# What is Apostol: Definition and 94 Discussions

Eugenia "Eggie" Apostol (born September 29, 1925) is a Filipino publisher who played pivotal roles in the peaceful overthrow of two Philippine presidents: Ferdinand Marcos in 1986 and Joseph Estrada in 2001. She was awarded the 2006 Ramon Magsaysay Award for Journalism, Literature & Creative Communication Arts.

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1. ### Calculus Can I skip the introduction in Apostol's calculus?

I'll be starting Apostol's calculus book in a little less than two weeks, as I'm finishing up an easier textbook. I looked ahead and I see that there is a four part introduction, and I was wondering if I could just skip that. I briefly skimmed it, and it just looks like a review of summation...
2. ### Calculus Does Apostol Calculus Volume 2 cover sufficient multivariate calculus?

Hello. I am currently doing a high school univariate calculus book, but I would like to go through Apostol's two volumes to get a strong foundation in calculus. His first volume seems great, and I've heard great things about his series, but I am not sure if his second volume contains sufficient...
3. ### Other Is it worth doing EVERY problem in a textbook?

I am a freshman Physics major currently working through Apostol's Calculus Volume 1 in my free time, somewhat to further develop my calculus knowledge, but mainly for fun. Apostol's text is proof-based, and as such has a number of problems that are just proofs. As a hopeful future Biophysicist...
4. ### Induction Proof, Apostol Calc Vol I, I.4.4.7

For calculating ##n_1## I had no problems as ##n_1 = 3##. PF: Show ##(1+x)^n > 1 + nx + nx^2 ## is true for all ##x \ge 3##. Let ##n = 3, (1+x)^3 > 3x^2 + 3x + 1 ## ##x^3 +3x^2 +3x + 3 > 3x^2 + 3x + 1 ## is true. Assume ## n = k ## such that ##(1+x)^k > 1+kx + kx^2, k \ge 3## is true. We...
5. ### Calculus Revisiting Single-Variable Calculus after Multi-

Hello all! I was thinking about strengthening my knowledge of Calculus after I finish the course I am taking in Multivariable Calculus. I am in a particularly unique situation, as I am only going to enter high school next year. I took the AP Calculus BC Exam last year and got a 5. The course I...
6. ### Calculus Calculus Textbook: Courant vs Apostol

Hi! I was looking for a Calculus textbook to buy and was debating between Courant and Apostol. I was planning on reading Spivak parallel to one of these. My question is does Apostol cover more material than Courant does or are they practically the same. Sorry if this has already been asked, I...
7. ### Analysis Real Analysis (Baby Rudin vs Apostol)

I am currently trying to self study Real analysis . I have completed Hubbard's Multivariable book & Strang's Linear algebra book. I have Apostol's Mathematical Analysis & Baby Rudin . I started with rudin yesterday and was making excellent headway until I encountered a theorem about 15 pages in...
8. ### MHB Complex Derivative .... Remark in Apostol, Section 16.1 .... ....

I am reading Tom M Apostol's book "Mathematical Analysis" (Second Edition) ... I am focused on Chapter 16: Cauchy's Theorem and the Residue Calculus ... I need help in order to fully understand a remark of Apostol in Section 16.1 ... The particular remark reads as follows: Could someone...
9. ### MHB Exploring Theorem 4.29: Compact Metric Spaces & Inverse Functions

I am reading Tom M Apostol's book "Mathematical Analysis" (Second Edition) ... I am focused on Chapter 4: Limits and Continuity ... ... I need help in order to fully understand the example given after Theorem 4.29 ... ... Theorem 4.29 (including its proof) and the following example read as...
10. ### MHB Functions Continuous on Comapct Sets .... Apostol, Theorem 4.25 ....

I am reading Tom M Apostol's book "Mathematical Analysis" (Second Edition) ... I am focused on Chapter 4: Limits and Continuity ... ... I need help in order to fully understand the proof of Theorem 4.25 ... ... Theorem 4.25 (including its proof) reads as follows: In the above proof by...
11. ### MHB Lindelof Covering Theorem .... Apostol, Theorem 3.28 ....

I am reading Tom M Apostol's book "Mathematical Analysis" (Second Edition) ... I am focused on Chapter 3: Elements of Point Set Topology ... ... I need help in order to fully understand Theorem 3.28 (Lindelof Covering Theorem ... ) .Theorem 3.28 (including its proof) reads as follows: In the...
12. ### I Lindelof Covering Theorem .... Apostol, Theorem 3.28 ....

