# 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. ### Calculus Calculus book between Stewart & Spivak levels

Hi, Are there calculus books that lie between Stewart (or Thomas) level and Spivak (Courant/Apostol) level? Thanks.
2. ### I Spivak, Ch. 20: Understanding a step in the proof of lemma

In Chapter 20 of Spivak's Calculus is the lemma shown below (used afterward to prove Taylor's Theorem). My question is about a step in the proof of this lemma. Here is the proof as it appears in the book My question is: how do we know that ##(R')^{n+1}## is defined in ##(2)##? Let me try to...
3. ### 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...
4. ### Spivak, Ch 5 Limits, Problems 10c: Proving limit relationship

c) Why is the assertion ##\lim\limits_{x \to 0} f(x) = \lim\limits_{x \to 0} f(x^3)## obvious? First of all I don't think it is obvious but here is an explanation of why the limits are the same. ##\lim\limits_{x\to0} f(x^3)=l_2## means we are looking at points with ##x## close to zero and...

8. ### Solution of a simple integral equation

I did the first part, it is part (b) that I'm having trouble understanding. For any ##x \lt b##, ##f(x)=0## and ##\int_0^x {f(t)} \, dt = 0## (since ##f## is 0 everywhere from 0 to ##b##), which turns the equation ##\int_0^x f(t) \, dt = (f(x))^2+C## into ##0=0+C##, which implies ##C=0##. But...
9. ### I Sum of Binomial Expansion | Spivak Chapter 2, Excercise 3 part d

Hello, I am working through Spivak for self study and sharpening my math skills. I have become stuck on an exercise. What I need to show is the following: $$(a + b) \sum_{j = 0}^{n} \binom nj a^{n-j}b^{j} = \sum_{j = 0}^{n + 1} \binom{n+1}{j} a^{n-j + 1}b^{j}$$ My attempt, starting from...
10. ### Understanding the solution to a calculus problem (removable discontinuities)

Homework Statement The problem (Spivak's Calculus, chapter 6, problem 17): "Let ##f## be a function with the property that every point of discontinuity is a removable discontinuity. This means that ##\underset {y \rightarrow x} {\lim} {f(y)}## exists for all ##x##, but ##f## may be...
11. ### Convergence of Roots at Infinity

Homework Statement Hi everyone, I'm currently making my way through Spivak's calculus and got stuck in question 41 of chapter 5. It's important to note that at this point, the book has only reached the subject of limits (haven't reached continuous functions, derivatives, integrals, series...
12. ### Single Point Continuity - Spivak Ch.6 Q5

Hey Guys, I posed this on Math Stackexchange but no one is offering a good answering. I though you guys might be able to help :) https://math.stackexchange.com/questions/3049661/single-point-continuity-spivak-ch-6-q5
13. ### Need help on a proof from Spivak's Calculus

Homework Statement I'm currently working through Spivak independently and have reached the problems at the end of ch. 1. The problem is: Prove that if 0 < a < b , then a < \sqrt{ab} < \frac{a+b}{2} < b Homework Equations Spivak's properties P1 - P12 The Attempt at a Solution I was...
14. ### B Spivak's Calculus as a Prerequisite for General Topology

High school student here... Recently, I've found an interest in topology and am trying to figure out the correct path for self-studying. I am familiar with set theory and some concepts of abstract algebra but have not really studied any form of analysis, which from what I've read is a...
15. ### I Spivak's Calculus: clarification on Conic Sections appendix

Hello everyone. This was originally a homework problem but I realized my misunderstanding stems from the explanation given before the problem so here we are. The thread deals with these 3 pages from Spivak's Calculus: https://ibb.co/kAKyVU https://ibb.co/jXVSPp https://ibb.co/kwRdVU I'm pretty...
16. ### Spivak: Conic Sections appendix, problem 1

Homework Statement "Consider a cylinder with a generator perpendicular to the horizontal plane; the only requirement for a point ##(x,y,z)## to lie on this cylinder is that ##(x,y## lies on a circle: ##x^2+y^2=C^2##. Show that the intersection of a plane with this cylinder can be described by...
17. ### I Topology vs Differential Geometry

Hello. I am studying Analysis on Manifolds by Munkres. My aim is to be able to study by myself Spivak's Differential Geometry books. The problems is that the proof in Analysis on Manifolds seem many times difficult to understand and I am having SERIOUS trouble picturing myself coming up with...
18. ### Spivak's "Calculus": AM-GM inequality problem.

