A Formal Background to Mathematics 2a

Inhaltsverzeichnis

• VII: Convergence of Sequences.
• Hidden hypotheses.
• VII.1 Sequences convergent inR.
• VII.2 Infinite limits.
• VII.3 Subsequences.
• VII.4 The Monotone Convergence Principle again.
• VII.5 Suprema and infima of sets of real numbers.
• VII.6 Exponential and logarithmic functions.
• VII.7 The General Principle of Convergence.
• VIII: Continuity and Limits of Functions.
• and hidden hypotheses.
• VIII.1 Continuous functions.
• VIII.2 Properties of continuous functions.
• VIII.3 General exponential, logarithmic and power functions.
• VIII.4 Limit of a function at a point.
• VIII.5 Uniform continuity.
• VIII.6 Convergence of sequences of functions.
• VIII.7 Polynomial approximation.
• VIII.8 Another approach to expa.
• IX: Convergence of Series.
• IX.1 Series and their convergence.
• IX.2 Absolute and conditional convergence.
• IX.3 Decimal expansions.
• IX.4 Convergence of series of functions.
• X: Differentiation.
• X.1 Derivatives.
• X.2 Rules for differentiation.
• X.3 The mean value theorem and its corollaries.
• X.4 Primitives.
• X.5 Higher order derivatives.
• X.6 Extrema and derivatives.
• X.7 A differential equation and the exponential function again.
• X.8 Calculus in several variables.
• XI: Integration.
• XI.1 Integration and area.
• XI.2 Analytic definition and study of integration.
• XI.3 Integrals and primitives.
• XI.4 Integration by parts.
• XI.5 Integration by change of variable (or by substitution).
• XI.6 Termwise integration of sequences of functions.
• XI.7 Improper integrals.
• XI.8 First order linear differential equations.
• XI.9 Integrals in several variables.
• XII: Complex Numbers: Complex Exponential and Trigonometric Functions.
• XII.1 Definition of complex numbers.
• XII.2 Groups, subgroups and homomorphisms.
• XII.3 Homomorphisms ofRinto?; complex exponentials.
• XII.4 The exponential function with domainC.
• XII.5 The trigonometric functions cosine and sine.
• XII.6 Further inverse trigonometric functions.
• XII.7 The simple harmonic equation.
• XII.8 Another differential equation.
• XII.9 Matrices and complex numbers.
• XII.10 A glance at Fourier series.
• XII.11 Linear differential equations with constant coefficients.
• XIII: Concerning Approximate Integration.
• XIII.1 Quotes from syllabus notes.
• XIII.2 Notation and preliminaries.
• XIII.3 Precise formulation of statements XIII.1.1 – XIII.1.3.
• XIII.4 Some corrected versions.
• XIII.5 Falsity of statements XIII.3.1 – XIII.3.3.
• XIII.6 The formulas applied to tabulated data.
• XIV: Differential Coefficients.
• XIV.1 The d-notation and differential coefficients.
• XIV.2 The simple harmonic equation.
• XV: Lengths of Curves.
• XV.1 Quotes and criticisms.
• XV.2 Paths.
• XV.3 Lengths of paths.
• XV.4 Path length as an integral.
• XV.5 Ratio of arc length to chord length.
• XV.6 Additivity of arc length.
• XV.7 Equivalent paths; simple paths.
• XV.8 Circular arcs; application to complex exponential and trigonometric functions.
• XV.9 Angles and arguments.
• XV.10 General remarks about curves.