Need recommendation on Advanced Calculus books

const451

I am looking for the supplementary textbook(s) for the Advanced Calculus course. The required is Advanced Calculus by Patrick Fitzpatrick.

Could someone recommend something else.

What about Wilfred Kaplan, is it any good?

THX!

Nope, Kaplan is no good (I just donated my copy to a charity bookshop yesterday: don't know why I bothered hanging onto it). At best, he covers some of the usual topics in AC in pedestrian style: partial derivatves, multiple integrals, vector calc, infinite series, Fourier series, ODEs, complex variables. The style is old, the presentation cramped and opaque.

Problem with AC is it's so ill-defined: it can run the gamut from multivariable calc to a course in real analysis, depending on the university. I don't know what Fitzpatrick covers (can't see the contents at Amazon). What's the syllabus?

const451

Right, other U. call the course Elementary Analysis and the content might be different.

Here is the Fitzpatrick's contents. For the midterm we are going to cover 1,2,3,4, and 6.

1 Tools for analysis
1.1 The completeness Axion
1.2 The distribution of the Integers and Rational
1.3 Inequalities and Identities.

2 Convergent Sequences
2.1 The convergence of Sequence
2.2 Sequences and sets
2.3 The monotone convergence theorem
2.4 the sequential compactness theorem
2.5 covering properties of sets

3 Continuous Functions
3.1 Continuity
3.2 The extreme value theorem
3.3. the intermediate value theorem
3.4 uniform continuith
3.5 the e-s criterion of continuity
3.6 images and inverses; monotone functions
3.7 Limits

4. Differentiation

5. Elementary Functions as solutions to diff. equations

6. Integration: two fundamental theorems

7 Integration further topics
7.1 Solutions of diff. eq.
7.2 Integration by parts and by Substitution
7.3 The convergence of Darboux and Riemann Sum
7.4 the approximation of integrals

8 Approximation by Taylor polynomials

9 Sequences and series of functions

10 The Euclidean Space Rn

11 Continuity, compactness, and connectdness

12 Metric spaces

13 Differentiating functions of several variables

14 Local approximation of real-valued functions

15 Approximating nonlinear mappings by linear mappings

16 Images and inverses: the inverse function theorem

17 the implicit function theorem

18 integrating functions of several variables

19 iterated integration and changes of variables

20 line and surface integrals

THX

Right, other U. call the course Elementary Analysis and the content might be different.

A couple of books for the entirely uninitiated would be:

1) A First Course in Mathematical Analysis, by Burkill, pub. Cambridge, and
2) Yet Another Introduction to Analysis, by Bryant, pub. Cambridge

I think Bryant is used at Baruch. Burkill used to be recommended reading for incoming math students at Cambridge (i.e., before they started). Neither even defines Cauchy sequences. 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).

If you want real books on analysis, you're spoilt for choice. Rudin's dated Principles of Mathematical Analysis is still used at American universities (my boy used it a couple of years back). I'm reluctant to recommend it to quant students. One I would recommend is Bartle and Sherbert's Introduction to Real Analysis, which is where I learnt my analysis from. 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. 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. 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. Our intuition leads us astray and hence the need for a sound foundation. Bressoud is ideal for explaining how and why the sound foundation came into being historically. So Bartle and Bressoud are a couple of recommendations. I may come up with one or two more.

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). Then from completeness of the reals to various properties of sequences. Then the epsilon-delta definition of continuity.

const451

Here is the full contents of Advanced Calculus by Fitzpatrick. It talks about Gauchy Mean Value.

1. TOOLS FOR ANALYSIS. The Completeness Axiom and Some of Its Consequences. The Distribution of the Integers and the Rational Numbers. Inequalities and Identities.

2. CONVERGENT SEQUENCES. The Convergence of Sequences. Sequences and Sets. The Monotone Convergence Theorem. The Sequential Compactness Theorem. Covering Properties of Sets.

3. CONTINUOUS FUNCTIONS. Continuity. The Extreme Value Theorem. The Intermediate Value Theorem. Uniform Continuity. The Epsilon-Delta Criterion for Continuity. Images and Inverses; Monotone Functions. Limits.

4. DIFFERENTIATION. The Algebra of Derivatives. Differentiating Inverses and Compositions. The Mean Value Theorem and Its Geometric Consequences. The Cauchy Mean Value Theorem and Its Analytic Consequences. The Notation of Leibnitz.

5. ELEMENTARY FUNCTIONS AS SOLUTIONS OF DIFFERENTIAL EQUATIONS. Solutions of Differential Equations. The Natural Logarithm and the Exponential Functions. The Trigonometric Functions. The Inverse Trigonometric Functions.

