Theorem Definition and 1000 Threads

I am reading Anderson and Feil - A First Course in Abstract Algebra. I am currently focused on Ch. 47: Galois Groups... ... I need some help with an aspect of the proof of Theorem 47.1 ... Theorem 47.1 and its proof read as follows: At the end of the above proof by Anderson and Feil, we...

Homework Statement Question attached: Homework Equations [/B] Using the result from two fields that ## T(\phi(x) \phi(y))= : \phi(x) \phi(y) : + G(x-y)## Where ##G(x-y) = [\phi(x)^+,\phi(y)^-] ## ## : ## denotes normal ordered and ##\phi(x)^+ ## is the annihilation operator part , and...

This proof is taken from this site: http://cmt.dur.ac.uk/sjc/thesis_ppr/node12.html I get that ##v_{ext}(r)## and ##N## determines ##H## from proof 1. But why is ##\Psi## determined by ##H##? Can someone derive a path to prove it mathematically?

I am reading Anderson and Feil - A First Course in Abstract Algebra. I am currently focused on Ch. 45: The Splitting Field ... ... I need some help with an aspect of the proof of Theorem 45.6 ... Theorem 45.6 and its proof read as follows: https://www.physicsforums.com/attachments/6701At...

I am reading Anderson and Feil - A First Course in Abstract Algebra. I am currently focused on Ch. 45: The Splitting Field ... ... I need some help with an aspect of the proof of Theorem 45.6 ... Theorem 45.6 and its proof read as follows: At the start of the proof of Theorem 45.6 we read...

I am reading Anderson and Feil - A First Course in Abstract Algebra. I am currently focused on Ch. 45: The Splitting Field ... ... I need some help with some aspects of the proof of Theorem 45.5 ... Theorem 45.5 and its proof read as follows: In the above text from Anderson and Feil we...

I am reading Anderson and Feil - A First Course in Abstract Algebra. I am currently focused on Ch. 45: The Splitting Field ... ... I need some help with some aspects of the proof of Theorem 45.4 ... Theorem 45.4 and its proof read as follows: My questions on the above proof are as...

I am reading Anderson and Feil - A First Course in Abstract Algebra. I am currently focused on Ch. 45: The Splitting Field ... ... I need some help with some aspects of the proof of Theorem 45.4 ... Theorem 45.4 and its proof read as follows: My questions on the above proof are as...

This is out of Jackson's electrodynamics, problem 1.15. I am trying to prove the theorem that if a number of surfaces are fixed in position and a given total charge is placed on each surface, then the electrostatic energy in the region bounded by the surfaces is an absolute minimum when the...

Hello! Why do the singularities in the Residue Theorem must be isolated? If we have let's say a disk around ##z_0##, ##D_{[z_0,R]}## where all the points are singularities for a function ##f:G \to C## with the disk in region G, but f is holomorphic in ##G-D_{[z_0,R]}##, we can still write f as a...

Homework Statement Hi guys, I have the following transmitted power signal: $$x(t)=\alpha_m \ cos[2\pi(f_c+f_m)t+\phi_m],$$ where: ##\alpha_m=constant, \ \ f_c,f_m: frequencies, \ \ \theta_m: initial \ phase.## The multipath channel is: $$h(t)=\sum_{l=1}^L \sqrt{g_l} \ \delta(t-\tau_l).$$...

Determine if the triangle with the given vertices is a right triangle. (7, -1), (-3, 5), (-12, -10) I must find the lengths of the sides using the distance formula for points on the xy-plane. The question then tells me to use the converse of the Pythagorean theorem. How do I use this...

The Weinberg-Witten theorem (https://en.wikipedia.org/wiki/Weinberg–Witten_theorem) states that A ##3 + 1##D QFT quantum field theory with a conserved ##4##-vector current ##J^\mu## which is Poincaré covariant does not admit massless particles with helicity ##|h| > 1/2##. A ##3 + 1##D QFT...

Hello! I came across Jordan Curve Theorem while reading something on Complex Analysis. I don't know much about topology and I apologize if my questions is silly, but from what I understand the theorem states that a closed curve in the complex plane separate the plane into an inner region and an...

Hello! I am reading a book on complex analysis and I came across this: If ##G \in \mathbb{C}## is a region, a function f is holomorphic in G and ##\gamma## is a piecewise smooth path with ##\gamma \sim_G 0## then ##\int_\gamma f = 0##. I want to make sure I understand. First of all, ##\gamma...