I am reading Tom M Apostol's book "Mathematical Analysis" (Second Edition) ... I am focused on Chapter 3: Elements of Point Set Topology ... ... I need help in order to fully understand Theorem 3.28 (Lindelof Covering Theorem ... ) .Theorem 3.28 (including its proof) reads as follows: In...
13. ### MHB Directional Derivatives .... Apostol, Section 12.2, Example 4 ....

I am reading Tom M Apostol's book "Mathematical Analysis" (Second Edition) ...I am focused on Chapter 12: Multivariable Differential Calculus ... and in particular on Section 12.2: The Directional Derivative ... ...I need help with part of Example 4, Section 12.2 ...Section 12.2, including the...
14. ### MHB Proof of Apostol's Definition 3.2 and Theorem 3.3: Help Appreciated

I am reading Tom M Apostol's book "Mathematical Analysis" (Second Edition) ...I am focuses on Chapter 3: Elements of Point Set Topology ... I need help regarding a remark of Apostol's made after Definition 3.2 and Theorem 3.3 ...Definition 3.2 and Theorem 3.3 read as follows: In a note at the...
15. ### Line integral of vector field from Apostol calculus

Homework Statement Here are the three problems that i couldn't solve from the book Calculus volume 2 by apostol 10.9 Exercise 2. Find the amount of work done by the force f(x,y)=(x^2-y^2)i+2xyj in moving a particle (in a counter clockwise direction) once around the square bounded by the...
16. ### Line integral problems in Apostol calculus

Homework Statement A two dimensional force field f is give by the equation f(x,y)=cxyi+x^6 y^2j, where c is a positive constant. This force acts on a particle which must move from (0,0) to the line x=1 along a curve of the form y=ax^b where a>0 and b>0 Homework Equations Find a value of a(in...
17. ### I The two equivalent parallel velocity vectors

This is an exercise from the textbook Apostol Vol 1 (page 525, second edition), and I do not know how to prove it: Suppose a curve C is described by two equivalent functions X and Y, where Y(t) = X[u(t)]. Prove that at each point of C the velocity vectors associated with X and Y are parallel...
18. ### Apostol Analysis vs Shilov Analysis

I will be taking analysis in the spring next semester. I am taking intro to proof, java programming, differential equations (did not get transfer credit), and either a probability course or a second semester linear, this semester. I feel I have a good understanding of differential equations at...
19. ### Tom Apostol: Preeminent Educator Passes

Sad news. See here. A pre-eminent educator.
20. ### I Need help understanding text from 'Calculus' by Apostol

Sorry if I'm posting in the wrong place. Since the difficulty I have is with the text of the book, and not the exercises, I posted here. In proving theorem I.2, how is theorem I.1 used to assert that 'there is at most one such x'? The first image below gives the background text. The text I have...
21. ### I Cantor's intersection theorem (Apostol)

Hi, I am reading "mathematical analysis" by Apostol right now for a course in analysis. Since I am trying to understand the author's proof of the above theorem(3.25 in the book), but I have something that I can't understand. He assumes that each of the nested sets contains infinitely many...
22. ### Calculus Deciding Between Apostol & Courant Books - Input Appreciated

Im really having a hard time on deciding which of the two multi-volume books to get. I haven't read either of the two, but I do have Spivak's "Calculus", great book, and I've been told both authors are somewhat similar to him. Any input would be much appreciated. (And yes I have considered just...
23. ### Self Studying Proof Based Calculus

Hello everyone. 'm studying proof based calculus in my spare time. I have already taking Multivariable calculus, Diff. Equations and Linear Algebra. I'm using the textbook Apostol Calculus 1, as it was recommended by these forums, but I'm having great difficulty on understanding some of the...
24. ### Calculus Changing calculus textbook from Tom Apostol to James Stewart

Hi I read T.Apostol calculus and linear algebra in summer until page 230. Now in the first year of university , the math teacher told us to study James Stewart calculus (one problem is that she said we have to read the 2nd edition because that's the version she studied during her years) Now I'm...
25. ### Proving ∫xn⋅(ax+b)½ using Integration by Parts | Integral from Apostol