6. INTEGRATION: TWO FUNDAMENTAL THEOREMS. Darboux Sums; Upper and Lower Integrals. The Archimedes-Riemann Theorem. Additivity, Monotonicity and Linearity. Continuity and Integrability. The First Fundamental Theorem: Integrating Derivatives. The Second Fundamental Theorem: Differentiating Integrals.

7. INTEGRATION: FURTHER TOPICS. Solutions of Differential Equations. Integration by Parts and by Substitution. The Convergence of Darboux and Riemann Sums. The Approximation of Integrals.

8. APPROXIMATION BY TAYLOR POLYNOMIALS. Taylor Polynomials. The Lagrange Remainder Theorem. The Convergence of Taylor Polynomials. A Power Series for the Logarithm. The Cauchy Integral Remainder Theorem. ANon-Analytic, Infinitely Differentiable Function. The Weierstrass Approximation Theorem.

9. SEQUENCES AND SERIES OF FUNCTIONS. Sequences and Series of Numbers. Pointwise Convergence of Sequences of Functions. Uniform Convergence of Sequences of Functions. The Uniform Limit of Functions. Power Series. A Continuous, Nowhere Differentiable Function.

10. THE EUCLIDEAN SPACE Rn. The Linear Structure of Rn and the Scalar Product. Convergence of Sequences in Rn. Open Sets and Closed Sets in Rn.

11. CONTINUITY, COMPACTNESS, AND CONNECTEDNESS. Continuous Functions and Mappings. Sequential Compactness, Extreme Values and Uniform Continuity. Pathwise Connectedness and the Intermediate Value Theorem. Connectedness and the Intermediate Value Property.

12. METRIC SPACES. Open Sets, Closed Sets, and Sequential Convergence. Completeness and the Contraction Mapping Principle. The Existence Theorem for Nonlinear Differential Equations. Continuous Mappings Between Metric Spaces. Sequentially Compactness and Connectedness.

13. DIFFERENTIATING FUNCTIONS OF SEVERAL VARIABLES. Limits. Partial Derivatives. The Mean Value Theorem and Directional Derivatives.

14. LOCAL APPROXIMATION OF REAL-VALUED FUNCTIONS. First-Order Approximation, Tangent Planes, and Affine Functions. Quadratic Functions, Hessian Matrices, and Second Derivatives. Second-Order Approximations and the Second-Derivative Test.

15. APPROXIMATING NONLINEAR MAPPINGS BY LINEAR MAPPINGS. Linear Mappings and Matrices. The Derivative Matrix and the Differential. The Chain Rule.

16. IMAGES AND INVERSES: THE INVERSE FUNCTION THEOREM. Functions of a Single Variable and Maps in the Plane. Stability of Nonlinear Mappings. A Minimization Principle and the General Inverse Function Theorem.

17. THE IMPLICIT FUNCTION THEOREM AND ITS APPLICATIONS. A Scalar Equation in Two Unknowns: Dini's Theorem. The General Implicit Function Theorem. Equations of Surfaces and Curves in R³. Constrained Extrema Problems and Lagrange Multipliers.

18. INTEGRATING FUNCTIONS OF SEVERAL VARIABLES. Integration of Functions on Generalized Rectangles. Continuity and Integrability. Integration over Jordan Domains.

19. ITERATED INTEGRATION AND CHANGES OF VARIABLES. Fubini's Theorem. The Change of Variables Theorem: Statements and Examples. Proof of the Change of Variables Theorem.

20. LINE AND SURFACE INTEGRALS. Arclength and Line Integrals. Surface Area and Surface Integrals. The Integral Formulas of Green and Stokes.

Appendix A: Consequences of the Field and Positivity Axioms. The Field Axioms and Their Consequences. The Positivity Axioms and Their Consequences. Appendix B: Linear Algebra.

const451

Fortunately we have these books at the library, I'll check them out once semester starts. That should be plenty.

BBW - thank you very much for your help with the books!!

Advanced Calculus, 3rd Edition, by R.C. Buck

A First Course in Mathematical Analysis, by Burkill, 1970

Yet Another Introduction to Analysis, by Bryant

Introduction to Real Analysis by Bartle and Sherbert

A Radical Approach to Real Analysis by Bressoud

Principles of Mathematical Analysis by Walter Rudin

Fortunately we have these books at the library, I'll check them out once semester starts. That should be plenty.

A First Course in Mathematical Analysis, by Burkill, 1970

Then start with Burkill immediately and try to finish it before class even starts. You will hit the ground running, so to speak, when term begins. Even now, you should know things like the irrationality of square root of 2 (why?), and that the real continuum is of a higher order of cardinality than the rationals (proof?), and that every monotone bounded sequence doesn't converge in the rationals (example?).

I've heard of it, but never seen it, much less used it. Seems to be another "me too" text, and falls between the two stools of being theoretical on the one hand and application-oriented on the other. Can't say more till I see it. Not really an analysis text.

