Theorem Definition and 1000 Threads

Inspired by the recent discussion on the mean value theorem, let us see how Bolzano's theorem works on linear spaces over endowed with a bilinear form.Suppose you have vectors x,y such that B(x,x)>0 and B(y,y)<0, where B is the bilinear form. Then there exists a non-zero vector z such that B(z,z)=0.

abf(x) x= g(b)-g(a). Some know this. But in my book, the example for a parabola is: "The area under a region with a curved boundary such as the graph of a function f between the numbers a and b, can be found by a method that does not use the limits of the sums of the areas of the rectangles...

Bloch's Theorem:For the Schrodinger Equation--If V(x+a) = V(x),then (x) = u(x)*eikx.Okay, so if (x) is a continuous function, then so is u(x). My question is where exactly does the eikxcome from? Does it imply that the wavefunction is simply out of phase with the potential by k... or what?eNtRopY

##0=1+a+b+c## ##20=8+4a+2b+c## it follows that, ##13=3a+b## and, ##0=k^3+ak^2+bk+c##...1 ##0=(k+1)^3+a(k+1)^2+(k+1)b+c##...2 subtracting 1 and 2, ##3k^2+k(3+2a)+14-2a=0##

Hi, My question pertains to the question in the image attached. My current method: Part (a) of the question was to state what Stokes' theorem was, so I am assuming that this part is using Stokes' Theorem in some way, but I fail to see all the steps. I noted that \nabla \times \vec F = \nabla...

I have a problem which consist in 1 bit RAM made of 3 MOSFETs. One of the questions is to calculate the maximum voltage that the memory element can receive. I have obtained the result by inspection (it is 4 Volts) but I'm unable to reach the same by applying the Thevenin Theorem. My...

I tried to derive the right hand side of the Radon-Nikodym derivative above but I got different result, here is my attempt: \begin{equation} \label{eq1} \begin{split} \frac{\mathrm d\mu_{\Theta\mid X}}{\mathrm d\mu_\Theta}(\theta \mid x) &= f_{\Theta\mid X}(\theta\mid x) \mathrm \space...

Although Stokes Theorem says that the line integral of a closed surface equals to zero why do we get a non-zero value out of this question 1.11 (and figure 1.33) in the Griffits Introduction to Eletrodynamics Book?

It all makes sense to me, but I don't know how to formalize it nicely. I wanted to divide it into two cases. First case where f is fixed in the segment. And a second case where f is not fixed in the segment. But I don't know how to prove it for the case where f i is not fixed

I am reading Manfred Stoll's book: Introduction to Real Analysis. I need further help with Stoll's proof of Theorem 3.1.16 Stoll's statement of Theorem 3.1.16 and its proof reads as follows: Can someone please help me to demonstrate a formal and rigorous proof of the following:If the subset U...

I am reading Manfred Stoll's book: Introduction to Real Analysis. I need help with Stoll's proof of Theorem 3.1.16 Stoll's statement of Theorem 3.1.16 and its proof reads as follows: Can someone please help me to demonstrate a formal and rigorous proof of the following:If U = X \cap O for some...

Could it be that the transformations keeping the wave equation invariant have other solutions than the usual Lorentz ones ?

Summary:: This question is about a Stokes' Theorem question that I saw on Khan Academy and I am trying to attempt to solve it a different way. The problem is as follows: Problem: Let \vec{F} = \begin{pmatrix} -y^2 \\ x \\ z^2 \end{pmatrix} . Evaluate \oint \vec F \cdot d \vec {r} over the...

I'm studying fluid and propulsion mechanics by myself. I stumbled upon this website from MIT: http://web.mit.edu/16.unified/www/SPRING/propulsion/UnifiedPropulsion2/UnifiedPropulsion2.htm#fallingblock It states that "Newton’s second law for a control volume of fixed mass" is $$\sum...

I am reading Andrew Browder's book: "Mathematical Analysis: An Introduction" ... ... I am currently reading Chapter 5: The Riemann Integral and am currently focused on Section 5.2 Existence Results ... ... I need some help in understanding the proof of Theorem 5.12 ...Theorem 5.12 and its...

