An Introductory Calculus Course

Articles

Introductory Calculus course.  Read tutorial guide and the chapters in text book and then do the tutorial.

∫ ln(x)^2/(1+e^x), ∫ xln(x)/(1+e^x),∫ xln(x)^2/(1+e^x), Gamma function and Eta function  New

Proofs of convergence of the improper integrals defining the Gamma function, justification of differentiation under the integration sign, derivatives of Gamma function and Eta function.

  Riemann Integral and Bounded Function

  Comment and Errata to Calculus, an introduction

Calculus, an introduction available from NUS Coop

Download and install the following 3 math fonts to view the review:Click to install  font a  font b font c

  L Hopital's Rule - And a Generalized Version 

  Concavity - Definitions and Equivalence  

Answer to each individual chapter's exercise is available upon request with your email 

Now ALL fourteen chapters come with exercise problems.  Intermediate to advanced entry to mathematical analysis

Comments welcomed

The links to each individual chapter below:

   Arc Length, Function of Bounded Variation and Total Variation 

   Sequences and Series 

Change of Variables Theorems in Integration   - a follow up of  "change of Var  in Riemann and Lebesgue Integration" shorter proof.

  Kestelman's Change of Variable Theorem 

Functions Having Finite Derivatives, Bounded Variation, Absolute Continuity, the Banach Zarecki Theorem and de La Vallee Poussin Theorem   Elementary Proof of de la Vallee Poussin Theorem

Function of Bounded Variation and Johnson Indicatrix

  Partial Fraction Expansion -- Its proof, a simple application of complex analysis

  When is a continuous function on a closed and bounded interval be of bounded variation, absolutely continuous?  Revised

 NUS MA3110 2011/12 Sem 2 Exam Sample Answer  

NUS MA3110 2010/11 SEm 1 Exam Comment and Answer

   Algebra refresher Inequalities

    Self Help Centre Quick Reference

   Self Help Centre Refresher

   Self Help Centre Work Books

Mathematics Assessments for Revision

    (Algebra and Calculus AO-A level)  

Cantor Lebesgue Function, Canonical Cantor type function between families of Cantor sets, Absolute Continuity and Arc Length    

Included are results on the derivatives of Cantor type functions over the fat Cantor set and their integrals.

Positive Borel Measure and Riesz Representation Theorem

Riesz Representation Theorem-positive measure version for positive linear functional. Detail step by step proofs and Lebesgue measure on R^k via Riemann integration and Lebesgue integral.

Convex Function, L^p spaces, Space of continuous function, Lusin's Theorem 

A detail introductory exposition of L^p spaces and a proof of Lusin's Theorem including the necessary topological ideas and concepts.    

  A short proof of the Kestelman change of variable Theorem for Riemann integral  Revised (To include a result that don't require the function f to be bounded on the whole of the domain.)

 A proof using only the properties of absolutely continuous function and the chain rule for the composition of functions having finite derivative almost everywhere.

  A general change of variable theorem for the Riemann integrable New

Generalized Kestelman change of variable theorem - applies to most situation

  An Introduction To Measure Theory 

A leisurely introduction to measure theory.  A learner's guide to   Lebesgue Monotone Convergence Theorem, Lebesgue Dominated Convergence Theorem, Fatou's Lemma and complete measure.

  Lebesgue Measure On The Real Numbers and Lebesgue Theorem On Riemann Integrability

A detail definition of Lebesgue measure on the real numbers is given. Show that Lebesgue measure is Borel and complete.  Define Riemann integral via step functions, show that it is equivalent to the Darboux integral and prove the Lebesgue characterization of Riemann integrability.

Complex Measure, Dual Space of L^p , Radon-Nikodym Theorem and Riesz RepresentationTheorems- Complex and real versions 

Identification of the dual of L^p spaces and the dual of C_c(X) with detail exposition and proofs.  Proofs for both real and complex versions when X is locally compact as well as the dual for BC(X) the space of bounded continuous real valued functions when X is normal and Hausdorff are presented.  A brief discussion when X is completely regular and Hausdorff is added.

Convergence In Measure 

Convergence in measure or in probability, a notion often used in probability theory. Convergence almost uniformly and convergence almost everywhere, Egoroff's theorem. As is expected, for a probability space, convergence almost everywhere implies convergence in measure.  Monotone Convergence theorem, Bounded Convergence Theorem and Dominated Convergence Theorem for Convergence in measure. Fatou's Lemma.

Product Measure and Fubini's Theorem  New

This completes the above article: An Introduction To Measure Theory.  A step by step construction of the product measure space and the definition of the positive product measure function is given, followed by a detailed elaboration of the proof of the Fubini's Theorem.  The special case when all measure spaces are required to be complete, is worked through with detail steps and intermediary results. 

Function of Bounded Variation on Arbitrary Set 

Following my previous article, on the image of the total variation function of a function of bounded variation on a closed interval, we now obtain the same result on general arbitrary domain.

Denjoy Saks Young Theorem

  Arbitrary Function, Lim Sup Inf, Dini Derivates,Lebesgue Density Theorem

Functions of Bounded Variation and de La Vallee Poussin's Theorem

  Denjoy Saks Young Theorem for Arbitrary Function

  Discourse on Monotone Functions

Absolutely Continuous Function on Arbitrary Domain and Function of Bounded Variation

A continuous function on a closed and bounded domain is absolutely continuous if and only if it is of bounded variation and a Lusin function.  A detail proof of this result is presented and other equivalent formulation with the integrability of the derivative function or the nullity of the image of the set, where the derivative is infinite, positive or negative.  All the technical intermediate results used and their proofs are deliberated.

Topology Article

Explicit decomposition of Steenrod Square 64 by stable secondary cohomology operations and an illustration of proof of non-immersion of quaternionic projective space with obstruction theory. 

  A de La Vallee Poussin's  Decomposition Theorem

  For arbitrary domain, the classical de La Vallee Poussin's  Decomposition for the outer measure of the image of measurable set under the total variation function.

  Lebesgue Stieltjes Measure, de La Vallee Poussin’s Decomposition, Change of Variable, Integration by Parts for Lebesgue Stieltjes Integrals  New with general Change of Variable and general integration by parts with correction terms for discontinuous functions for Lebesgue Stieltjes integrals     

Lebesgue Stieltjes signed measure generated by a function of bounded variation, its total variation measure and Lebesgue Stieltjes measure generated by the total variation of the function. Detail proofs of the de La Vallee Poussin's decomposition including the component given by the Jump function of the function of bounded variation. Decompositions in terms of the Lebesgue measure of the image of the total variation function, positive and negative variation functions of the function of bounded variation are deliberated and proved. Integration by parts and versions of change of variable for Lebesgue Stieltjes integrals.