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. mishima

    Proving x=y or x=-y from x2=y2.

    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...
  2. H

    [Req] Calculus Michael Spivak 4th Edition

    Hello. I'm not sure if I'm making the topic in the right place but I'd like to have the fourth edition of Calculus (Spivak). I've got the 3rd one which is quite good but I'd like to see the 4th one. If anyone has it or knows where I can download it from it would be much appreciated if you can...
  3. W

    Analysis Learning: Spivak or Stewart? Self-Taught Student

    I have realized that almost everyone on this forum talks about the calculus by spivak, and from the arguments in it's favour make it unbeatable, it seems that nobody can learn analysis without reading that book. I am studying ONLINE at Paul's notes and Calculus by James Stewart, is my...
  4. T

    How long can I expect to work on Spivak?

    Hey guys, I'm an electrical engineering student on my first year. I'm from Canada and I think our first year of University is the equivalent of 2nd year in the USA. I've taken calc I, II and III (multi-variable calculus, PDE's...), statistics and linear algebra. I've also learned some discrete...
  5. Saladsamurai

    Solving Spivak's Problem 1.18.b: Showing x2+bx+c>0 for all x

    Homework Statement In problem 1.18.b of Calculus, he says: . (Bold mine). I assume that is supposed to say in fact x2+bx+c>0 ? However, I am having trouble proving that for all choices of b and c. If I complete the square on x2+bx+c>0 I get x2+bx+c = (x+b/2)2 + k > 0 => k > 0. So I...
  6. Saladsamurai

    Spivak Problem 6(i): x<y => x^n < y^n

    Homework Statement Prove that if 0≤x<y then xn < ynHomework Equations 12 properties of numbers.The Attempt at a Solution For the case: 0 =x < y is trivial. So now we just need y > x > 0 : Now, I am having a bit of trouble with this. Take the case where n=2: x < y so x*x < x*y AND x*y...
  7. Saladsamurai

    Spivak 4 (xiv): Inequality

    Homework Statement Find all x for which \frac{x-1}{x+1}>0 \qquad(1)Homework Equations (2) AB > 0 if A,B >0 OR A,B < 0 (3) 1/Z > 0 => Z > 0 The Attempt at a Solution Since (1) holds if: (x-1) > 0 \text{ and } (x+1) > 0 \qquad x\ne -1 then we must have x>1 AND x>-1 and since (1) also...
  8. Saladsamurai

    Solve Ineq. in Spivak Calculus 4(ii): 5-x2>8

    Homework Statement I am doing the HW in Spivak's calculus (problem 4 (ii) ) on inequalities. The problem statement is: find all x for which 5-x2 > 8The Attempt at a Solution I know this is a simple problem, but bear with me for a moment. I want someone who is familiar with Spivak to tell...
  9. Saladsamurai

    Spivak Calculus: Problem 1-(iii)

    Homework Statement Prove that if x2 = y2 then x=y or x = -y Homework Equations The 12 Properties of numbers The Attempt at a Solution I think I should do this case-wise: Case (a) if x=y then x2 = x*x=y*y=y2. Simple enough. Case (b) if x = -y then x2 = x*x=(-y)*(-y)...
  10. A

    Spivak Calc: Is my proof correct?

    Homework Statement Prologue Chapter 1. Problem 8. (P10) Trichotomy law: For every number a, one and only one of the following holds: (i) a = 0 (ii) a is in the collection P (iii) -a is in the collection P (P11) If a and b are in P, then a + b are in P (P12) If a and b are in P, then a x b is...
  11. M

    Help with a few questions about spivak calculus

    Question if a < b, then -b < -a proof if a < b then a-b<0 and b-a>0 so a-b<0<b-a so -b<-a Question if a<b & c<0, then ab>bc proof if ac<ab then ab>bc then ac<ab>bc then ac>bc Question if a>1 then a^2>a proof a*a > 1*a lemma 1: a*a = a^2...
  12. M

    Strategies for Tackling Proofs in Spivak Calculus

    I am having a really difficult time doing these proofs and want to know how i need to confront them and if there is a good strategy to solving these. I am in first year university. I understood the problem the prof went over but when i try to do on my own, I am just not able to prove them. I...
  13. Saladsamurai

    Spivak: Is his how to approach these proofs?