Hi everyone, I was wondering if it was possible to calculate a double integral by converting it to a line integral, using the greens theorem, and if so is it possible to get a non zero answer. if we were working on a rectangular region

I am reading Andrew Browder's book: "Mathematical Analysis: An Introduction" ... ... I am currently reading Chapter 5: The Riemann Integral and am currently focused on Section 5.2 Existence Results ... ... I need some help in understanding the proof of Theorem 5.12 ...Theorem 5.12 and its...

Noether's theorem tells us that an invariance of the Lagrangian yields a constant of motion. In this problem, that constant is: $$Q_v = p^a \Big( \frac{\partial q_a^{\lambda}}{\partial \lambda}\Big)_{\lambda = 0} + p^b \Big( \frac{\partial q_b^{\lambda}}{\partial \lambda}\Big)_{\lambda = 0}=...

The Poincare's recurrence theorem : This theorem implies the following: Suppose a container is divided in two by a wall. Half of it contains particles and the other none. If you were to remove the wall and wait a very very long time, the particles would eventually be found in the same half...

I am reading Andrew Browder's book: "Mathematical Analysis: An Introduction" ... ... I am currently reading Chapter 5: The Riemann Integral and am currently focused on Section 5.1 Riemann Sums ... ... I need some help in understanding the proof of Theorem 5.10 ...Theorem 5.10 and its proof...

I am reading Andrew Browder's book: "Mathematical Analysis: An Introduction" ... ... I am currently reading Chapter 5: The Riemann Integral and am currently focused on Section 5.1 Riemann Sums ... ... I need some help in understanding the proof of Theorem 5.10 ...Theorem 5.10 and its proof...

Hi All. Does anybody have a reference, (book, internet site) - besides those books of Paulo Ribenboim - where one can find a compilation of demonstrations of the Euclid's theorem on the infinitude of primes? As a suggestion, if the known proofs are neither too many not too long, it would be nice...

Hi all I was wondering if someone could help clear up some confusion about the Parallel Axis Theorem. I am trying to understand the purpose/benefit of applying the Parallel Axis Theorem with respect too the Second Moment Of Area. For example I have a beam that is under load. I have found its...

a) Proof: By theorem above, there exists a ##a \in \mathbb{R}## such that for all ##x \in I## we have ##f'(x) = a##. Let ##x, y \in I##. Then, by Mean Value Theorem, $$a = \frac{f(x) - f(y)}{x - y}$$ This can be rewritten as ##f(x) = ax - ay + f(y)##. Now, let ##g(y) = -ay + f(y)##. Then...

I just need the meaning of In.

How do I prove that: If X and Y are two compact Hausdorff spaces and f : X × Y → R is a continuous function, then f is approximable by ∑ fi gi , wheret  f1, ...,  fn  in X and g1, ..., gn in Y are continuous functions. As far as I read I need to use the Stone-Weierstarss Theorem to prove...

I am reading Andrew Browder's book: "Mathematical Analysis: An Introduction" ... ... I am currently reading Chapter 8: Differentiable Maps and am specifically focused on Section 8.2 Differentials ... ... I need some further help in order to fully understand the proof of Theorem 8.15 ...

I am reading Andrew Browder's book: "Mathematical Analysis: An Introduction" ... ... I am currently reading Chapter 8: Differentiable Maps and am specifically focused on Section 8.2 Differentials ... ... I need some help in order to fully understand the proof of Theorem 8.15 ... Theorem 8.15...

Can anyone explain what the positive energy theorem is layman terms ?

Anyone able to help if I provide you with more details?

Curious, is there any useful reason to translate the 4d curved Lorentzian manifold in GR to, if i read this right, either a 46 or 230 dimensional flat Euclidian space, depending whether the manifold is compact or not? (although another source listed a 39 dimensional flat embedding). ( from...

Hi All I am a bit exasperated right now. On another forum a person claimed Bell's second theorem proved QM was not local. I carefully explained what local causality was, and what the theorem states: There exist quantum phenomena for which there is no theory satisfying local causality. It...

Given a function F(t) $$ F(t) = \int_{-\infty}^{\infty} C(\omega)cos(\omega t) d \omega + \int_{-\infty}^{\infty} S(\omega)sin(\omega t) d \omega $$ I am looking for a proof of the following: $$ \int_{-\infty}^{\infty} F^{2}(t) dt= 2\pi\int_{-\infty}^{\infty} (C^{2}(\omega) + S^{2}(\omega)) d...