That should be plenty.

One more -- a personal favorite. Basic Real Analysis, by Sohrab, pub. Birkhauser. Don't buy it until you've inspected it (it may be too difficult for you). This is rigorous but written in a limpid style, and its 560 pages cover not only the real line but metric spaces and Banach and Hilbert spaces as well, and not only the Riemann integral but also the Lebesgue integral. Plus, if you think you're a big boy now, there's a compact and fast-paced 17-page presentation of measure-theoretic probability. A book to cherish. If only all mathematicians could write like this ....

const451

I just received Kreider! It is in pretty good condition, especially considering its age. I wish they never invented highlighters, students use them rather enthusiastically.

I liked the first few pages of Burkill I read from Amazon, he does seem to explain well and thorough.

const451

Unfortunately, we do not have Basic Real Analysis by Sohrab in our lib. From the contents it does look like a bit of overkill for me. Like Rudin, he talks about Topology and Spaces. You said as a quant wannabe I do not need Spaces - too theoretical. Measure-theoretic probability would be useful - I am planning to study it next semester.

const451

My required Advanced Calculus by Fitzpatrick does not have answers to exercises - neither even nor odd - none. How am I supposed to practice - I guess prof. has the plan for us.

You said as a quant wannabe I do not need Spaces - too theoretical.

You need to know what a vector space is. Finite-dimensional vector spaces arise in differential equations right from the start (e.g. the space of solutions to y + y" = 0 is two dimensional, with basis vectors cos x and sin x). Infinite-dimensional vector spaces arise equally naturally in PDEs (e.g., a Fourier series expansion). At some stage you will have to grapple with them (but not when you are just beginning to learn PDEs). Then it's handy to know that some of the concepts you learnt from intro real analysis can be generalised to an infinite-dimensional setting. But you are right: Sohrab is too difficult for a newbie.

One final book and I'm through: Introduction to Real Analysis, by Schramm, pub. Prentice-Hall. What I like about this treatment is the careful way the author explains different and equivalent aspects of completeness: Cauchy sequences, least upper bound, connectedness of the reals, nested intervals property, Bolzano-Weierstrass theorem, and the Heine-Borel theorem. I also like the way he emphasises the theoretical importance of the Extreme Value and Intermediate Value Theorems to the development of calculus, and his stress on the applications of the Mean Value theorem (which follows, incidentally, from the Extreme Value theorem) to results in calculus. It's pedagogically well-motivated. There, I'm through with recommending analysis books.

const451

Since my original post I've completed the first semester in Advanced Calculus. I've looked through at least a dozen of books, and the winner is...

Introduction to Real Analysis by Bartle and Sherbert. It skips some minor explanations but it is the most comprehensive textbook on the subject. I wish it had more hints or answers to problems to make it even more suitable for self-study. The notation seems outdated. It is not cheap even used but it is nice to own it.

The other two books I liked and used:
Analysis: With an Introduction to Proof by Steven Lay. This one has less material but more explanations.

A First Course in Mathematical Analysis, by Burkill. It is very basic but the explanations are excellent.

IMHO, the combination of these three books is the best "real line" treatment on analysis.

The textbook required for my class:
Advanced Calculus by Patrick M. Fitzpatrick. It is terrible. You cannot possibly learn from it, expecially without help.

Thanks everyone, esp. bbw for help!!

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myawr

I recommend "Fundamentals of real analysis" - Sterling K. Berberian - Springer
Most of Springer's books are simply great.

const451

I recommend "Fundamentals of real analysis" - Sterling K. Berberian - Springer
Most of Springer's books are simply great.

This book is over my head. I need advanced calculus - real line introductory approach to analysis, not spaces. thank you though.

This book is over my head. I need advanced calculus - real line introductory approach to analysis, not spaces. thank you though.

You are right. This book is designed for an intro grad course in analysis for mathematicians. Berberian is a good writer, though, and I have his book "Lectures in Functional Analysis and Operator Theory."

Postscript: I have a sneaking suspicion the poster mean Berberian's "A First Course in Real Analysis," which seems to be pitched at the level of Bartle and Sherbert (I've not seen the book and so I glanced at the table of contents at Amazon).

const451

Berberian's "A First Course in Real Analysis" does look very interesting from the table of contests. Unfortunately, my second semester will be Advanced Calculus of Several variables (the first semester was on Single Variable) but this book does not have Several variables. It's only \$12 used but I do not think I need it.

CGiuliano

For anyone taking a first course in advanced calculus I would strongly recommend an Introduction to Analysis -Arthur Mattuck. An absolutely beautiful textbook that can even be used for self-study.

This may draw criticism from grad students, but for novices: exceptional. If I had my hands on this book at the beginning of the summer, it would be finished.

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