What is Spivak: Definition and 166 Discussions

Spivak or Spivack is a surname of Ukrainian and Polish origin, meaning singer. It is also common among Ashkenazi Jews. The name may refer to:

Charlie Spivak (1905 or 1907–1982), American trumpeter and bandleader
David Spivak (born 1978), American mathematician
Elye Spivak (1890–1950), Soviet linguist
Gayatri Chakravorty Spivak (born 1942), Indian literary critic and professor at Columbia University
Gordon Spivack (1928–2000), American antitrust lawyer and Justice Department official
John L. Spivak (1897–1981), American communist reporter and author
Lawrence E. Spivak (1900–1994), American journalist and publisher
Lori Spivak (contemporary), Canadian jurist from Manitoba
Marla Spivak (born 1955), American entomologist and winner of the MacArthur Fellowship
Maryana Spivak (born 1985), Russian actress
Michael Spivak (born 1940), American mathematician
Mira Spivak (born 1934), Canadian politician from Manitoba; member of the Canadian Senate
Nissan Spivak (1824–1906), Bessarabian cantor and composer
Nova Spivack (born 1969), American internet entrepreneur
Oleksandr Spivak (born 1975), Russian football player of the FC Zenit Saint Petersburg Russian football club
Sidney Spivak (1928–2002), Canadian politician from Manitoba

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  1. R

    Starting with Spivak - question

    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...
  2. X

    Calculus Michael Spivak 3th ed. Pag 46: Function as a rule

    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...
  3. X

    Calculus Michael Spivak 3th Edition Pag 42 Rational Function

    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?
  4. P

    Calculus Calculus by Spivak: Understanding Problems in Chapter 1

    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...
  5. U

    Are Directional Derivatives Worth Pursuing in Spivak's Calculus on Manifolds?

    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...
  6. U

    Calculus Spivak Calculus on Manifolds and Epsilon delta proofs

    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...
  7. T

    Thinking of getting this book in conjuction with Spivak

    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...
  8. S

    Spivak Calculus Chapter 1 Problem 3.i

    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...
  9. A

    Question of "min" function from Spivak

    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...
  10. E

    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...
  11. T

    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 :)
  12. T

    Are Assumptions in Spivak's Calculus a Common Theme Throughout the Book?

    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...
  13. J

    Spivak Chapter 2 problem 10

    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...
  14. J

    How Can I Improve My Understanding of Proofs in Spivak's Calculus Text?

    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...
  15. W

    What is the Best Way to Prepare for Spivak's 'Calculus'?

    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...
  16. I

    Study Calculus: Lang or Spivak?

    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...
  17. I

    What Should I Study Before Tackling Spivak's Calculus?

    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...
  18. tridianprime

    Where to get spivak calculus solutions?

    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.
  19. tridianprime

    What Are the Possible Values of C for f(cx) = f(x) to Hold True?

    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...
  20. B

    Calculus by Spivak Trichotomy Law

    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...
  21. A

    Need clarification in proof (P9) from Chap 1 Spivak

    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...
  22. T

    Start with Spivak's Calculus: A Guide to Choosing the Best Resource

    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?
  23. V

    Solve Spivak Inequality for x: 0<x<1

    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...
  24. H

    How Does Fitzpatrick Compare to Spivak?

    Hi all, I'll be taking analysis in the fall, and I would like to do some preparation for the course over the summer. It's pretty clear that this forum recommends Spivak's Calculus text as the ideal for such a task, but I'm wondering how it compares to Patrick Fitzpatrick's Advanced Calculus...
  25. Astrum

    Where can I find a challenging but rewarding follow-up to Spivak's Calculus?

    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...
  26. A

    Is Spivak not enough of a cookbook?

    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...
  27. micromass

    Analysis Calculus on Manifolds by Spivak

    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...
  28. micromass

    Geometry A Comprehensive Introduction to Differential Geometry series by Spivak

    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...
  29. Q

    Spivak Ch. 5 Limits, problem 6

    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}...
  30. Greg Bernhardt

    Calculus Is Calculus by Michael Spivak: A Must-Read for Strong Students?

    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...
  31. H

    Understanding the Ellipsis in Algebraic Operations

    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
  32. E

    Help With Spivak: Learning Calculus from First Principles

    Mod note: post split of from this thread: https://www.physicsforums.com/showthread.php?p=4187253 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 ed. Now...
  33. P

    Difference Between Calculus 3rd and 4th Edition by Spivak

    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.
  34. B

    M. Spivak, problem 25 chapter 2

    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
  35. C

    Spivak Exercice, Least Upper Bounds Chapter, Proof

    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...
  36. A

    Spivak - Radioactivity and the Exponential Function

    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...
  37. C

    Straight Question, Spivak, Integrals

    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...
  38. dgm5028

    Spivak calculus notation

    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...
  39. U

    Programs Wasting time with Spivak? Engi major to-be

    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...
  40. U

    Binomial Theorem proof by induction, Spivak

    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...
  41. V

    Shilov, Tenenbaum and Edwards Dover Books after Spivak.

    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...
  42. U

    Show that [properties] can be deduced as a theorems, Spivak

    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...
  43. I

    What to read after Spivak? And Physics books

    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...
  44. N

    Spivak or LA this Summer? [independent study]

    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...
  45. T

    Learning Spivak's Calculus Edition 3 - Different Approaches?

    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...
  46. B

    Proof for Angle Preserving Transformation: Eigenvalues Same Magnitude

    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...
  47. QuarkCharmer

    Is Micromass Spivak the Ultimate Solution to All Scientific Problems?

  48. A

    Which Book to Start With for Mathematical Self-Study: Rudin or Spivak?

    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...
  49. F

    Spivak Calculus - Chapter 1 Problems

    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...
  50. J

    Flawed Proof: x + y = y in Algebra - Spivak Calculus (1994) Homework

    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...
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