Theorem Definition and 1000 Threads

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 ...

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...

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...

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...

In Bernoulli's theorem I understood the gravitational potential and kinetic energy but I can't understand what is pressure energy.

Hello good folks! I'm stuck trying to solve the problem b). In the theory book examples they are skipping steps and shortly states 'use algebra' and parsevals theorem to rewrite the Fourier series into the answer that is given. So I've tried to use parsevals theorem but I still can't rewrite...

Verify that $\dfrac{d}{dx}\ln(x)=\dfrac{1}{x}$ <br> by applying Theorem 7<br> Theorem 7 states that: If f is a one-to-one differentiable function with inverse $f^{-1}$ and $f'(f^{-1}(a))$ then the inverse function is differtiable at a and <br> $$\dfrac{dy}{dx}=\dfrac{1}{\dfrac{dx}{dy}}$$<br>...

Veritasium posted a video, featuring a visualization of an "intuitive" explanation of the Intermediate Axis Theorem by Terry Tao, based on centrifugal forces in a rotating frame of reference: Unfortunately, the animation is just as incomplete, as Tao's original explanation from 2011, and...

Hey guys, Sorry that it's been a decent amount of time since my last posting on here. Just want to say upfront that I am extremely appreciative of all the support that you all have given me over my last three or four posts. Words cannot express it and I am more than grateful for it all. But, in...

A recent paper analyzing LIGO data suggests - though not conclusively - that the no-hair theorem likely holds, with implication for the information paradox and I was wondering where a confirmed no-hair outcome leaves us with regards going beyond the Standard Model...

Hi! I am given the lagrangian: ## L = \dot q_1 \dot q_2 - \omega q_1 q_2 ## (Which corresponds to a 2D harmonic oscillator) And I am given two transformations and I am asked to say if there is a constant of motion associated to each transformation and to find it (if that's the case). I am...

This is Jackson's 3rd edition 1.12 problem. So, for both ## \phi ## and ## \phi' ##, I started from Green's second identity: ## \int_V ( \phi \nabla^2 \phi' - \phi' \nabla^2 \phi )dV = \int_S ( \phi \frac {\partial \phi'} {\partial n} - \phi' \frac {\partial \phi} {\partial n} ) dS ## And...

In the derivation of energy conservation, there is the transformation ##q(t)\rightarrow q'(t)=q(t+\epsilon)##, whose end points are kind of fuzzy. The original path ##q(t)## is only defined from ##t_1## to ##t_2##. If this transformation rule is imposed, ##q'(t_2-\epsilon)=q(t_2)## to...

Hi. Is the binomial theorem ##(1+x)^n = 1+nx+(n(n-1)/2)x^2 + ….## valid for x replaced by an infinite series such as ##x+x^2+x^3+...## with every x in the formula replaced by the infinite series ? If so , does the modulus of the infinite series have to be less than one for the series to...

I am reading Charles G. Denlinger's book: "Elements of Real Analysis". I am focused on Chapter 2: Sequences ... ... I need help with the proof of Theorem 2.9.6 (a)Theorem 2.9.6 reads as follows: In the above proof of part (a) we read the following: " ... $$ \forall \ m, n \in \mathbb{N}, \...

A question about Bell's theorem : Consider the ##CHSH=AB-AB'+A'B+A'B'## Then the theorem states : ##-2\leq CHSH\leq 2## Implying ##|<CHSH>|\leq 2##. We could repeat the average : ##\langle |\langle CHSH\rangle|\rangle\leq 2## Now Bell's theorem deals with large numbers average, in order to...

With the Liouville's theorem $$\frac{{d\rho }}{{dt}} = \frac{{\partial \rho }}{{\partial t}} + \sum\limits_{a = 1}^{3N} {(\frac{{\partial \rho }}{{\partial {p_a}}}\frac{{d{p_a}}}{{dt}} + \frac{{\partial \rho }}{{\partial {q_a}}}\frac{{d{q_a}}}{{dt}})} = 0$$ when we calculate the time evolution...

I am reading Tom M Apostol's book "Mathematical Analysis" (Second Edition) ... I am focused on Chapter 4: Limits and Continuity ... ... I need help in order to fully understand the example given after Theorem 4.29 ... ... Theorem 4.29 (including its proof) and the following example read as...

I am reading Tom M Apostol's book "Mathematical Analysis" (Second Edition) ... I am focused on Chapter 4: Limits and Continuity ... ... I need help in order to fully understand the proof of Theorem 4.25 ... ... Theorem 4.25 (including its proof) reads as follows: In the above proof by...

Summary: Rudin theorem 1.21 He has said that as t=X/(X+1) then t^n<t<1 then maximum value of t is 1. then in the next part he has given that t^n<t<x. as maximum value of t is less than 1 why has he given that t<x ?

Suppose a wave function is a linear combination of 2 stationary states: ##\psi(x)=c_1\psi_1(x)+c_2\psi_2(x)##. By [5.52] and [5.53], we have ##\psi(x+a)=e^{iK_1a}c_1\psi_1(x)+e^{iK_2a}c_2\psi_2(x)##. But to prove [5.49], we need ##K_1=K_2##. That means all the eigenvalues of the "displacement"...

I am reading the book: Complex Analysis: A First Course with Applications (Third Edition) by Dennis G. Zill and Patrick D. Shanahan ... I need some help with an aspect of the proof of Theorem 3.1.1 (also named Theorem A1 and proved in Appendix 1) ... The statement of Theorem 3.1.1 (A1) reads...

At the end of appendix C (concerning the non-interaction theorem of classical relativistic hamiltonian systems) of the book "Classical Relativistic Many-Body Dynamics" by Trump and Schieve it is stated that"It follows that Currie's equation (C.21), and subsequently the assertion of vanishing...

count(ℝ) > count(ℚ) ; count(ℚ) == count(ℕ) But still in-between any members of ℝ, we are quarantine to find element of ℚ Can someone help me understand: were are these members of ℝ we cannot account for? For reference: https://en.wikipedia.org/wiki/Rational_number "The rationals are a dense...

i know its pretty basic but please give some insight for triangle law of vector addition and pythgoras theorem. becuase ofcourse if you use traingle law to find resultant it will be different from what is pythagoras theorem

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 3.55 on page 110 ... ...Theorem 3.55 and its proof read as follows: At the start...

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 3.55 on page 110 ... ... Theorem 3.55 and its proof read as follows: At the...

Hi to all! The ordinary Gauss theorem states that ##\Phi\left(\vec{E}\right)\,=\, \frac{\sum_{i=1}^{n}q_{i}}{\varepsilon_{0}}## where ##\sum_{i=1}^{n}q_{i}## is the sum of all charges internal of a closed surface and ##\varepsilon_{0}## is the dielectric constant in the empty. Now I ask to 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 further help in order to fully understand the proof of Theorem 3.47 on page 107 ... ... Theorem 3.47 and its proof read as follows: In...

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 3.47 on page 107 ... ... Theorem 3.47 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 further help in order to fully understand the proof of Theorem 3.43 on pages 105-106 ... ... Theorem 3.43 and its proof read as follows...