The Mean Value Theorem and Directional Derivatives.ġ4. DIFFERENTIATING FUNCTIONS OF SEVERAL VARIABLES. Sequentially Compactness and Connectedness.ġ3. Continuous Mappings Between Metric Spaces. ![]() The Existence Theorem for Nonlinear Differential Equations. Completeness and the Contraction Mapping Principle. ![]() Open Sets, Closed Sets, and Sequential Convergence. Connectedness and the Intermediate Value Property.ġ2. Pathwise Connectedness and the Intermediate Value Theorem. Sequential Compactness, Extreme Values and Uniform Continuity. CONTINUITY, COMPACTNESS, AND CONNECTEDNESS. The Linear Structure of Rn and the Scalar Product. A Continuous, Nowhere Differentiable Function.ġ0. Uniform Convergence of Sequences of Functions. Pointwise Convergence of Sequences of Functions. ANon-Analytic, Infinitely Differentiable Function. The Convergence of Darboux and Riemann Sums. Integration by Parts and by Substitution. The Second Fundamental Theorem: Differentiating Integrals.ħ. The First Fundamental Theorem: Integrating Derivatives. The Natural Logarithm and the Exponential Functions. ELEMENTARY FUNCTIONS AS SOLUTIONS OF DIFFERENTIAL EQUATIONS. The Cauchy Mean Value Theorem and Its Analytic Consequences. The Mean Value Theorem and Its Geometric Consequences. Differentiating Inverses and Compositions. The Epsilon-Delta Criterion for Continuity. The Distribution of the Integers and the Rational Numbers. The Completeness Axiom and Some of Its Consequences. Here is the full contents of Advanced Calculus by Fitzpatrick. Then the epsilon-delta definition of continuity. Then from completeness of the reals to various properties of sequences. The starting point is the completeness of the reals, and different ways of saying this (e.g., Dedekind cuts, convergence of Cauchy sequences, least upper bound). So Bartle and Bressoud are a couple of recommendations. Bressoud is ideal for explaining how and why the sound foundation came into being historically. Our intuition leads us astray and hence the need for a sound foundation. mean-value theorem, definition and basic properties of the Riemann integral): why bother proving all this obvious material? The reasons are rooted in history: how to prove that the ratio and root tests work, for example, or why every continuous function isn't differentiable. Why is this important? For beginners, it's not clear why we go through a careful construction of the real numbers, and the result that every Cauchy sequence converges in the reals: after all, the results that we eventually come to are already known from calculus (e.g. Another I recommend is Bressoud's A Radical Approach to Real Analysis: this is written by a master pedagogue who presents analysis through a historical framework. ![]() One I would recommend is Bartle and Sherbert's Introduction to Real Analysis, which is where I learnt my analysis from. I'm reluctant to recommend it to quant students. Rudin's dated Principles of Mathematical Analysis is still used at American universities (my boy used it a couple of years back). If you want real books on analysis, you're spoilt for choice. I prefer Burkill myself because it's a nifty little book and explains how analysis can elucidate the properties of the exponential and trig functions, and how completeness is essential for proving the intermediate-value and mean-value theorems (so does every other book, but it's done here in pedagogically simple style). Burkill used to be recommended reading for incoming math students at Cambridge (i.e., before they started). Cambridge, andĢ) Yet Another Introduction to Analysis, by Bryant, pub. equationsħ.2 Integration by parts and by Substitutionħ.3 The convergence of Darboux and Riemann Sumġ1 Continuity, compactness, and connectdnessġ3 Differentiating functions of several variablesġ4 Local approximation of real-valued functionsġ5 Approximating nonlinear mappings by linear mappingsġ6 Images and inverses: the inverse function theoremġ8 integrating functions of several variablesġ9 iterated integration and changes of variablesĪ couple of books for the entirely uninitiated would be:ġ) A First Course in Mathematical Analysis, by Burkill, pub. Elementary Functions as solutions to diff. For the midterm we are going to cover 1,2,3,4, and 6.ġ.2 The distribution of the Integers and Rationalģ.6 images and inverses monotone functionsĥ. call the course Elementary Analysis and the content might be different.
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