Homework Statement The problem is stated as follows: "The result in Problem 1-7 has an important generalization: If ##a_1,...,a_n≥0##, then the "arithmetic mean" ##A_n=\frac {a_1+...+a_n} {n}## and "geometric mean" ##G_n=\sqrt[n] {a_1...a_n}## Satisfy ##G_n≤A_n## Suppose that ##a_1\lt A_n##...
19. ### Spivak Ch1, Q8: Deducing Basic Properties of Numbers

Homework Statement Although the basic properties of inequalities were stated in terms of the collection P of all positive numbers, and < was defined in terms of P, this procedure can be reversed. Suppose that P10-P12 are replaced by (P'10) For any numbers a and b one, and only one, of the...
20. ### Spivak chapter 3 problem 24 - proof of a composition

Homework Statement Suppose g is a function with the property that g(x) =/= g(y) if x=/=y. Prove that there is a function f such that f( g(x) ) = x. (The composition) Homework Equations Definition of a function, collection of ordered pairs; g(x) =/= g(y) if x=/=y; x → g(x) → x (The composition...
21. ### Spivak Chapter 5 Problem 26) a

Homework Statement Give an example to show that the given "definition" of limx→aƒ(x) = L is incorrect. Definition: For each 0<δ there is an 0<ε such that if 0< l x-a I < δ , then I ƒ(x) - L I < ε . Homework EquationsThe Attempt at a Solution I considered the piece-wise function: ƒ(x) = (0 if...
22. ### If f(x+y)=f(x)+f(y) and f(x.y)=f(x)f(y) then f(x)=x , x in R

Homework Statement [/B] Suppose that the function ##f## satisfies the two properties. ##f(x+y)=f(x)+f(y)## and ##f(x\cdot y)=f(x)\cdot f(y)##, but that ##f## is not always ##0##. Prove that ##\forall x~ f(x) = x##, as follows: (a) Prove that ##f(1)=1## (b) Prove that ##f(x)=x## if ##x## is...
23. ### I How Does Summing Cubic Expansions Reveal the Formula for Sum of Squares?

I found a deduction to determinate de sum of the first n squares. However there is a part on it that i didn't understood. We use the next definition: (k+1)^3 - k^3 = 3k^2 + 3k +1, then we define k= 1, ... , n and then we sum... (n+1)^3 -1 = 3\sum_{k=0}^{n}k^{2} +3\sum_{k=0}^{n}k+ n The...
24. ### Are there typos in Spivak "Calculus on Manifolds " ?

1. The problem statement, all variables and given Before I try to work through the book, it would be great to have a list of typos, if there are any. Homework EquationsThe Attempt at a Solution
25. ### Did I do this problem from Spivak correctly?

Homework Statement Prove induction from the well-ordering principle. Homework EquationsThe Attempt at a Solution So my attempt is similar to what Spivak uses to prove well-ordering from induction. Let A be the set equipped with the following properties: 1. 1 is in A 2. For every k in A, k+1...
26. ### I Trying to understand the mistake in my logical reasoning

Hello I am in middle on solving problem 17 from Chapter 3 of Spivak's Calculus. We have a function f(x), which is a non-zero function and it obeys the following properties. \forall \, x \, y \, [f(x+y) = f(x) + f(y)] \forall \,x \, y \, \left[f(x \cdot y) = f(x)\cdot f(y)\right] We have to...
27. ### I Spivak - Proof of f(x) = c on [a, b]

