Catálogo de la biblioteca Antonio Monteiro - INMABB

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Chapter 1 NUMBERS [1] --
Sets. Real numbers. Decimal representation of real numbers. Geometric representation of real numbers. Operations with real numbers. Inequalities. Absolute value of real numbers. Exponents and roots. Logarithms. Axiomatic foundations of the real number system. Point sets. Intervals. Countability. Neighborhoods. Limit points. Bounds. Weierstrass-Bolzano theorem. Algebraic and transcendental numbers. The complex number system. Polar form of complex numbers. Mathematical induction. --
Chapter 2 FUNCTIONS, LIMITS AND CONTINUITY [20] --
Functions. Graph of a function. Bounded functions. Monotonic functions. Inverse functions. Principal values. Maxima and minima. Types of functions. Special transcendental functions. Limits of functions. Right and left hand limits. Theorems on limits. Infinity. Special limits. Continuity. Right and left hand continuity. Continuity in an interval. Theorems on continuity. Sectional continuity. Uniform continuity. --
Chapter 3 SEQUENCES [41] --
Definition of a sequence. Limit of a sequence. Theorems on limits of sequences. Infinity. Bounded, monotonic sequences. Least upper bound and greatest lower bound of a sequence. Limit superior. Limit inferior. Nested intervals. Cauchy’s convergence criterion. Infinite series. --
Chapter 4 DERIVATIVES [57] --
Definition of a derivative. Right and left hand derivatives. Differentiability in an interval. Sectional differentiability. Graphical interpretation of the derivative. Differentials. Rules for differentiation. Derivatives of special functions. Higher order derivatives. Mean value theorems. Rolle’s theorem. The theorem of the mean. Cauchy’s generalized theorem of the mean. Taylor’s theorem of the mean. Special expansions. L’Hospital’s rules. Applications. --
Chapter 5 INTEGRALS [80] --
Definition of a definite integral. Measure zero. Properties of definite integrals. Mean value theorems for integrals. Indefinite integrals. Fundamental theorem of integral calculus. Definite integrals with variable limits of integration. Change of variable of integration. Integrals of special functions. Special methods of integration. Improper integrals. Numerical methods for evaluating definite integrals. Applications. --
Chapter 6 Page PARTIAL DERIVATIVES [101] --
Functions of two or more variables. Dependent and independent variables. Domain of a function. Three dimensional rectangular coordinate systems. Neighborhoods. Regions. Limits. Iterated limits. Continuity. Uniform continuity. Partial derivatives. Higher order partial derivatives. Differentials. Theorems on differentials. Differentiation of composite functions. Euler’s theorem on, homogeneous functions. Implicit functions. Jacobians. Partial derivatives using Jacobians. Theorems on Jacobians. Transformations. Curvilinear coordinates. Mean value theorems. --
Chapter 7 VECTORS [134] --
Vectors and scalars. Vector algebra. Laws of vector algebra. Unit vectors. Rectangular unit vectors. Components of a vector. Dot or scalar product. Cross or vector product. Triple products. Axiomatic approach to vector analysis. Vector functions. Limits, continuity and derivatives of vector functions. Geometric interpretation of a vector derivative. Gradient, divergence and curl. Formulas involving V. Vector interpretation of Jacobians. Orthogonal curvilinear coordinates. Gradient, divergence, curl and Laplacian in orthogonal curvilinear coordinates. Special curvilinear coordinates. --
Chapter 8 APPLICATIONS OF PARTIAL DERIVATIVES [161] --
Applications to geometry. Tangent plane to a surface. Normal line to a surface. Tangent line to a curve. Normal plane to a curve. Envelopes. Directional derivatives. Differentiation under the integral sign. Maxima and minima. Method of Lagrange multipliers for maxima and minima. Applications to errors. --
Chapter 9 MULTIPLE INTEGRALS [180] --
Double integrals. Iterated integrals. Triple integrals. Transformations of multiple integrals. --
Chapter 10 LINE INTEGRALS, SURFACE INTEGRALS AND INTEGRAL THEOREMS [195] --
Line integrals. Vector notation for line integrals. Evaluation of line integrals. Properties of line integrals. Simple closed curves. Simply and multiply-connected regions. Green’s theorem in the plane. Conditions for a line integral to be independent of the path. Surface integrals. The divergence theorem. Stokes’ theorem. --
Chapter 11 INFINITE SERIES [224] --
Convergence and divergence of infinite series. Fundamental facts concerning infinite series. Special series. Geometric series. The p series. Tests for convergence and divergence of series of constants. Comparison test. Quotient test. Integral test. Alternating series test Absolute and conditional convergence. Ratio test. The nth root test Raabe’s test, Gauss’ test. Theorems on absolutely convergent series. Infinite sequences and series of functions. Uniform convergence. Special tests for uniform convergence of series. Weierstrass M test Dirichlet’s test Theorems on uniformly convergent series. Power series. Theorems on power series. Operations with power series. Expansion of functions in power series. Some important power series. Special topics. Functions defined by series. Bessel and hypergeometric functions. Infinite series of complex terms. Infinite series of functions of two (or more) variables. Double series. Infinite products. Summability. Asymptotic series. --
Chapter 12 IMPROPER INTEGRALS [260] --
Definition of an improper integral. Improper integrals of the first kind. Special improper integrals of the first kind. Geometric or exponential integral. The p integral of the first kind. Convergence tests for improper integrals of the first kind. Comparison test. Quotient test. Series test. Absolute and conditional convergence. Improper integrals of the second kind. Cauchy principal value. Special improper integrals of the second kind. Convergence tests for improper integrals of the second kind. Improper integrals of the third kind. Improper integrals containing a parameter. Uniform convergence. Special tests for uniform convergence of integrals. Weierstrass M test. Dirichlet’s test. Theorems on uniformly convergent integrals. Evaluation of definite integrals. Laplace transforms. Improper multiple integrals. --
Chapter 13 GAMMA AND BETA FUNCTIONS [285] --
Gamma function. Table of values and graph of the gamma function. Asymptotic formula for Γ(n). Miscellaneous results involving the gamma function. Beta function. Dirichlet integrals. --
Chapter 14 FOURIER SERIES [298] --
Periodic functions. Fourier series. Dirichlet conditions. Odd and even functions. Half range Fourier sine or cosine series. Parseval’s identity. Differentiation and integration of Fourier series. Complex notation for Fourier series. Boundary-value problems. Orthogonal functions. --
Chapter 15 FOURIER INTEGRALS [321] --
The Fourier integral. Equivalent forms of Fourier’s integral theorem. Fourier transforms. Parseval’s identities for Fourier integrals. The convolution theorem. --
Chapter 16 ELLIPTIC INTEGRALS [331] --
The incomplete elliptic integral of the first kind. The incomplete elliptic integral of the second kind. The incomplete elliptic integral of the third kind. Jacobi’s forms for the elliptic integrals. Integrals reducible to elliptic type. Jacobi’s elliptic functions. Landen’s transformation. --
Chapter 17 FUNCTIONS OF A COMPLEX VARIABLE [345] --
Functions. Limits and continuity. Derivatives. Cauchy-Riemann equations. Integrals. Cauchy’s theorem. Cauchy’s integral formulas. Taylor’s series. Singular points. Poles. Laurent’s series. Residues. Residue theorem. Evaluation of definite integrals. --

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