Spivak Definition and 161 Threads

Hey, I'm just starting out with calculus and am giving the Spivak book a try per threads on here. In the introduction to functions chapter spivak defines a function f(x) =x^2. for all x such that -17 ≤ x ≤ π/3 Then says you should be able to check the following assertion about the function...

Pag 46 says: He begins to do a rethink about what is a function. Function is a rule? If it is a rule then: f(x) = x² and f(x) = x² +3x + 3 -3(x+1) are different. But that was not what we wanted. But he rewrites, this way: f(x) = x² and f(x) = x² +3x +3 -c(x+1) Apparently this was a small...

Pag 42 says: Seems inconsistent. in (9): p = x +x²+x.sin²x q = x.sin x + x.sin²x But by definition p and q are not polynomial functions. It is a mistake in the book?

So recently, I started reading Michael Spivak's "Calculus" to shore up my understanding of the subject, after being told that this text is well-known for its rigor. In chapter one, he lists 12 properties (or axioms) of numbers (distributive law, trichotomy law, and closure under addition, to...

I have one question about Spivak's Calculus on Manifolds book. I have not learned directional derivatives and understand that these are left as exercises in his book, which would make one think these are not that important whereas he focuses on total derivatives or what you may name them...

I am currently having some issue understanding, what you may find trivial, epsilon-delta proofs. I have worked through Apostol Vol.1 and ran through Spivak and I found Apostol just uses neighborhoods in proofs instead of the epsilon-delta approach, while nesting neighborhoods is 'acceptable' I...

https://www.amazon.com/dp/1425589138/?tag=pfamazon01-20 Calculus and Linear Algebra V.1 (Spo Scholarly Monograph) https://www.amazon.com/dp/1425589340/?tag=pfamazon01-20 Calculus and Linear Algebra Vol. 2: Vector Spaces, Many-Variable Calculus, and Differential Equations Has anyone had any...

Homework Statement Prove the following: ##a / b = ac / bc##, if ##b, c \neq 0##. Homework Equations P1-12 The Attempt at a Solution ##a/b = a*b^{-1}## ##1 = c*c^{-1}## ##a/b*1 = (a*b^{-1})(c*c^{-1})## ##a/b = (a*c)(b^{-1}*c^{-1})## Now, if ##b^{-1}c^{-1} = (bc)^{-1}##, then the problem is...

Hi, Suppose you want to prove |x - a||x + a| < \epsilon You know |x - a| < (2|a| + 1) You need to prove |x + a| < \frac{\epsilon}{2|a| + 1} So that |x - a||x + a| < \epsilon Why does Michael Spivak do this: He says you have to prove --> |x + a| < min(1, \frac{\epsilon}{2|a| + 1}) in...

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...

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 :)

Hi everybody! I have started working through Spivak's "Calculus". I was slightly upset with the first chapter. The book made a few assumptions. (The concept of a number and that if a=b then a+c=b+c, stuff like that) I am fine with it for now because I know where the gap in knowledge is but...

1. Prove the principle of mathematical induction from the well ordering principle. I didn't get very far, but here's my attempt at it... Let A be a set that contains 1 and contains n whenever it contains n+1. Now, let there be a non empty set B that contains all natural numbers not in A...

I'm having a hard time seeing how spivak derived this formula... "6. The formula for 12+...+n2 may be derived as follows... (k+1)3 - k3 = 3k2+3k+1 where k=1,...,n Therefore, (n+1)3 - n3 = 3n2+3n+1 But this is where I am confused...He then presents this, (n+1)3 - 1 = 3[12...

Hello. I wanted to complete my preparation in algebra 2 and precalculus so that I can tackle Spivak's "Calculus." I have a few options in front of me. I can either go: 1. through the Art of Problem Solving series rather quickly to be prepared. The issue with that is that some members here...

I'm looking to begin an independent study of Calculus, having read Lang's Basic Mathematics. I would like to know if I should move straight on to Spivak's Calculus, or start with Lang's A First Course in Calculus? If it affects the answer in any way, I have had some rudimentary experience with...

Hello, I am a high school student in the UK who is looking to start studying Calculus from Spivak. My question is as follows: What topics might I need to cover in greater detail before doing so? In terms of work done thus far, I have completed my Mathematics IGCSE (1 year early) and am now...