Homework Statement Hi guys,can anyone help me prove the following using integration by parts ∫xn⋅(ax+b)½=2/a(2n+3)(xn⋅(ax+b)3/2-nb∫xn-1⋅(ax+b)½)Homework EquationsThe Attempt at a Solution Setting u=xn,dv=(ax+b)1/2dx du=n⋅xn-1 and v = 2/3⋅(ax+b)3/2 i can never get b in the equation and the term...
26. ### Method of Exhuastion for the area of a parabolic segment

Hello everyone, I'm using the book Apostol- Calculus Vol. 1 for self-studying to get a better understanding of proof based Calculus. They said this book was good for self studying, but I am already stuck in the first chapter. I'm trying to understand how he got the identity: 12+22+...+n(2)=...
27. ### Calculus Using Apostol as my first calculus book?

Hello, I'm currently learning pre calculus from various sources. I'm preparing myself to hopefully be able to learn calculus which I feel is an essential tool for Physics. I've just purchased Apostol's Calculus Vol.1. Enough research for great introductory calculus textbooks brought me to...
28. ### Calculus Courant/John vs. Apostol for In-Depth Studying of Calculus?

Dear Physics Forum personnel, I would like to investigate the depth of single-variable calculus by reading either Courant/John's "Calculus and Analysis" or Apostol's "Calculus Vol.1". Of course, I know the best route of action is to go to mu university library, but I just found out that all...
29. ### Calculus Why "Transition Books (Apostol, Spivak)" are necessary?

Dear Physics Forum friends, Why so many people recommend Spivak, Apostol, and Courant calculus textbooks, especially as a preparation toward the advanced courses like analysis and abstract algebra? Are they really necessary? I have been studying Apostol's Mathematical Analysis, Rudin's PMA...
30. ### Calculus Is There an Official Errata for Apostol's Calculus Textbook?

I am studying calculus using apostol, I found that some answers at the back are wrong. I tried to search for an errata copy but can't find any. Does anyone know how to get one or has a list of errors?
31. ### Calculus by Apostol Exercise 2.8 number 30

Homework Statement Homework EquationsThe Attempt at a Solution I have no idea on how to start proving this, but I know the theorem is stating that the integral of a translated periodic function is the same with the integral of the periodic function without translation, is this concept...
32. ### Calculus by Apostol Exercise 1.7

Homework Statement A point (x,y) in the plane is called a lattice point if both coordinates x and y are integers. Let P be a polygon whose vertices are lattice points. The area of P is I + B/2 - 1, where I denotes the number of lattice points inside the polygon and B denotes the number on the...
33. ### MHB Apostol - Continuity & Differentiabilty

I need help with the proof of Apostol Theorem 5.2. Theorem 5.2 and its proof read as follows: https://www.physicsforums.com/attachments/3910 In the above proof, Apostol gives an expression or formula for f^* and then states the following: " ... ... Then f^* is continuous at c ... ... "I need...
34. ### MHB Metric Spaces - Fixed Point Theorem (Apostol, Theorem 4.48)

I need help with the proof of the Fixed Point Theorem for a metric space (S,d) (Apostol Theorem 4.48) The Fixed Point Theorem and its proof read as follows: In the above proof Apostol writes: " ... ... Using the triangle inequality we find for m \gt n, d(p_m, p_n) \le \sum_{k=n}^{m-1}...
35. ### MHB Metric Spaces & Compactness - Apostol Theorem 4.28

I need help with the proof of Theorem 4.28 in Tom Apostol's book: Mathematical Analysis (2nd Edition). Theorem 4.28 reads as follows:In the proof of the above theorem, Apostol writes: " ... ... Let m = \text{ inf } f(X). Then m is adherent to f(X) ... ... " Can someone please explain to me...
36. ### Finding Solutions to a Step Function Integral

Homework Statement This is from Apostol's Calculus Vol. 1. Exercise 1.15, problem 6.(c) Find all x>0 for which the integral of [t]2 dt from 0 to x = 2(x-1) Homework Equations [t] represents the greatest integer function of t. The Attempt at a Solution [/B] Integral of [t]2 dt from 0 to x...
37. ### Apostol Calulus Voume1 I 2.4 question 9

1. Compute the area of the region S between the graphs of f and g over the interal [a,b] specified in each case Make a sketch of the two graphs and indicate S by shading. Question 9 -F(x)=x^2 g(x)=x+1,a =-1, b = (1+√5)/2 Homework Equations The Attempt at a Solution I think...
38. ### Using Spivak and Apostol Calculus in parallel?

