My Calculus Web |
|
|
|
A Place to Learn and to Explore |
Most notes and articles are in Adobe Acrobat (PDF) format. Download the free Adobe - Adobe Acrobat Reader for viewing and printing these files.
This web site is dedicated to mathematics enthusiasts, beginners, those who love to learn, to seek the truth in beauty or in form or just simply thirsting for answers and/or knowledge.
An Introductory Calculus Course |
Articles |
||
General Information
Introductory Calculus course. Read tutorial guide and the chapters in text book and then do the tutorial. |
Substitution in Integral and Differentials | Integral of x^(x^n) and x^(-x^n) | |
Objectives | Do we need Mean Value Theorem? | Integration using Differentiation under integral sign Examples of how to check the condition for using the Theorem on differentiaion under the integral. | |
Syllabus | Extreme Value Theorem | Integration Using Power Series Examples of evaluation of (ln(sin(x))^2, (ln(tan(x))^2, etc. | |
References | Intermediate Value Theorem | Evaluation of ∫ ln(1+x^4)/(1+x^2) Four methods, with one using Fubini's Theorem. | |
Notes on Derive | Boundedness Theorem |
∫ 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. |
|
General Advice and Learning Guide | Monotone Function and Continuity | On the integrals, ∫ x^s/(1+e^x)^n and ∫ x^s ln(x)/(1+e^x)^n A recursive relation ie presented , using the relation, you can compute the integral as a finite sum involving the Gamma and eta functions. | |
Problem-Solving Process | Injective Function and Monotone Function | Integrating x/(1-e^x) from 0 to infinity without using power series | |
Tutorials | |||
Guide and Comment for Tutorial Assignment | Riemann Integral and Infinite Series | ||
Example Sessions | Derived functions and Derivative | ||
Online Quizzes | Continuity, Differentiability, Weierstrass' Function | ||
Precalculus Online Quiz | Intermediate Value Theorem for Derived Function | ||
A Formula of Euler and Appreciating Calculus | Monotone Function, Bounded Variation, Fundamental Theorem of Calculus | ||
Tests and Past Exam Papers | Heine-Borel, Bolzano Weierstrass Theorems, Uniform Continuity and Riemann Integrability | ||
Letter to Students | Composition and Riemann Integrability | ||
Letter to and from a fellow teacher | Composition and Lebesgue Integrability | ||
Link to other Calculus Web sites | Change of Variable in Riemann and Lebesgue Integration | ||
Comment and Errata to
Calculus, an introduction Calculus, an introduction available from NUS Coop |
The Cantor Set | ||
Darboux's Fundamental Theorem of Calculus | |||
Review of 1999-2000 1st Semester Exam
|
L Hopital's Rule - And a Generalized Version | ||
Books on web: | Integration By Parts | ||
Real Numbers?
|
Application of integration - arc length, volume of solid of revolution, area of surface of revolution | ||
Mathematical Analysis, An Introduction
With some tutorials for self study. 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:
The real numbers,
Sequences,
Continuous functions,
Differenmtiable functions,
Integration, Series, Series of functions and Power Series, Uniform Convergence and differentiation, Uniform Convergence, Integration and Power Series. Weierstrass Approximation Theorem Special Test for Convergence - Kummer, Raabe, Gauss and Bertrand's Tests |
Arc Length, Function of Bounded Variation and Total Variation
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 |
||
On the primitive of product of two functions | |||
A gentle course introducing mathematical analysis Including a week by week study plan and guide. | Convergence of ∑ sin(√(n)x)/n and other problems | ||
Advanced Calculus | Fourier Cosine and Sine Series and Their Convergence | ||
Ideas of Lebesgue and Perron integration in Uniqueness of Fourier and Trigonometric series | |||
Mathematics Diagnostic Testing | Convergence and summability of Fourier Series | ||
Basic Skills help: Algebra refresher Inequalities |
Second Mean Value Theorem for Integrals and Bonnet Mean Value Theorem | ||
Abel-summability of Fourier Series and its Derived Series | |||
Mathematics Assessments for Revision (Algebra and Calculus AO-A level) |
Fourier Series for Even and Odd Functions | ||
Assessment Gallery | Riemann Summable everywhere Series, Two Special Cosine series | ||
Comment on A 2019 PSLE Math Question | An improperly Riemann integrable function that does not give the conclusion of the Riemann Lebesgue Lemma | ||
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. |
All About Lim Sup and Lim Inf | ||
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 Rk via Riemann integration and Lebesgue integral. |
Convex Function, Lp spaces,
Space of continuous function, Lusin's Theorem A detail introductory exposition of Lp 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 Lp , Radon-Nikodym Theorem and Riesz
RepresentationTheorems- Complex and real versions
Identification of the dual of Lp spaces and the dual of Cc(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 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 |
||
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 |
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. |
COMMENT OR SUGGESTION??
If you have any comment or suggestion regarding this website (for example, you feel that you like something (other than lecture notes!!) to be put on the web. Or if you have found some interesting websites. Or, you particularly like something we have put here.) I shall be glad to hear from you. You may e-mail Ng Tze Beng at tbengng@gmail.com
This page was last updated on 12/02/2024
By Ng Tze Beng