Teaching GuideTerm Higher Technical University College of Civil Engineering |
Grao en Tecnoloxía da Enxeñaría Civil |
Subjects |
Cálculo infinitesimal II |
Contents |
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Identifying Data | 2023/24 | |||||||||||||
Subject | Cálculo infinitesimal II | Code | 632G02002 | |||||||||||
Study programme |
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Descriptors | Cycle | Period | Year | Type | Credits | |||||||||
Graduate | 2nd four-month period |
First | Basic training | 6 | ||||||||||
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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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