Apostol Definition and 94 Threads

Hello, I am new to doing mathematics of the type in Apostol. I wish to study analysis. Hence I have started with Apostol Calculus volume 1 so as to build my fundamentals. I need a tip in solving the exercise problems which one encounters in first few chapters. I am presently on the...

Homework Statement Apostol, Vol 1: Section 4.12 Problem 26 Water flows into a hemispherical tank of radius 10 feet (flat side up). At any instant, let h denote the depth of the water, measured from the bottom, r the radius of the surface of the water, and V the volume of the water in the...

Apostol in his "Mathematica Analysis" defines something called a "component interval". However, I cannot find it anywhere on google or in other books I have on analysis, and I really would like to see a picture of what he means.. Apostol's definition is that the component interval of an open...

S is a set such that S\subseteqRn point a is in S: a\inS The point "a" is an interior point of S if there is an open n-ball with center "a", all of whose points belong to S. ie., every interior point of S can be surrounded by an n-ball such that B(a)\subseteqS, where B(a) is the set of...

I found a torrent online of Apostol's "Mathematical Analysis" 1st edition and I think I found a typo, or whoever scanned the book cut off the edge a bit... Apostol writes that the extended real number system R* is denoted by [-∞, +∞] while the regular real number system R is denoted by (-∞...

Okay, so I'm struggling with understanding where I went wrong. The instructor feels like I don't understand the material and when she presented my explanation to a colleague, he too agreed with her. I would really appreciate if someone could tell me the first part of where I went wrong in my...

Homework Statement \int\frac{\sqrt{2-x-x^2}}{x^2}dx Hint: multiply the numerator and denominator by \sqrt{2-x-x^2} Homework Equations This is in the Integration using Partial Fractions section, but the last few have not been using Partial Fractions. The Attempt at a Solution...

Homework Statement \int\frac{4x^5-1}{(x^5+x+1)^2}dx Homework Equations This is in the section on Partial Fractions. The main idea in this section was that you get the integral down to a sum integrals of the following forms: \int\frac{dx}{(x+a)^n} , \int \frac{x dx}{(x^2 + bx + c)^m} ...

Does Apostol ever introduce "Trig Substitutions"? I took Calculus before, but I am going over Apostol's Calculus Vol. 1 book. In section 5.7 he introduces Integration by Substitution, but never really discusses what was commonly referred to as "Trig Substitutions" in my Calc classes. For...

A solid has a circular base of radius 2. Each cross section cut by a plane perpendicular to a fixed diameter is an equilateral triangle. Compute the volume of the solid. First, we find a way to define a the distance of a chord of the circle perpendicular to the fixed diameter. The equation...

Here is the question verbatim: The radius of a right-circular cylinder increases at a constant rate. Its altitude is a linear function of the radius and increases three times as fast as the radius. When the radius is 1 foot, the altitude is 6 feet. When the radius is 6 feet, the volume is...

Hi, so basically i have just finished reading spivak's calculus and quite a lot people in this forum suggested that apostol's vol.2 is a good 'next book' after spivak calculus. I was just wondering whether apostol's volume 2 presupposes knowledge of linear algebra, because i am not too...

Hello, I am currently using Apostol for self-study. It seems to only have answers for computational problems. Some of the other problems are hard! And I see no student solution manual to guide me through these types of problems. I have no idea where to begin. Homework Statement a) (Found in...

Homework Statement Continuing with my Apostol efforts. From Section I 2.5: These exercises go over some of the absolute basics of sets. In 3. I'm given A = {1}, B = {1,2} and asked to decide whether some statements are true or false, proving the ones that are true. Seeing which ones are...

Homework Statement Hi all, I'm starting from page one of Apostol's Calculus Vol. 1 (ed. 2) to try and refresh and advance my maths while I'm on exchange and can't take maths courses. I'm an economics undergrad but I'm planning on switching to Maths/Computer Science if I can show myself I've...

Hi, I am doing self study with some special attention to number theory with application to computer science. I have Knuth vol1 and Apostol but I am finding Knuth exercises too tough and explanations too short and very less basic material to base proofs on. I find that simply reading...

Hi, I'm curious in learning calculus again but through a more rigorous fashion than plug and chug that I did last year. I'm trying to go slowly through Apostol's calculus but the way he's deriving the integral for a parabolic function x^2 is kinda troubling me. I'm fine until he reaches...

Heard many people say that there are three good cal textbooks: the ones by Apostol, Spivak, and Courant. I own Apostol's and Spivak's. The major difference between the two is the degree of rigor and logical order, in which Apostol's apparently beats Spivak's, although Spivak's is far better than...

Homework Statement The notation [x] denotes the greatest integer <= x "integral sign" with a lower limit of -1 and a upper limit of 3 [x] dx Homework Equations I would like to know how to enter symbols for certain problems. I am lost when Apostol says "the notation [x] denotes the...

Has anyone used Apostol's 2nd edition textbooks for Calculus? They contain lots of information about linear algebra and differential equations. But do they contain enough to be equivalent to a first and/or second course in them? If they do not, what supplementary texts do you recommend to...

Hey guys. Seeing that the Calculus textbooks of Apostol and Spivak have been mentioned and recommended numerous times in this forum, I was wondering which one is better? Currently, I am senior taking calculus. I feel that my calculus class is too easy and will not help me in college where I...

Let me further specify: Which of the following texts would be the best for understanding REAL WORLD applications of calculus, and approaching it in a practical manner? Okay, so maybe Spivak and Apostol can be removed from the list as they are a rigorous theoretical approach. How about the rest...

