; quotient; Raabe; logarithmic; condensation.
7. Series of positive and negative terms: absolute and unconditional convergence and divergence; Riemann, Dirichlet and Leibnitz theorems.
8. Methods of addition of series: by decomposition; from the the harmonic; from the exponential of x expansion; hypergeometric series.
IV. FUNCTIONAL SEQUENCES AND SERIES. 1. Functional sequences: definition; simple and uniform convergence; sequences of continuous functions.
2. Functional series: definition; simple and uniform convergence; majorant criterion; continuity; integration; derivation.
3. Power series: definition; Cauchy-Hadamard theorem; continuity, derivation and integration; Abel's theorems; power series expansion of a function, Taylor series.
V. COMPLEX NUMBERS. 1. Introduction.
2. Definition, binomial form and basic operations.
3. Trigonometric form; graphic representation.
4. Conjugate, additive and multiplicative inverse; division.
5. Exponential of a complex; Euler's formula.
6. Natural power of a complex; Moivre's formula.
7. Root of a complex.
8. Fundamental Theorem of Algebra.
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