    Hello all :smile: I have started the problem set for Chapter one (basic properties of numbers) in Spivak's Calculus (self study). I think I am doing these right, but I have some questions. As a solid example, problem 1-(iv) says to prove the following: x^3 - y^3 =...
  14. Saladsamurai

    Spivak: Basic properties of numbers

    Hello all! :smile: I am (painfully) going through the first chapter of Spivak's Calculus. At one point he introduces the property: if a, b, and c are any numbers, then a\cdot(b+c) = a\cdot b +a\cdot c. He then uses this property in an example in which he shows that the only time that a - b =...
  15. C

    Understanding Induction and Finding Formulas for Summations

    Homework Statement 1b) Prove by induction: 1^{3}+...+n^{3}=(1+...+n)^{2} 2a) Find a formula for: \sum^{n}_{i=1}(2i-1) Homework Equations There's a Hint for 2a): 'What to this expression have to do with 1+2+3+...+2n?' The Attempt at a Solution In 2a) I've got near the answer...
  16. C

    How can I solve inequalities involving absolute values?

    Homework Statement d) Show that \left|x-y\right| \leq \left|x\right|+\left|y\right| e) Show that \left|x\right|-\left|y\right| \leq \left|x-y\right| The Attempt at a Solution For item d) I've tried some approaches but none was promising. For item e), I tried squaring...
  17. S

    Problem understanding a proof in Spivak Vol. 4

    Homework Statement This is from Spivak, Vol. 4 Page 102-103 Given |x-x_0| < 1, |x-x0| < Epsilon/(2(|y_0|+1)) Also given |y-y_0| < Epsilon/(2(|x_0| + 1)) Prove |xy-x_0y_0| < Epsilon Homework Equations See above The Attempt at a Solution The proof proceeds clearly enough...
  18. K

    Get Spivak Calculus < 50$ | Gift Card Accepted

    Where can I get a copy of Spivak (new) for <= 50$ as I have a gift card for $50?
  19. S

    Proving AM-GM Inequality using Induction: Spivak 2-22

    Using the fact that the Arithmetic Mean of n numbers is greater than the Geometric Mean of those n numbers when n=2, prove by induction on k that the Arithmetic Mean is greater than the Geometric Mean for n=2^k. I know how to prove the AM-GM inequality in general, but I can't figure out how...
  20. O

    Help with some questions from spivak calc on manifolds

    Hi everyone, been away for a while I got bogged down with my classes so didn't have time to work on this book and haven't been on the forums much. Was getting caught back up to where I was before in here and I ran into a problem that I can't figure out the notation on. I am only looking for...
  21. D

    Which Calculus Book? Stewart or Spivak?

    Hey all, I'm going to be reading a calculus book after I finish my book on Trig. Problem is, i can't decide which book I want to read. I'm wondering if anyone has any insight on either book, or could recommend a better book. I want to use my time the most effectively so any input will be a big...
  22. R

    Spivak Calculus edition 3 solutions?

    I have been trying to find them all over the place, but have had no such luck. Does anybody have the skinny on this book? I just got through ch. 1, and I am hooked! Spivak is a bad mamajamma, and I want to verify my answers. Thanks.
  23. F

    Spivak Calculus 3rd Ed. or 2nd Ed.?

    Spivak "Calculus" 3rd Ed. or 2nd Ed.? I've found a pretty interesting price for the 3rd Ed. of Spivak "Calculus". I'm wondering if you suggest me to buy it or to look for the second edition. My concern is related to the fact that almost every scientific textbook is watered down edition...
  24. H

    Spivak Chapter 1 Problem 1 (i) is my proof correct?