Last year I took ap calculus ab, and I am taking ap calculus bc next year. So i decided to refresh my memory this summer and went through The Calculus Lifesaver by Adrian Banner, and thoroughly enjoyed it since i didn't get the depth that it went into in class. Now i know there are more "in...

Is there a big difference between the 1st and the 2nd edition? I'm considering buying the book, and the 1st edition is much cheaper, but I suspect it may not be that good.

Homework Statement If x and y are arbitrary real numbers with x<y, prove that there is at least one real z satisfying x<z<y Homework Equations The Attempt at a Solution The problem arises from my inexperience in rigorously proving anything. If possible a general explanation of...

I am looking for a calculus book that gives the reader deep understanding of how calculus works, not just rote memorization. I have heard that the books by Micahel Spivak and Tom Apostol are good. Which of the books provides the best understanding of calculus: Spivak's or Apostol's? My parents...

Some time back I made a thread saying I would be buying a calculus book. I ended up buying Apostol as the Indian edition is only 6$. I have made a thread about this because I can see that there are going to be lots of problems I cannot do and I haven't a solution manual. 1.\int (a2 - x2)ndx...

So, for instance, maybe Baby Rudin is the paradigm of "how to do it right" which includes proving stuff like Baire's Theorem. By "rigorous", I don't necessarily mean "Baby Rudin". On the other hand, Thomas and Finney does, in fact, have the formal definition of a limit, for instance, on p70 of...

im just starting to work through vol 1 of apostol and these questions are kind of dumbfounding? #3 on page 15 Let A={1}, B={1,2} Discuss the validity of the following statements (prove the ones that are true) (a) A\subset B (b) A\subseteq B (c) A \in B (d) 1 \in A (e) 1 \subseteq B (f) 1...

Hello all, I've been reading Apostol's Calculus online (with Amazon's "search inside this book" feature - it's cumbersome but it works.) However, I recently decided to purchase the physical book. I got a very good deal from a used bookseller advertising the 2nd edition of the book (which I...

Hello all, In exercise 31, page 106 of Calculus by Tom Apostol (volume 1), the reader is asked to prove that the definite integral from zero to 2*pi of sin(nx) * cos(mx) = 0 (where n and m are arbitrary integers). The book hints that the reader should use the fact that the definite integral...

I'm looking into buying a cheap & introductory/mid-level textbooks on Differential Equations and Linear Algebra. My question is, as looking through Apostol's contents for his Multivariable Calculus textbook. I notice sections on DEs as well, would his his text sufficient for those two subjects...

Hello, I am working my way through Cal vo1 by Apostol and working the problems and was wondering if someone could assist. The question is on pg 19 to prove theorems i.5-i.15 using axioms and theorems i.1-i.4. Theorem i.5 states a(b-c) = ab - bc. I started the proof but am stuck at at...

Homework Statement (10.4 in Mathematical Analysis by Apostol) This exercise gives an example of an upper function f on the interval I = [0,1] such that -f (not a member of) U(I). Let {r1, r2, ...} denote the set of rational numbers in [0,1] and let I_n = [r_n - 4^-n, r_n + 4^-n] (intersect)...

Quality of International Books? i found Apostol's textbook for very cheap on Apebooks.com. I also found a numerous others but they are int'l editions. Does anyone know by chance if the quality of the page is good, or the overall quality of the books bc I will definitely be using it a lot and...

I've started with Apostol's first volume of calculus and I'm already having a problem with the first exercise in the book. Weak. It asks to use Archimedes' method to find the area under a parabola from 0<x<b for a parabola where y = a*b^2 + c. Using the summation of upper/lower rectangles of...

If you were a maniac of calculus what would it be? Tom M. Apostol’s or Richard Courant’s Calculus?

Hi, I'm working through some questions in Apostol's Calculus text (vol. 1). The problem is with ex. 7 on p.64. Here, he states that "by changing the index of summation, note that [tex]\sum_{n=a}^{b-1} [\frac{na}{b}] = \sum_{n=a}^{b-1} [\frac{a(b-n)}{b}][\tex] " I'm comfortable with...

I own both volumes of Apostol, but I must confess that when I bought them I had never heard of Spivak's book. I think it's pretty much a toss up for me now. What about you guys?

I've been going over the book and everything was going well until I got to this question: \frac{1} {2 + \sqrt{4 - x^2}} = \frac{2 - \sqrt{4 - x^2}} {x^2} Tried with partial fractions but that does not work. I also tried to split the fraction but I don't know how to get the x^2 at the...

I'm looking into buying both volumes of Apostol's Calculus, to "read around my subject" as Ian Stewart says, while taking Calculus courses that use James Stewart's text. I've found the hardcover versions used for around $80-90, but I've also found a paperback edition where both volumes come...

Prove that every triangular region is measurable and its area is one half the product of its base and altitude. Let Q be a set that can be enclosed between two step regions S and T , so that S \subseteq Q \subseteq T If there is one number c such that a(S) \leq c \leq a(T) , then Q...

Prove that each of the following sets is measurable, and has zero area: (a) a set consisting of a single point (b) a set consisting of a finite number of points in a plane (c) the union of a finite collection of line segments in a plane (a) To prove that a set is measurable you have to say: Let...

Hello! Anyone read Apostol's Calculus vol. 1. On p. 28 the exercises feels very hard. Can somebody help me with nr. 2?