Pdf - Text of NPTEL IIT Video Lectures

... We will first start by interpreting a line integral. So, let us for simplicity, assume that the gamma is a certain arc like that, it is a curve. So, in terms of what we have been describing as gamma, what I have a drawn here is the trace of a certain gamma, which is a parameterized curve. So, rememb ...

... We will first start by interpreting a line integral. So, let us for simplicity, assume that the gamma is a certain arc like that, it is a curve. So, in terms of what we have been describing as gamma, what I have a drawn here is the trace of a certain gamma, which is a parameterized curve. So, rememb ...

A Tour with Constructive Real Numbers

... (equivalence classes of) Cauchy sequences [TvD88] and co-inductive streams of digits [CDG00] — and will be compared to other proposals of the literature [Bri99, GN01]. In particular we will prove that our axiomatization has a sufficient deductive power. We have formalized and used our axioms inside ...

... (equivalence classes of) Cauchy sequences [TvD88] and co-inductive streams of digits [CDG00] — and will be compared to other proposals of the literature [Bri99, GN01]. In particular we will prove that our axiomatization has a sufficient deductive power. We have formalized and used our axioms inside ...

Developing the Calculus

... conclusion. He did not actually use the functional notation, nor did he realize the importance of the calculations that he was making. Therefore, it is not proper to say that Barrow invented Calculus. He merely stumbled upon the fundamental theorem by utilizing his knowledge of geometry [8, p. 503]. ...

... conclusion. He did not actually use the functional notation, nor did he realize the importance of the calculations that he was making. Therefore, it is not proper to say that Barrow invented Calculus. He merely stumbled upon the fundamental theorem by utilizing his knowledge of geometry [8, p. 503]. ...

Cauchy sequences. Definition: A sequence (xn) is said to be a

... Now suppose S is a nonempty set of equivalence classes of Cauchy sequences that is bounded above. For each fixed n ∈ N, consider the set ( recall that we have identified constant sequences (r) with rational numbers r) Tn = {k ∈ Z : k/2n is an upper bound of S}. First, let us show that it is nonempty ...

... Now suppose S is a nonempty set of equivalence classes of Cauchy sequences that is bounded above. For each fixed n ∈ N, consider the set ( recall that we have identified constant sequences (r) with rational numbers r) Tn = {k ∈ Z : k/2n is an upper bound of S}. First, let us show that it is nonempty ...

MATH 409, Fall 2013 [3mm] Advanced Calculus I

... limit L. We claim that the limit L depends only on c and does not depend on the choice of the sequence {xn }. Indeed, let {x̃n } ⊂ E0 be another sequence converging to c. Then a sequence x1 , x̃1 , x2 , x̃2, . . . also converges to c. Consequently, the sequence f (x1 ), f (x̃1 ), f (x2 ), f (x̃2 ), ...

... limit L. We claim that the limit L depends only on c and does not depend on the choice of the sequence {xn }. Indeed, let {x̃n } ⊂ E0 be another sequence converging to c. Then a sequence x1 , x̃1 , x2 , x̃2, . . . also converges to c. Consequently, the sequence f (x1 ), f (x̃1 ), f (x2 ), f (x̃2 ), ...

Statistical convergence of sequences of fuzzy numbers

... with integer sets having natural density zero. To facilitate this, we introduce the following notation: If X = {Xk} is a sequence that satisfies some property P for all k except a set of natural density zero, then we say that Xk satisfies P for "almost all fc" and we abbreviate this by "a.a. k". The ...

... with integer sets having natural density zero. To facilitate this, we introduce the following notation: If X = {Xk} is a sequence that satisfies some property P for all k except a set of natural density zero, then we say that Xk satisfies P for "almost all fc" and we abbreviate this by "a.a. k". The ...

Infinitesimal Complex Calculus

... Calculus to obtain results that are beyond the reach of the Complex Calculus of Limits. 1) In the Calculus of Limits, Cauchy’s Theorem that any loop integral of a Complex f (z ) on a Simply-Connected domain, vanishes, requires only Continuity of f (z ) . Then, the derivation of the Cauchy Formula re ...

