Theorem Definition and 1000 Threads

I am reading Karl R. Stromberg's book: "An Introduction to Classical Real Analysis". ... ... I am focused on Chapter 3: Limits and Continuity ... ... I need help in order to fully understand the proof of Theorem 3.40 on page 104 ... ... Theorem 3.40 and its proof read as follows: In the...

I am reading Karl R. Stromberg's book: "An Introduction to Classical Real Analysis". ... ... I am focused on Chapter 3: Limits and Continuity ... ... I need help in order to fully understand the proof of Theorem 3.36 on page 102 ... ... Theorem 3.36 and its proof read as follows: In the...

I recently found the centre of mass of a semicircle using polar coordinates, by first finding the centre of mass of a sector, and then summing all the sectors from 0 to pi to get the centre of mass of the semicircle. However, being a beginner at integrals, I struggled for a long time getting the...

I am reading Karl R. Stromberg's book: "An Introduction to Classical Real Analysis". ... ... I am focused on Chapter 3: Limits and Continuity ... ... I need help in order to fully understand the proof of Theorem 3.6 on page 94 ... ... Theorem 3.6 and its proof read as follows: In the above...

is there a rigorous version of this proof of fundamental theorem of calculus?if yes,what is it?and who came up with it? i sort of knew this short proof of the fundamental theorem of calculus since a long while...but never actually saw it anywhere in books or any name associated with it. i know...

I am reading Houshang H. Sohrab's book: "Basic Real Analysis" (Second Edition). I am focused on Chapter 4: Topology of R and Continuity ... ... I need help in order to fully understand the proof of Theorem 4.3.4 ... ... Theorem 4.3.4 and its proof read as follows: In the above proof by...

Stokes' Theorem states that: $$\int (\nabla \times \mathbf v) \cdot d \mathbf a = \oint \mathbf v \cdot d \mathbf l$$ Now, if for a specific situation, I can work out the RHS and it's equal to zero, does it necessarily mean that ##\nabla \times \mathbf v = 0##? I mean all that tells me is that...

##I_{AB} = I_{GXX} + A.(y^{2})## Same applies to CD; ##I_{CD} = I_{GYY} + A.(x^{2})## In the above statement, "any axis in its plane" where does the plane exist in this sketch?

I am reading Houshang H. Sohrab's book: "Basic Real Analysis" (Second Edition). I am focused on Chapter 4: Topology of [FONT=MathJax_AMS]R and Continuity ... ... I need help in order to fully understand the proof of Theorem 4.1.10 ... ... Theorem 4.1.10 and its proof read as follows: In the...

To be honest i don't know from where to start. I know how i can test the stokes theorem if i have a cylindrical shape and a cylindrical vector or spherical vector and a spherical shape but here I am out of ideals. The first thing i tried was to compute the left part of the stokes theorem but i...

State where in the ty-plane the hypotheses of Theorem 2.4.2 are satisfied $\displaystyle y^\prime= \frac{t-y}{2t+5y}$ ok I don't see how this book answer was derived since not sure how to separate varibles $2t+5y>0 \textit{ or }2t+5y<0$

I am reading Tom M Apostol's book "Mathematical Analysis" (Second Edition) ... I am focused on Chapter 3: Elements of Point Set Topology ... ... I need help in order to fully understand Theorem 3.28 (Lindelof Covering Theorem ... ) .Theorem 3.28 (including its proof) reads as follows: In the...

I am reading Tom M Apostol's book "Mathematical Analysis" (Second Edition) ... I am focused on Chapter 3: Elements of Point Set Topology ... ... I need help in order to fully understand Theorem 3.28 (Lindelof Covering Theorem ... ) .Theorem 3.28 (including its proof) reads as follows: In...

Hello guys, is it possible to "see" the mean value theorem when one is only thinking of numerical values without visualizing a graph? Perhaps through a real world problem?

This theorem is from the stewart calculus book 11.1.6 If $$ \lim_{n\to\infty} |a_n| = 0$$, then $$\lim_{n\to\infty} a_n = 0$$ I wonder whether converse of this theorem true or not

I am reading Houshang H. Sohrab's book: "Basic Real Analysis" (Second Edition). I am focused on Chapter 2: Sequences and Series of Real Numbers ... ... I need help with the proof of Theorem 2.3.9 (a) ... Theorem 2.3.9 reads as follows: Now, we can prove Theorem 2.3.9 (a) using the Cauchy...

