Spivak Definition and 161 Threads

Hi, Are there calculus books that lie between Stewart (or Thomas) level and Spivak (Courant/Apostol) level? Thanks.

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...

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...

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...

Consider item ##vii##, which specifies the function ##f(x)=\sqrt{|x|}## with ##a=0## Case 1: ##\forall \epsilon: 0<\epsilon<1## $$\implies \epsilon^2<\epsilon<1$$ $$|x|<\epsilon^2\implies \sqrt{|x|}<\epsilon$$ Case 2: ##\forall \epsilon: 1\leq \epsilon < \infty## $$\epsilon\leq\epsilon^2...

Hello everyone. I'm about to take Calc 3 next semester and am looking for a rigorous book to work with on multivariable calculus. I've gone through Spivak's "Calculus" from cover to cover and am hoping to find something with the same degree of rigor, if possible, and preferably with a solution...

I am 100% reading too much into this, but I am curious which of the properties provided by Spivak allow one to justify a specific argument. For reference/context, the properties are: P1: If a, b, and c are any numbers, then $$a +(b + c) = (a + b) +c$$ P2: If a is any number, then $$a + 0 = 0 +...

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...

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...

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...

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...

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

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...

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...

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...

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...

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...

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##...

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...

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...

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...

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...

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...

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

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...

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...

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...

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...

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...

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...

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...

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...

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...

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...

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...

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...

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...

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...

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...

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...

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...

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...

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 ...

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...

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...

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...

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...

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...

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...

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...