... Calculus to obtain results that are beyond the reach of the Complex Calculus of Limits. 1) In the Calculus of Limits, Cauchy’s Theorem that any loop integral of a Complex f (z ) on a Simply-Connected domain, vanishes, requires only Continuity of f (z ) . Then, the derivation of the Cauchy Formula re ...

Prerequisites and some problems

... But don’t let the higher-dimensional cases intimidate you; when you’ve seen one you’ve seen them all. ...

... But don’t let the higher-dimensional cases intimidate you; when you’ve seen one you’ve seen them all. ...

Notes for 11th Jan (Wednesday)

... that g(Lx ) is bounded above. (Why ?) We claim that this map is injective, surjective, and preserves all the properties. (Why?) ...

... that g(Lx ) is bounded above. (Why ?) We claim that this map is injective, surjective, and preserves all the properties. (Why?) ...

MAT 371 - Test 1 Solution

... (a) The least upper bound property Every non-empty set of real numbers which is bounded above has a least upper bound. (b) Bolzano-Weierstrass theorem Ever bounded infinite set of real numbers has at least one accumulation point. 3. Give an example of each of the following. You do not have to give p ...

... (a) The least upper bound property Every non-empty set of real numbers which is bounded above has a least upper bound. (b) Bolzano-Weierstrass theorem Ever bounded infinite set of real numbers has at least one accumulation point. 3. Give an example of each of the following. You do not have to give p ...

What if there was no Law? What if There Were No Law of Large

... 2.5 and standard deviation 1.5 shown for comparison. ...

... 2.5 and standard deviation 1.5 shown for comparison. ...

Lecture 4: Cauchy sequences, Bolzano

... certain conditions under which we are guaranteed that limits of sequences converge. Definition We say that a sequence of real numbers {an } is a Cauchy sequence provided that for every > 0, there is a natural number N so that when n, m ≥ N , we have that |an − am | ≤ . Example 1 Let x be a real n ...

... certain conditions under which we are guaranteed that limits of sequences converge. Definition We say that a sequence of real numbers {an } is a Cauchy sequence provided that for every > 0, there is a natural number N so that when n, m ≥ N , we have that |an − am | ≤ . Example 1 Let x be a real n ...

Presentation on Weierstrass M-Test

... Any sequence that satisfies the Cauchy Criterion is known as a Cauchy sequence. The above theorem also shows that every convergent sequence is Cauchy, and every Cauchy sequence is convergent. COROLLARY 1: If is a Cauchy sequence that converges to Z, and N is chosen such that every such that , then f ...

... Any sequence that satisfies the Cauchy Criterion is known as a Cauchy sequence. The above theorem also shows that every convergent sequence is Cauchy, and every Cauchy sequence is convergent. COROLLARY 1: If is a Cauchy sequence that converges to Z, and N is chosen such that every such that , then f ...

The Meaning of Integration

... Einstein.Sincerelativity, our view of physical sciences is not what it was before. Without the work of Riemann this revolutionary scientific thought would have been impossible; unless someone had created the concepts and the mathematical methods that Riemann created. As A SEddington says (see nn "A ...

... Einstein.Sincerelativity, our view of physical sciences is not what it was before. Without the work of Riemann this revolutionary scientific thought would have been impossible; unless someone had created the concepts and the mathematical methods that Riemann created. As A SEddington says (see nn "A ...

Cauchy Sequences

... In any metric space S, a divergent Cauchy sequence, because it “converges to a hole,” detects a hole into which S could fit another point. A metric space that has no such holes is called a complete metric space: Definition 4 A metric space S is complete iff every Cauchy sequence in S has a limit in ...

... In any metric space S, a divergent Cauchy sequence, because it “converges to a hole,” detects a hole into which S could fit another point. A metric space that has no such holes is called a complete metric space: Definition 4 A metric space S is complete iff every Cauchy sequence in S has a limit in ...

Ch 5.1 Fundamental Concepts of Infinite Series

... Therefore, it is not convergent but oscillatory. ...

... Therefore, it is not convergent but oscillatory. ...

Real Analysis Lecture 14

... other, |sn − sm | < ε, rather than to a preditermined number. One may suspect that being close to a fixed number s would make terms close to each other as well, that is that all convergent sequences are Cauchy. One would be right. Proposition 1.2. If sn converges to s, then sn is Cauchy. Proof. For ...