This weekend I was trying to calculate the work-energy theorem, considering a body that can be treated like a particle, and has its mass varying in time. I searched through a lot of sites if such thing existed, and didn´t find anything. Then I found a thread...

Hi All, According to the fundamental theorem of algebra: "every non-zero, single-variable, degree n polynomial with complex coefficients has, counted with multiplicity, exactly n complex roots". My question is: what about polynomials with degree say 2.3 or 3.02, as in the polynomial: ## p(x) =...

Hello. Uh, I'm trying to undestand how to prove Thevenin's theorem. The Sadiku book puts an independent current source where the load used to be in order to reach the equation: V = Vth + I*Rth. I do understand how he reaches that conclusion after putting the source, what I don't understand is...

Hi. I'm not a physicist, but I’m intrigued by Bell's theorem and I've been stumbling with "superdeterminism." My understanding of the concept is that everything is not just predetermined, but the initial conditions of the universe are fine-tuned and "conspire" so choices of which versions of...

[Mentor's note: This thread has was forked off from another thread because it was a digression there. This is false. Bell's theorem is based on a certain assumption (the statistical independence or free-will assumption). Some local and realistic theories that contradict this assumption exist...

I've managed to derive the form of Reynolds transport theorem as a bilance of linear momentum of the system: \left (\frac{\vec{\mathrm{d} p}}{\mathrm{d} \tau} \right )_{system}=\frac{\mathrm{d} }{\mathrm{d} x}(\int_{V}^{ }\vec{v}\cdot \rho dV)+\int_{A}^{ }\vec{a}dm+\int_{A}^{ }\vec{v}\cdot \rho...

Homework Statement F(x,y,z)=4x i - 2y^2 j +z^2 k S is the cylinder x^2+y^2<=4, The plane 0<=z<=6-x-y Find the flux of F Homework Equations The Attempt at a Solution What is the difference after if I change the equation to inequality? For example : x^2+y^2<=4, z=0 x^2+y^2<=4 , z=6-x-y...

I am reading "Abstract Algebra: Structures and Applications" by Stephen Lovett ... I am currently focused on Chapter 7: Field Extensions ... ... I need help with another aspect of the proof of Theorem 7.1.10 ...Theorem 7.1.10, and its proof, reads as follows: In the proof of the above...

Homework Statement Using the power series for ln(x + 1) and the Estimation Theorem for the Alternating Series, we conclude that the least number of terms in the series needed to approximate ln 2 with error < 3/1000 is: (i) 333 (ii) 534 (iii) 100 (iv) 9 (v) 201 Homework Equations ln(x+1) =...

Homework Statement Are the groups ##\mathbb{Z}_8 \times \mathbb{Z}_{10} \times \mathbb{Z}_{24}## and ##\mathbb{Z}_4 \times \mathbb{Z}_{12} \times \mathbb{Z}_{40}## isomorphic? Why or why not? Homework EquationsThe Attempt at a Solution I think I am misunderstanding the Theorem of Finitely...

Homework Statement Carnot theorem states that no engine working between two temperatures T1 of source and T2 of sink can have a greater efficiency than that of the Carnot engine. Second law of thermodynamics:it is impossible for a self acting machine to transfer heat from a body at a higher...

I cannot understand the the relation between Reynolds Transport Theorem and Volume Calculation. Volume calculation is an simple, straightforward process which, I think, have much connection between Reynolds Transport Theorem. We calculate volumes in thermodynamics, heat transfer and fluid...

I have encountered this theorem in Serge Lang's linear algebra: Theorem 3.1. Let F: V --> W be a linear map whose kernel is {O}, then If v1 , ... ,vn are linearly independent elements of V, then F(v1), ... ,F(vn) are linearly independent elements of W. In the proof he starts with C1F(v1) +...

I am reading Abstract Algebra: Structures and Applications" by Stephen Lovett ... I am currently focused on Chapter 7: Field Extensions ... ... I need help with an aspect of the proof of Theorem 7.1.10 ...Theorem 7.1.10, and the start of its proof, reads as follows: In the above text from...