In my textbook when proving continuity implies uniform continuity (which is very similar to the proof given here), BWT is used to find a converging subsequence. I cannot see why this is needed. Referring to the linked proof, if we open up the inequality ##|x_n-y_n|<\frac{1}{n}##, isn't by the...

I am reading D. J. H. Garling's book: "A Course in Mathematical Analysis: Volume I: Foundations and Elementary Real Analysis ... ... I am focused on Chapter 3: Convergent Sequences I need some help to fully understand some remarks by Garling made after the proof of Theorem 3.1.1...

I am reading D. J. H. Garling's book: "A Course in Mathematical Analysis: Volume I: Foundations and Elementary Real Analysis ... ... I am focused on Chapter 3: Convergent Sequences I need some help to fully understand some remarks by Garling made after the proof of Theorem 3.1.1...

Ans is 3. I know basic t-test but I have no clue to solve this question. Thanks.

Section ##3.8## talks about the gradient and smooth surfaces, defining when the directional derivative ##(\partial f/\partial\mathbf{u})(\mathbf{p})## takes maximum value and that when it equals ##0##, then ##\mathbf{u}## is a unit vector orthogonal to ##(grad\ f)(\mathbf{p})##.It also says that...

I am working with a simulation which generates an arbitrary number ##n## of identical curves with different phases and calculates their (normalized) sum. As expected, the fluctuation depth of the curves decreases as we increase ##n##. Here is an example of my simulation (when ##n>1##, the...

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My question is: What is wrong with my working/ method (in the attached pictures) to find i_{sc} ? I can get the Norton equivalent from there, but seem to get the same answer as the solution scheme. Context: we are given the circuit depicted in the picture (initially with no connection between...

Picture of the circuit is posted below. Apologies, the voltage source on the left should read 24 V. My question is: What is wrong with this method? [Edit: Sorry if it wasn't clear- the method in the picture yields the wrong answer] When I originally did the question, I just turned the LHS into...

I am reading Andrew McInerney's book: First Steps in Differential Geometry: Riemannian, Contact, Symplectic ... and I am focused on Chapter 3: Advanced Calculus ... and in particular on Section 3.3: Geometric Sets and Subspaces of ##T_p ( \mathbb{R}^n )## ... ... I need help with...

I am reading Andrew McInerney's book: First Steps in Differential Geometry: Riemannian, Contact, Symplectic ... and I am focused on Chapter 3: Advanced Calculus ... and in particular on Section 3.3: Geometric Sets and Subspaces of ##T_p ( \mathbb{R}^n )## ... ... I need help with an aspect...

I want to check Stokes' theorem for the following exercise: Consider the vector field ##\vec F = ye^x \hat i + (x^2 + e^x) \hat j + z^2e^z \hat k##. A closed curve ##C## lies in the plane ##x + y + z = 3##, oriented counterclockwise. The parametric representation of this curve is defined as...

As far as I can tell the divergence theorem might be one of the most used theorems in physics. I have found it in electrodynamics, fluid mechanics, reactor theory, just to name a few fields... it's literally everywhere. Usually the divergence theorem is used to change a law from integral form to...

I don't understand proof of uniqueness theorem for polynomial factorization, as described in Stewart's "Galois Theory", Theorem 3.16, p. 38. "For any subfield K of C, factorization of polynomials over K into irreducible polynomials in unique up to constant factors and the order in which the...

Since it is known than the number N of primitive Pythagorean triples up to a given hypotenuse length A is given on average by N = Int(A/(2.pi)) and according to my calculations with primitive triples and A = B + C, I get on average N = Int(0.152 A^2) (for A = 10^3 I get N = 152,095 compare...

I am checking the divergence theorem for the vector field: $$v = 9y\hat{i} + 9xy\hat{j} -6z\hat{k}$$ The region is inside the cylinder ##x^2 + y^2 = 4## and between ##z = 0## and ##z = x^2 + y^2## This is my set up for the integral of the derivative (##\nabla \cdot v##) over the region...

From gauss divergence theorem it is known that ##\int_v(\nabla • u)dv=\int_s(u•ds)## but what will be then ##\int_v(\nabla ×u)dv## Any hint?? The result is given as ##\int_s (ds×u)##

A monic polynomial of degree N has N number of coefficients. The product of N number of linear factors has N number of free terms. A complex number has 2 DOF. Therefore, both a monic polynomial and the product of free terms have 2N number of DOF of real values. Thus, it must be possible to...