    1. Spivak 4th Edition Problem 1 (i) Chapter 1 If ax = a for some a not equal to 0, then x = 1. 2. P7 where a*a^-1=a^-1*a=1 3. Using P7 Then (x^-1)ax=a then (X^-1 * x)a=a then 1*a=a then x=1 Am I approaching this correct or am I supposed to prove P7 as well or prove this a whole different...
  25. L

    Spivak Calculus Problem 5-23

    Spivak "Calculus" Problem 5-23 Spivak "Calculus" Problem 5-23: For your convenience, here is the problem: Prove that if neither of the following two holds : 1. lim x ->0 f(x) exists and is not 0 2. lim x ->0 |f(x)| = infinity then there is a function g such that...
  26. S

    Where Can I Find a PDF Copy of Spivak Calculus on Manifolds?

    I hope this is not the wrong place to ask this... Can anybody tell me if it is possible to find "Spivak calculus on manifolds" on line (a PDF copy for example) Thanks
  27. K

    Proving the Equality of Fractions Using Multiplicative Properties

    Homework Statement Prove that a/b=c/d if and only if ad=bc Homework Equations Multiplicative inverse property: (a)(a^1) = 1 Commutativity: ab = ba Associativity: (ab)c = a(bc) Transitivity: If a = b and b = c then a = c The Attempt at a Solution a/b=c/d=ab^1= cd^1. Multiplying both sides by...
  28. S

    Is Adding Zero Valid in Inequality Proofs?

    Hi I'm doing the first chapter of Spivak's Calculus and just a little concerned about a particular thing he does in the chapter. He is talking about the trichotomy axiom and that if a > b then a - b, this can be understood as expressing (a - b) > 0 and then the axiom can be interpreted...
  29. I

    Apostol vs Spivak: A Calculus Textbook Comparison

    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...
  30. Z

    Question About Spivak Proof Of Bounds

    Homework Statement Let A and B be two nonempty sets of numbers which are bounded above, and let A+B denote the set of all numbers x+y with x in A and y in B. Prove that sup(A+B) = sup(A) + sup(B). Hint: The inequality sup(A+B) <= sup(A) + sup(B) is easy. To prove that sup(A) + sup(B)<=...
  31. G

    What to study to prepare me for Spivak?

    I'm currently studying Euclid's Elements and Elements of Algebra by Euler. I'd like to know what others recommend I study before Spivak/Apostol. I've been researching a lot, and have had major difficulty in discerning proper texts.
  32. D

    Spivak (Calculus on Manifolds) proof of stolkes theorem

    http://planetmath.org/?op=getobj&from=objects&id=4370 that's pretty much the proof of Stolkes Theorem given in Spivak but I'm having a lot of difficulty understanding the details specifically...when the piecewise function is defined for j>1 the integral is 0 and for j=1 the integral is...
  33. F

    Another Spivak Calculus question

    Homework Statement |x + y + z| \le |x| + |y| + |z|. Indicate when the equality holds, and prove your statement. Homework Equations Answer in the books says it hold only when x, y, and z are all of the same sign. The Attempt at a Solution The value on the rhs of the eq will keep...
  34. B

    Does Michael Spivak's Calculus explain why x²sin(1/x) < ε when ε > 1?

    Is Michael Spivak Wrong!? My text says, In general, if ε > 0, to ensure that |x²sin(1/x)| < ε, we need only require that |x| < ε and x ≠ 0, provided that ε ≤ 1. If we are given an ε which is greater than 1 (it might be, even thought it is "small" ε's which are of interest), then it does not...
  35. J

    Solve in spivak's comprehensive intro to smooth manifolds

    Homework Statement im trying to solve in spivak's comprehensive intro to smooth manifolds, p.103 num. 31 it's a pretty long question but i am stuck at 1 specific part. i have a matrix A in GL(n,R) and i showed that it can be written uniquely as A = A1.A2 where A1 in O(n) (ie A.A^t = I) and...
  36. S

    Phys book on level of Halliday in style of Spivak

    I've been trying to learn introductory physics for about a month now from a copy of Halliday 6th ed with some supplements from MITs 8.01 and an electronic copy of Feynman's lectures (I'm at about chapter/lecture 10 give or take in all of them) and I'm finding the presentation in Halliday rather...
  37. S

    Which Calculus Textbook is Better: Apostol or Spivak?

