A Formal Background to Mathematics 2a

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. - and hidden hypotheses. - 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. - and hidden hypotheses. - 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. Language: English