Hello, I'm newly discovering the world of the Energy. My question is about the equation ##U=\int \vec{F}\times d\vec{r}=-\int \vec{F}_{s}\times d\vec{r}##. Can you tell me what does this equation means? Thanks!

Hello, Just refining my understanding of the work-KE theorem and seeking some validation: The work-kinetic energy theorem states that the mechanical work done by a force (be it conservative or nonconservative) is always equal to the change in kinetic energy of the body. The net force in a...

Let f be a function twice-differentiable function defined on [0, 1] such that f(0)=0, f′(0)=0, and f(1)=0. (a) Explain why there is a point c1 in (0,1) such that f′(c1) = 0. (b) Explain why there is a point c2 in (0,c1) such that f′′(c2) = 0. If you use a major theorem, then cite the theorem...

I am reading Michael Field's book: "Essential Real Analysis" ... ... I am currently reading Chapter 9: Differential Calculus in \mathbb{R}^m and am specifically focused on Section 9.2.1 Normed Vector Spaces of Linear Maps ... I need some help in fully understanding Theorem 9.2.9 (3) ...

I have to admit that my understanding of the Wigner-Eckart is quite precarious. In my grad studies I try to learn it using Sakurai and I suspect that might have been the wrong choice. So, what books-articles better explain the Wigner-Eckart theorem?

I am reading "Complex Analysis for Mathematics and Engineering" by John H. Mathews and Russel W. Howell (M&H) [Fifth Edition] ... ... I am focused on Section 3.2 The Cauchy Riemann Equations ... I need help in fully understanding the Proof of Theorem 3.4 ...The start of Theorem 3.4 and its...

Reduced graph states are characterized as follows from page 46 of this paper: Proposition: Let ##A \subseteq V## be a subset of vertices for a graph ##G = (V,E)## and ##B = V\setminus A## the corresponding complement in ##V##. The reduced state ##\rho_{G}^{A}:= tr_{B}(|G\rangle\langle G|)## is...

Brahmagupta's theorem: A cyclic quadrilateral is orthodiagonal (diagonals are perpendicular) if and only if the perpendicular to a side from the point of intersection of the diagonals bisects the opposite side. But I don't understand the first step of the proof for the necessary condition...

Hello, This term in university I'm taking a second year intro to astrophysics course and my professor talks a lot about different situations and then solves a problem using the virial theorem. The reason I'm confused is because the range of topics that he applies this theorem to vary in many...

I am reading the book: "Theory of Functions of a Complex Variable" by A. I. Markushevich (Part 1) ... I need some help with an aspect of the proof of Theorem 7.1 ...The statement of Theorem 7.1 reads as follows: At the start of the above proof by Markushevich we read the following: "If f(z)...

Thought following might be of interest - application of Ptolemy's Theorem from "De Revolutionibus Orbium Coelestium: Liber Primus".

Show that if ##f## is a shrinking map ##d(f(x),f(y)) < d(x,y)## and ##X## is compact, then ##f## has a unique fixed point. Hint. Let ##A_n=f^n(X)## and ##A=\cap A_n##. Given ##x\in A##, choose ##x_n## so that ##x=f^{n+1}(x_n)##. If ##a## is the limit of some subsequence of the sequence...

In the chemical engineering text of Smith, VanNess, and Abbott, there is a section on partial molar volume. It states that Gibbs theorem applies to any partial molar property with the exception of volume. Why is volume different? In other words, when evaluating the partial molar volume of a...

I found this video showing an elementary proof of the FTA.

first state whether it can be solved using the Master Theorem, and if it can then use that. Otherwise, use the Akra-Bazzi formula. 1. T(n) = 3T([n/3])+n 2. T(n) = T([n/4])+T([n/3])+n 3. T(n) = 2T([n/4])+√n

I read on the Internet that the work done by a (rigid) body = the change in Kinetic energy. What if I lift a rigid body slowly and vertically by 1 meter above the Earth's surface so that the initial velocity = final velocity =0? According to the Work Energy theorem as stated on many sites on...