    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...
  38. S

    Spivak 2=1 Proof fallacy

    Hi, I've gotten Spivak's calculus and I have a question on the second proof in the first chapter What is wrong with the following "proof"? suppose x=y 1. x² = xy 2. x² - y²= xy - y² 3. (x + y)(x - y)=y(x - y) 4...
  39. K

    Spivak Inverse Function Theorem Proof

    On p. 36 of "Calculus on Manifolds" Spivak writes: "If the theorem is true for (\lambda^{-1})\circf , it is clearly true for f." This far I understand. However, he next says: "Therefore we may assume at the outset that \lambda is the identity." I don't understand how this follows...
  40. Saladsamurai

    Spivak Error? Definition: Absolute Value

    Homework Statement In Spivak's Calculus 4e, he defines absolute value as: |a|= a,\qquad a\ge 0 \qquad \text{ and } \qquad -a,\qquad a \le 0 Did he really mean to include the '\le ' and not just '<' ? I know it does not affect the answer, but I didn't think that you could...
  41. M

    Spivak Calculus- Chapter 1, Q1(v)

    Homework Statement Prove the following: xn-yn= (x-y)(xn-1+xn-2y+ … + xyn-2+yn-1) The Attempt at a Solution What do the 3 full stops (...) mean? I didnt know where to start, so I tried to subsitute a number for n, I said n=5, I then multiplied out the brackets and removed terms that...
  42. B

    Spivak calculus ch.5 #24 delta epsilon proof for limit of peicewise function

    Homework Statement Suppose A_n is, for all natural numbers n, some finite set of numbers in [0,1] and A_n intersect A_m={ } if m!=n Define f as follows: f(x) = 1/n if x is in A_n and 0 if x is not in A_n for all n. Prove that the limit as x goes to a of f(x) = 0 for all a in [0,1]...
  43. P

    Spivak Calculus chapter 1: absolute values?

    Homework Statement http://img3.imageshack.us/i/0902091724.jpg/ http://img3.imageshack.us/i/0902091724.jpg/Homework Equations The Attempt at a Solution My problem is that I don't even know where to start on this! My first problem is always forgetting what I can and can't use, because we can...
  44. P

    Spivak v. Anton v. Stewart v. Kline v. Apostol v. Adam. Which is the best?

    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...
  45. F

    Which Calculus Book Will Give Me the True Understanding I'm Looking For?

    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...
  46. K

    Inverse Function Theorem in Spivak

    In his proof of the IFT, on p. 36 of "Calculus on Manifolds," Spivak states: "If the theorem is true for \lambda^{-1} \circf, it is clearly true for f. Therefore we may assume at the outset that \lambda is the identity. I don't understand why we may assume that. thanks for your help...
  47. S

    Spivak Calculus Proof: exponents distribute

    Homework Statement Prove: $\left(ab^{-1}\right)\left(cd^{-1}\right)^{-1}=\left(ad\right)\left(cb\right)^{-1}$. I know how to distribute exponents to get both sides to look identical... Homework Equations ...but a step in the solution requires distributing exponents. But how do...
  48. K

    Please confirm problem in Spivak

    Working my way through Spivak "Calculus on Manifolds." On p. 34, problem 2-33, the problem asks "show that the continuity of D1f^{i} at a can be eliminated from the hypothesis of Theorem 2-8. Is this a typo? Is he saying that there is no need for continuity of ONE partial derivative, or...
  49. J

    Spivak as Preperation for Mathematics Degree

    Hi, I got a 5 on the AP Calculus (BC?) exam, so I have a basic knowledge of calculus (probably on the level that a Stewart book would teach). However, I'm planning to major in mathematics and computer engineering/science, so I'm looking for a rigorous introduction to Calculus. I've read the...
  50. B

    Spivak Calculus on Manifolds

    Homework Statement Given a Jordan-measurable set in the yz-plane, use Fubini's Thm to derive an expression for the volume of the set in R3 obtained by revolving the set about the z-axisHomework Equations The Attempt at a Solution I solved this problem very easily using change of variable...
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