Hi all, So I got 2 months exactly before my exams and since I want to start from 0 again because I realized my base is not very well founded. So I'll take this weekend to go through my pre-calculus again and start my Spivak-Apostol schedule Monday. I need to polish my maths, start with the...
39. ### Apostol vs Spivak for physicists

Hello PF! I was wondering which would be more helpful for a physicist. Also a little bit of a tangent, if I learn calculus from one of the "popular" books, how would I go back to rigorous math? And where would it be useful for a physicist? Thanks :)
40. ### Help with integral from Apostol Calculus

Help with integral from Apostol "Calculus" Homework Statement I seem to be stuck trying to prove the following integral from Apostol "Calculus" Volume 1 Section 5.11, Question 33. \int\frac{\cos^mx}{\sin^nx}dx =...
41. ### Deriving the Integral Property from Theorems 1-45 and 1-46

Hello there. I recently installed the Physics Forums app and could not find the homework section in the categories so I beg your pardon from the following question. I am using Apostol's Calculus Volume 1 and have encountered a problem on page 72 number 21 (for those in possession of the text)...
42. ### Apostol calculus volume 1, Solutions to the exercises in the book.

Hi guys! I am studying Apostol calculus volume 1 on my own. I tried many times to find a solution manual to exercises in the book but I couldn't. I would appreciate your help in this matter. Any other advice will be much appreciated. Thanks, Abe
43. ### Geometry New Horizons in Geometry by Apostol, Mnatsakanian

Author: Tom Apostol, Mamikon Mnatsakanian Title: New Horizons in Geometry Amazon Link: https://www.amazon.com/dp/088385354X/?tag=pfamazon01-20
44. ### Understanding Limits in Calculus to Apostol Book - Page 129

Hello guys, I am stuck in page 129 on Calculus Vol I - Apostol book. I would like to know if there is anybody here who can help me. I am not a mathematician, so It might be a simple transformation but I am not going through it. He states that: lim(x->p) f(x) = A is equivalent to say that...
45. ### Calculus Apostol Calculus: Historical Intro, Concepts & Applications

Author: Apostol Title: Calculus Amazon Link: https://www.amazon.com/dp/0471000051/?tag=pfamazon01-20 https://www.amazon.com/dp/0471000078/?tag=pfamazon01-20 Prerequisities: PreCalculus/Algebra & Trig ; Ideal with a basic knowledge of calculus however, it's fine without it. Table of Contents...
46. ### Apostol Calculus Vol1 10.4.29 Sequences

Homework Statement Assume that \{ a_n\}\rightarrow 0 . Use the definition of limit to prove that \{ a_n^2\} \rightarrow 0. Homework Equations Definition of limit. For all ε>0 there exists N s.t. n>N implies |a_n - L|<ε. The Attempt at a Solution I know why this is true... if the sequence...
47. ### Apostol 1 vs Ross's Elementary Analysis

Hi everyone, I left college many years ago and now I've forgotten a lot about multivariate calculus. I've also never taken a real analysis course of any kind before. Now I'm planning to go back to school for a graduate degree in econ. The school says that I must have good skills in real...
48. ### Need help in Apostol Calculus proof

let b denote a fixed positive integer. Prove the following statement by induction: for every integer n≥0, there exist nonnegative integers q and r such that n= qb+r, 0≤r<b. Can someone help me on how to solve this question? and how does induction works here? thank you
49. ### Currently working through Apostol need a check on my logic

Homework Statement Prove the following theorem. If a≠0, then (a-1)-1=a Theorems proven before/axioms I am allowed to use: Existence of reciprocals axiom: there exist real numbers x and y where x≠0 such that xy=1 Possibility of division Thm: basically a-1=1/a Homework Equations I just need a...
50. ### Congruence as explained by Apostol in Calculus Volume 1.

1) I am having trouble understanding some parts of congruence as is explained by T Apostol in the book Calculus volume 1. In the chapter on integration , while discussing area functions he explains congruence as follows :- Now I can appreciate that concepts such as 'area' , 'congruence' are...