The title basically says it all but basically the 3rd edition of the book is missing a lot of solutions i would like to know to check my results and do you know where i could find them. I tried the solutions manual but that misses a lot as well.

Hello, I have been backtracking recently on Spivak problems to ensure I am fluent and I'm doing this one and I just cannot solve it: For which numbers C is there a number X such that f(cx) = f(x). Hint: there are a lot more than you might think at first glance. I get this isn't really calculus...

In the third edition, on page 9, I am reading about the Trichotomy law. It says, for every number a, one and only one of the following properties holds (i) a = 0 (ii) a is in the collection P, (iii) -a is in the collection Before stating this, though, the author said that P is the...

I am taking Calculus at the end of the month for the first time but I have started working through Spivak (wanted to get a headstart and didn't have the class textbook yet) and I have been grasping his beautiful description of some the properties of real numbers yet there is a very minor detail...

The question is if I can start with Spivak's calculus without reading a book like calculus made easy or the calculus lifesaver?and If I can't which is better calculus made easy or the calculus life saver or another?

Question: Find all numbers x for which \frac{1}{x}+\frac{1}{1-x}>0. Solution: If \frac{1}{x}+\frac{1}{1-x}>0, then \frac{1-x}{x(1-x)}+\frac{x}{x(1-x)}>0; hence \frac{1}{x(1-x)}>0. Now we note that \frac{1}{x(1-x)} \rightarrow ∞ as x \rightarrow 0 and \frac{1}{x(1-x)} \rightarrow 0 as x...

I'm finishing up Spivak, after a break from "pure" math. I'm looking for a good follow up on multivariate analysis. I've heard that "Calculus on Manifolds" uses a lot of linear algebra (which I know very little of), so I'm on the look out for another suggestion. I'm looking for a book on the...

Hey there, I'm a high school student learning some basic linear algebra and multivariable calculus. I just ordered Spivak's Calculus on Manifolds, since the book that I've been going through was not as interesting (not a horrible cookbook, but most of the problems could be done without a lot...

Author: Michael Spivak Title: Calculus on Manifolds Amazon link: https://www.amazon.com/dp/0805390219/?tag=pfamazon01-20 Prerequisities: Rigorous Calculus Level: Undergrad Table of Contents: Foreword Preface Functions on Euclidean Space Norm and Inner Product Subsets of Euclidean...

Author: Michael Spivak Title: A Comprehensive Introduction to Differential Geometry Amazon Link: https://www.amazon.com/dp/0914098705/?tag=pfamazon01-20 https://www.amazon.com/dp/0914098713/?tag=pfamazon01-20 https://www.amazon.com/dp/0914098721/?tag=pfamazon01-20...

Homework Statement Suppose the functions f and g have the following property: for all ε > 0 and all x, If 0 < \left| x-2 \right| < \sin^{2} \left( \frac{\varepsilon^{2}}{9} \right) + \varepsilon, then \left| f(x) - 2 \right| < \varepsilon. If 0 < \left| x - 2 \right| < \varepsilon^{2}...

Author: Spivak Title: Calculus Amazon Link: https://www.amazon.com/dp/0914098918/?tag=pfamazon01-20 Prerequisities: High-school mathematics Table of Contents: Preface Prologue Basic Properties of Numbers Numbers of Various Sorts Foundations Functions Appendix: Ordered Pairs...

Homework Statement Prove xn-yn=(x-y)(xn-1+xn-2y+…+xyn-2+yn-1 Homework Equations 12 basic properties of numbers The Attempt at a Solution I'm all good except I don't see why x2yn-2-xn-2y2=0

Mod note: post split of from this thread: https://www.physicsforums.com/showthread.php?p=4187253[/color] Hi. I know this thread hasn't been active in a year, but it still seems the most appropriate place to post. On the advice in this thread, I've bought Michael Spivak's book, Calculus, 4th...

What are the differences? I know it won't be very different but how about the problems? Will it be ok if I get the 3rd edition for class instead of 4th? I don't want to do the wrong questions. Thanks in advance.