Sorry for the misspelling, but this forum doesn't allow enough characters for the title. The title should be: For the topological proof of the Fundamental Theorem of Algebra, what is the deal when the roots are at the same magnitude, either at different complex angles, or repeated roots? I...

I am reading Gerard Walschap's book: "Multivariable Calculus and Differential Geometry" and am focused on Chapter 1: Euclidean Space ... ... I need help with an aspect of the proof of Theorem 1.3.1 ... The start of Theorem 1.3.1 and its proof read as follows: I tried to understand how/why...

Homework Statement Find all $$n \in Z$$, for which $$ (\sqrt 3+i)^n = 2^{n-1} (-1+\sqrt 3 i)$$ Homework Equations $$ (a+b i)^n = |a+b i|^n e^{i n (\theta + 2 \pi k)} $$ The Attempt at a Solution First I convert everything to it`s complex exponential form: $$ 2^n e^{i n (\frac {\pi}{3}+ 2\pi...

I'll have to think a bit about what you've written, but just to note this is not true due to Haag's theorem.

Hello everybody! I have a question regarding the first step of the quantistic proof of the Goldstone's theorem. Defining $$a(t) = \lim_{V \rightarrow +\infty} {\langle \Omega|[Q_v(\vec{x},t),A(\vec{y})]| \Omega \rangle}$$ where ##|\Omega\rangle## is the vacuum state of the Fock space, ##Q_v##...

Hello. In chapter 3 (Quantum Black Holes) of this book... https://www.amazon.com/dp/069116844X/?tag=pfamazon01-20 ...Stephen Hawking writes... "The no-hair theorem, proved by the combined work of Israel, Carter, Robinson and myself, shows that the only stationary black holes in the absence of...

Following my instructor's notes the statement of the Uniqueness Theorem(s) are as follows "If ##\rho_{inside}## and ##\phi_{boundary}## (OR ##\frac{d \phi_{boundary}}{dn}## ) are known then ##\phi_{inside}## is uniquely determined" A few paragraphs later the notes state "For the field inside...

Hello! I have been searching the web and textbooks for a certain theorem which generalizes the value of the integral around a infinitesimal contour in the real axis, or also called indented contour over a nth order pole. It is easy to prove that if the pole is of simple order, the value of the...

Homework Statement In comparison with the sampling sine wave, in order to reconstruct a square wave, do we need to increase or decrease sampling frequency? Homework Equations Aliasing effect Leakage effect The Attempt at a Solution No matter square wave or sine wave, the experimental results...

I am reading Andrew Browder's book: "Mathematical Analysis: An Introduction" ... ... I am currently reading Chapter 12: Multilinear Algebra and am specifically focused on Section 12.1: Vectors and Tensors ... I need help in fully understanding Corollary 12.4 to Theorem 12.2 ... ... Theorem...

How do you prove 1.85 is valid for all closed surface containing the origin? (i.e. the line integral = 4pi for any closed surface including the origin)

Homework Statement Given: sin(Πx/a)e6Πix/Na and e2Πi/a(7/N+4)x can these equations be represented in Bloch form?[/B] Homework Equations Given that Bloch form can be represented as: Ψ(x) = u(x) eikx[/B] The Attempt at a Solution sin(Πx/a)eikx w/n = 3 and...

Hey! :o We have the system \begin{align*}&x_1=\left (5+x_1^2+x_2^2\right )^{-1} \\ &x_2=\left (x_1+x_2\right )^{\frac{1}{4}}\end{align*} and the set $G=\{\vec{x}\in \mathbb{R}^2: \|\vec{x}-\vec{c}\|_{\infty}\leq 0.2\}$ where $\vec{c}=(0.2,1)^T$. I want to show with the Banach fixed-point...

One version of Liouville’s Theorem for non-dissipative classical systems, governed by a conserved Hamiltonian, is that the volume in phase space (position-momentum space) of an ensemble of such systems (the volume is the Lebesgue measure of the set of points where the ensemble’s density is...

Homework Statement Let a and b be integers, and let m be a positive integer. Then a ≡ b (mod m) if and only if a mod m = b mod m. Homework EquationsThe Attempt at a Solution By definition a ≡ b (mod m) => m| (a-b) mx = a -b => mx + b = a => b = a mod m b = a - mx => b = m(-x) + a => a = b...

I am wondering if it existes some discret version of the Noether symmetry for potential with discrete symmetry (like $C_n$ ). The purpose is to describe the possible evolution of the phase space over the time without having to solve equations numerically (since even if the potential may have...