... other, |sn − sm | < ε, rather than to a preditermined number. One may suspect that being close to a fixed number s would make terms close to each other as well, that is that all convergent sequences are Cauchy. One would be right. Proposition 1.2. If sn converges to s, then sn is Cauchy. Proof. For ...

Analysis

... “generality of algebra”, which had led C18 mathematicians to manipulate infinite sequences using ordinary algebra, sometimes getting absurd results. He replaced this with a rigorous notion called a “limit”, an idea already used by Newton and others but never made explicitly clear. Limits allowed ...

... “generality of algebra”, which had led C18 mathematicians to manipulate infinite sequences using ordinary algebra, sometimes getting absurd results. He replaced this with a rigorous notion called a “limit”, an idea already used by Newton and others but never made explicitly clear. Limits allowed ...

solutions - Math-UMN

... Now, using the fact that {yj } was Cauchy, you can show that the subsequence {yMk } is also a Cauchy sequence, and it represents the same real number x. Similarly, you can (and should!) show that the sequence defined by zk ≡ yMk − k1 is Cauchy, and represents x as well! Finally, it follows (provide ...

... Now, using the fact that {yj } was Cauchy, you can show that the subsequence {yMk } is also a Cauchy sequence, and it represents the same real number x. Similarly, you can (and should!) show that the sequence defined by zk ≡ yMk − k1 is Cauchy, and represents x as well! Finally, it follows (provide ...

Blog #2 - Professor Fekete

... of March 7, 2010, entitled FINDING YOUR ROOTS (blog #230). The gap was left for the sake of simplicity of presentation. The reader may pick up the thread by going to the fourth paragraph of my earlier blog. We have seen that, although the two direct operations addition and multiplication could alway ...

... of March 7, 2010, entitled FINDING YOUR ROOTS (blog #230). The gap was left for the sake of simplicity of presentation. The reader may pick up the thread by going to the fourth paragraph of my earlier blog. We have seen that, although the two direct operations addition and multiplication could alway ...

H6

... Note: You can use the previous homework assignment to take care of the case n = −1. For the remaining cases you should be able to use some of the theorems we know to compute the answer without explicitly integrating the function. 3. Let ...

... Note: You can use the previous homework assignment to take care of the case n = −1. For the remaining cases you should be able to use some of the theorems we know to compute the answer without explicitly integrating the function. 3. Let ...

Solutions 2

... y. From this (infinite) class of Cauchy sequences, it is easy to find one that will be strictly increasing (making all terms distinct) and such that the distance between all terms beyond a certain point will be less than |y − x|. The longer solution: First let’s show that between any two real number ...

... y. From this (infinite) class of Cauchy sequences, it is easy to find one that will be strictly increasing (making all terms distinct) and such that the distance between all terms beyond a certain point will be less than |y − x|. The longer solution: First let’s show that between any two real number ...

Lecture notes - UT Computer Science

... to move slightly right from xmed to cover another m balls. Notice that in total we have m balls, that is saying, the increased mass due to moving xmed to the right should be approximately ...

... to move slightly right from xmed to cover another m balls. Notice that in total we have m balls, that is saying, the increased mass due to moving xmed to the right should be approximately ...

Baron Augustin-Louis Cauchy FRS FRSE (French: [oɡysˈtɛ̃ lwi koˈʃi]; 21 August 1789 – 23 May 1857) was a French mathematician reputed as a pioneer of analysis. He was one of the first to state and prove theorems of calculus rigorously, rejecting the heuristic principle of the generality of algebra of earlier authors. He almost singlehandedly founded complex analysis and the study of permutation groups in abstract algebra. A profound mathematician, Cauchy had a great influence over his contemporaries and successors. His writings range widely in mathematics and mathematical physics.""More concepts and theorems have been named for Cauchy than for any other mathematician (in elasticity alone there are sixteen concepts and theorems named for Cauchy)."" Cauchy was a prolific writer; he wrote approximately eight hundred research articles and five complete textbooks. He was a devout Roman Catholic, strict Bourbon royalist, and a close associate of the Jesuit order.