How can I show that the set {(x,|x|) , x in real numbers} is not the image of any immersion of R into R^2 ? problem 25 chapter 2 differential geometry M. Spivak

Hi there! I always think whether I am posting this correctly, or this belongs to the homework section. If so, my apologies. I am trying to understand the solutions for a problem in Spicak Calculus, 3^{rd} edition. #8-13 The Problem: "Let A and B bt two nonempty sets of numbers...

Homework Statement A radioactive substance diminishes at a rate proportional to the amount present (since all atoms have equal probability of disintegrating, ...). If A(t) is the amount at time t, this means that A'(t)= p * A(t) for some p representing the probability that an atom will...

Homework Statement Exercice 14-14, 3^{rd} edition, Spivak: Find a function f such that f^{```}(x) = \frac{1}{\sqrt{1 + sin^{2}x}}. The answer acording to the book is : f(x) = \int\left(\int\left( \int \frac{1}{\sqrt{1+sin^{2}x}} dt \right) dz...

I've found a lot of help on this forum for awhile but just registered. I'd like to retroactively say thanks for a lot of help the past few semesters. Im using the 4th edition of spivak calculus. I tried looking for what the solutions to *'ed problems but could not find anything. I have the...

Hey, I am going to major in engineering physics come fall and purchased Spivaks' "Calculus" after seeing it recommended a lot (here amongst other places). I expected it to be hard but it's excruciatingly so. I have done some HS calculus before but that was more memorizing how to solve a...

Homework Statement Prove the binomial theorem by induction. The attempt at a solution http://desmond.imageshack.us/Himg35/scaled.php?server=35&filename=sumu.png&res=landing Hi, running into trouble with this proof and google hasn't helped me. I don't understand the jump here, and as...

Hello. This is my first time posting on the forum but I have been an avid reader since about last year (Didn't have an account). Well let me get to the point. I just finished working through Spivak's Calculus book through self-study. I have been looking around the forum and I have read...

Homework Statement http://desmond.imageshack.us/Himg228/scaled.php?server=228&filename=theorem.png&res=landing Picked up Spivak's Calculus, 3rd ed. and just started. Got to this question and I'm honestly not sure how to start, I looked in the answer book which didn't really clue me in any...

Hello I am about to finished working though Spivak's Calculus. I am wondering if you guys could give me some recommendations on what to read after I'm done with the book. I am looking for something as rigorous as Spivak. Also a bit unrelated but I have also worked through an old edition of...

I have taken Calc 1 (computational) and Discrete Math (low-rigor proofs), and am currently taking Calc 2 (calc is a 2 semester course at my school), in the Fall I will take Vector Cal (mostly computational), probability (mostly computational), and LA (supposed to transition students to the rigor...

I recently obtained a copy of Spivak' Calculus edition 3 and started going through the book. I've found that there is more than one way to prove what he is asking in his problem set and I'm wondering if this is a problem as to how he is trying to teach the subject? I would think not, but I'd...

Please read the italic part at the top of https://www.physicsforums.com/showthread.php?t=243852 Is it true if we generalize to the complex case that an angle preserving transformation's eigenvalues are all the same magnitude? Is there a simple proof for the real case? Obviously its not a...

I want to do some self-study both over the summer and during the school year, and I've chosen my books. They are: Rudin, Principles of Mathematical Analysis Spivak, Calculus + Calculus on Manifolds I will likely be taking a yearlong sequence starting next fall in analysis using Rudin, either...

I have begun in my spare time to work through Spivak's Calculus. Although I have been stuck on many problems, I am most troubled by my general clumsiness and nonelegance of answers (particularly proofs). It always seems that there likely is a much simpler route available, and yet I forgo this...

Homework Statement 2. What is wrong with the following "proof"? Let x = y. Then x2 =xy, x2 - y2 =xy - y , (x + y)(x -y) = y(x - y), x + y = y, 2y = y, 2= 1.Homework Equations It's obvious that x + y =/= y, but I do not know how to "prove" this, i.e. which proof from algebra is applicable...

Homework Statement Prove that if x2=y2, then x=y or x=-y. This is from Spivak's Calculus, problem 1(iii). Homework Equations Distributive law. If a, b, and c are any numbers, then a(b+c)=a*b+a*c. Existence of additive inverse. If a is any number, then a+0=0+a. Existence of multiplicative...