In Spivak's Calculus, on page 121 there is this theorem Then he generalizes that theorem: I tried proving theorem 4 on my own, before looking at Spivak's proof. Thus I let c = 0 and then by theorem 1, my proof would be completed. Is this a correct proof? Spivak's proof for theorem 4...
28. ### I Proving Theorem 1 in Spivak's Calculus: Tips & Tricks

Hello I am struggling with proving theorem 1, pages 98-99, in Spivak's Calculus book: "A function f cannot approach two different limits near a." I understand the fact that this theorem is correct. I can easily convince myself by drawing a function in a coordinate system and trying to find two...
29. ### Studying Struggling with end chapter problems (Spivak)

I'm a physics student trying to get a more in-depth understanding of math. A few weeks ago, I started studying from two textbooks, Spivak's Calculus on Manifolds and Hubbard's Vector Calculus, Linear Algebra, and Differential Forms. So far, the stuff from Hubbard's text is pretty straight...
30. ### Adequate proof? Spivak's Calculus ; Dense sets

Homework Statement Let A be a dense set**. Prove that if f is continuous and f(x) = 0 for all x in A, then f(x) = 0 for all x. **A dense set is defined, in the book, as a set which contains a point in every open interval, such as the set of all irrational or all rational numbers.Homework...
31. ### Spivak Calculus Chapter 7 problem 1(v)

Homework Statement Decide whether the given function is bounded above or below on the given interval, and which take on their maximum or minimum value. (Notice that ƒ might have these properties even if ƒ is not continuous, and even if the interval** isn't closed) **The interval is (-a-1...
32. ### Proving that a function can't take exactly same value twice

Homework Statement Prove that there does not exist a continuous function f, defined on R which takes on every value exactly twice. Homework Equations It uses this property: 1... If f is continuous on [a,b], then there exists some y in [a,b], such that f(y)≥f(x), for all x in [a,b]The Attempt...
33. ### I Root of nth Degree Polynomial f(x) in Spivak Calculus Ch. 7

In chapter 7 of Spivak calculus, it is proved that if n is odd, then the 'n'th degree polynomial equation f(x) has a root. I do understand what goes into the proof and can follow steps easily. But, my question is 1.How did they think of a proof like that? 2.By trial and error, did they find...
34. ### Spivak Calculus 4th ed (Ch1 problem 7)

Prove that if 0<a<b, then a < \sqrt{ab} < \frac {a+b} {2} < b Please excuse if format is messy, this is my first time writing in Latex. Suppose 0<a<b then 0<a+O =a (1). Since a<b we can can rewrite this using (1) which is a+0<b (2). Adding a to both sides of (2) (closed under addition) we...
35. ### Other Preparing for GRE Math Subject Test: Engineering Background

I am undergrad doing third year of my engineering. I looked into websites of many US universities for MS Applied math. Most of them require GRE math subject ( as far as I searched)test. The problem is am from engineering background and find it difficult to grasp seemingly simple concepts to math...
36. ### Understanding Limits - Spivak Calculus

I have read Spivak's Calculus up to chapter 5, which is on Limits. Up until this point, the majority has been very straightforward and easy to understand. However, I am having trouble grasping the concept of limits in the style/method that Spivak describes them. Can anyone elaborate in a more...
37. ### Spivak & Dimension of Manifold

1. Homework Statement I'm taking a swing at Spivak's Differential Geometry, and a question that Spivak asks his reader to show is that if ##x\in M## for ##M## a manifold and there is a neighborhood (Note that Spivak requires neighborhoods to be sets which contain an open set containing the...
38. ### Why Does Factoring Change the Existence of a Limit in Spivak's Problem?

