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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. BolzanoWeierstrass theorem. Algebraic and transcendental numbers. The complex number system. Polar form of complex numbers. Mathematical induction.
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SEQUENCES
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De nition of a sequence. Limit of a sequence. Theorems on limits of sequences. In nity. Bounded, monotonic sequences. Least upper bound and greatest lower bound of a sequence. Limit superior, limit inferior. Nested intervals. Cauchy s convergence criterion. In nite series.
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FUNCTIONS, LIMITS, AND CONTINUITY
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Functions. Graph of a function. Bounded functions. Montonic functions. Inverse functions. Principal values. Maxima and minima. Types of functions. Transcendental functions. Limits of functions. Right- and left-hand limits. Theorems on limits. In nity. Special limits. Continuity. Right- and left-hand continuity. Continuity in an interval. Theorems on continuity. Piecewise continuity. Uniform continuity.
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DERIVATIVES
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The concept and de nition of a derivative. Right- and left-hand derivatives. Di erentiability in an interval. Piecewise di erentiability. Di erentials. The di erentiation of composite functions. Implicit di erentiation. Rules for di erentiation. Derivatives of elementary functions. Higher order derivatives. Mean value theorems. L Hospital s rules. Applications.
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CONTENTS
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INTEGRALS
Introduction of the de nite integral. Measure zero. Properties of de nite integrals. Mean value theorems for integrals. Connecting integral and di erential calculus. The fundamental theorem of the calculus. Generalization of the limits of integration. Change of variable of integration. Integrals of elementary functions. Special methods of integration. Improper integrals. Numerical methods for evaluating de nite integrals. Applications. Arc length. Area. Volumes of revolution.
PARTIAL DERIVATIVES
Functions of two or more variables. Three-dimensional rectangular coordinate systems. Neighborhoods. Regions. Limits. Iterated limits. Continuity. Uniform continuity. Partial derivatives. Higher order partial derivatives. Di erentials. Theorems on di erentials. Di erentiation of composite functions. Euler s theorem on homogeneous functions. Implicit functions. Jacobians. Partial derivatives using Jacobians. Theorems on Jacobians. Transformation. Curvilinear coordinates. Mean value theorems.
VECTORS
Vectors. Geometric properties. Algebraic properties of vectors. Linear independence and linear dependence of a set of vectors. Unit vectors. Rectangular (orthogonal 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 r. Vector interpretation of Jacobians, Orthogonal curvilinear coordinates. Gradient, divergence, curl, and Laplacian in orthogonal curvilinear coordinates. Special curvilinear coordinates.
APPLICATIONS OF PARTIAL DERIVATIVES
Applications to geometry. Directional derivatives. Di erentiation under the integral sign. Integration under the integral sign. Maxima and minima. Method of Lagrange multipliers for maxima and minima. Applications to errors.
MULTIPLE INTEGRALS
Double integrals. Iterated integrals. Triple integrals. Transformations of multiple integrals. The di erential element of area in polar coordinates, di erential elements of area in cylindrical and spherical coordinates.
CONTENTS
LINE INTEGRALS, SURFACE INTEGRALS, AND INTEGRAL THEOREMS
Line integrals. Evaluation of line integrals for plane curves. Properties of line integrals expressed for plane curves. 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. Stoke s theorem.
INFINITE SERIES
De nitions of in nite series and their convergence and divergence. Fundamental facts concerning in nite series. Special series. Tests for convergence and divergence of series of constants. Theorems on absolutely convergent series. In nite sequences and series of functions, uniform convergence. Special tests for uniform convergence of series. Theorems on uniformly convergent series. Power series. Theorems on power series. Operations with power series. Expansion of functions in power series. Taylor s theorem. Some important power series. Special topics. Taylor s theorem (for two variables).
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