Consider the limit lim f(x)g(x) x→a Spivak has proved that this is equal to lim f(x) multlied by x→a lim g(x) x→a And also if lim g(x) = k and k≠0, x→a Then. lim 1/g(x) = 1/k x→a Now the...
39. ### Spivak "root 2 is irrational number" problem

Am using Spivak. Spivak elegantly proves that √2 is irrational. The proof is convincing. For that he takes 2 natural numbers, p and q ( p, q> 0)...and proves it. He defines irrational number which can't be expressed in m/n form (n is not zero). Here he defines m and n as integers. But in the...
40. ### Spivak Thomae's Function proof explanation

I am using Spivak calculus. Now Iam in the chapter limits. In pages 97-98, he has given the example of Thomaes function. What he intends to do is prove that the limit exists. He goes on to define the thomae's function as f(x)=1/q, if x is rational in interval 0<x<1 here x is of the form p/q...
41. ### Spivak calculus, page 22( 3rd edition).

Iam using Spivak these days for learning calculus. In page 22, I have difficulty understanding. He speaks about natural numbers. Do natural numbers always start with 1? He talks about the definition of a set of natural numbers as having 1. Always 1 in set. 2. If k is present, k+1is also...
42. ### Spivak's Calculus on Manifolds: Theorem 5-3

I am trying to finish the last chapter of Spivak's Calculus on Manifolds book. I am stuck in trying to understand something that seems like it's supposed to be trivial but I can't figure it out. Suppose M is a manifold and \omega is a p-form on M. If f: W \rightarrow \mathbb{R}^n is a...
43. ### Calculus by Spivak, Chapter 2, Problem 6, Part 3

In this problem, Spivak shows how to derive formulas to summations. They start by showing the method for 1^2 + 2^2 + ... + n^2 as follows: (k + 1)^3 - k^3 = 3k^2 + 3k + 1 Writing this formula for k = 1, 2, ..., n and adding, we obtain 2^3 - 1^3 = 3*1^2 + 3*1 + 1 3^3 - 2^3 = 3*2^2 + 3*2 + 1 ...
44. ### Calculus Difference between Calculus 4th edition and Calculus 3rd edi

Difference between Calculus 4th edition and Calculus 3rd edition by Michael Spivak? I currently possesses Calculus 3rd edition by Michael Spivak in it's electronic form. However, I am considering buying a hard copy and have the option of buying either a used 3rd edition or a new 4th edition...
45. ### Spivak Calculus Summation problem

Hi, I've enclosed my problem and attempt at solution below. I had problems with the latex so I photographed a picture of my work. The first top half is my attempt at the solution and the bottom is the solution that Spivak provides. I don't understand how he reached his solution and I don't...
46. ### 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...
47. ### Analysis Preparing for B.S. in Electrical Engineering: Analysis After Spivak

I'm using gap year to prepare for B.S. in electrical engineering. Currently I'm solving through Spivak's "Calculus", Lang's "Introduction to Linear Algebra" and Velleman's "How To Prove It." I have three books on analysis, Rudin's "Principles of Mathematical Analysis", Abbott's "Understanding...
48. ### Calculus What makes Mike Spivak's math textbooks popular and difficult?

Hi, everyone I was directed here by a poster in another thread and thought I'd post my question to you guys in this area of the forums. I had some questions about Mike Spivak's math textbooks. It was alluded to in another thread that his books are quite difficult. Yet, from what I can...
49. ### Calculus Why should I read Spivak's Calculus?

I'm currently an A-Level Maths and Physics student looking to get ahead before university maths/ physics. Looking on the internet I see people making a big deal of Spivak's Calculus but on looking thought it I can't see how it could possibly be useful for future maths and physics. To me it seems...
50. ### I'm a little rusty, cant solve x+e^x=b

Homework Statement Hi, I'm new here. I'm really rusty, I resume my career this year, and I'm reading 'the spivak book', (for Calculus 1). Making some exercises, I get curious about how to solve this: x+e^x=4 I would love if someone could give me any trick Homework EquationsThe Attempt at a...