I am reading Abstract Algebra: Structures and Applications" by Stephen Lovett ... I am currently focused on Chapter 7: Field Extensions ... ... I need help with an aspect of the proof of Theorem 7.1.10 ...Theorem 7.1.10, and the start of its proof, reads as follows: In the above text from...

I have proved for myself the following theorem, generalizing Galois theorem to general algebraic extensions. My question is: is it true, and is there some reference to this theorem in the literature? Theorem: Recall that a subfield ##M## of a field ##L## is a perfect closure in ##L## if there...

I am now looking at a physics problem that should be a use of stokes' theorem on a torus. The picture (b) here is a torus that the upper and bottom sides are identified as the same, so are the left and right sides. ##A## is a 1-form and ##F = dA## is the corresponding curvature. As is shown in...

I am reading Anderson and Feil - A First Course in Abstract Algebra. I am currently focused on Ch. 42: Field Extensions and Kronecker's Theorem ... I need some help with an aspect of the proof of Theorem 42.1 ( Kronecker's Theorem) ... Theorem 42.1 and its proof read as follows...

I am reading Anderson and Feil - A First Course in Abstract Algebra. I am currently focused on Ch. 42: Field Extensions and Kronecker's Theorem ... I need some help with an aspect of the proof of Theorem 42.1 ( Kronecker's Theorem) ... Theorem 42.1 and its proof read as follows: In the above...

What is the most motivating way to introduce Wilson’s Theorem? Why is Wilson’s theorem useful? With Fermat’s little Theorem we can say that working with residue 1 modulo prime p makes life easier but apart from working with a particular (p-1) factorial of a prime what other reasons are there for...

What is the best way to introduce Fermat’s Little Theorem (FLT) to students? What can I use as an opening paragraph which will motivate and have an impact on why students should learn this theorem and what are the applications of FLT? Are there any good resources on this topic?

If we consider a transformation of a field ##\Phi \rightarrow \Phi + \alpha \frac{\partial \Phi}{\partial \alpha}## which is not a symmetry of a lagrangian then one can show that the Noether current is not conserved but that instead ##\partial_{\mu}J^{\mu} = \frac{\partial L}{\partial \alpha}##...

Homework Statement There is a planet (spherical) with a hollow that is concentric with the planet.if the inner radius is r and outer radius is R and mass of the planet is M what would the gravity be outside of the planet at distance x from the center ? Homework Equations Shell theorem...

Homework Statement :[/B] Homework Equations W= \integral(Fxdx) W = \delta KEThe Attempt at a Solution I used the definition of work to find the final velocity at 2m, and then work theorem together with integration for the changing force. I ended up having different solutions at first but...

Homework Statement Consider a cylindrical solenoid(of radius a,and length h>>a,having n turns of wire for unity of lenght),The solenoid is connected to a resistance R and,at instant t=0 a current i(0)=i0 is flowing in the wire.Prove that poynting's theorem is verified for t>0[/B]Homework...

In the virial theorem the numerical value of the average potential energy within a system is exactly twice that of the average kinetic energy. I know the theorem is proved mathematically but to me it seems a coincidence that one value is exactly twice the other value. I find that interesting. I...

How would one prove the Stokes' theorem for general cases? Namely that $$ \int_{\partial M} W = \int_M \partial W$$ where ##M## is the manifold.

Homework Statement Find the point "c" that satisfies the Mean Value Theorem For Derivatives for the function ## f(x) = \frac {x-1} {x+1}## on the interval [4,5]. Answer - c = 4.48 Homework Equations ##x = \frac {-b \pm \sqrt{b^2 -4ac}} {2a}## ##f'(c) = \frac { f(b) - f(a)} {b-a}## The Attempt...

Hi. I was trying to translate the divergence theorem and the Green's theorem to tensor notation that we use in Relativity. For the divergence theorem, it was easy (please tell me if I'm wrong in the below derivation). I'm using the standard electromagnetic tensor ##F_{\mu \nu}## in place of the...

Great article showing you're never too old to do math: https://www.quantamagazine.org/20170328-statistician-proves-gaussian-correlation-inequality/

Most discussions about Bell's theorem meaning get at some point entangled in semantic and philosophic debates that end up in confusion and disagreement. I wonder if it could be possible to avoid this by reducing the premise, the basic assumption to its bare-bones math content in algebraic/group...

I am doing a panel study with multiple linear regression. When I want to make sure that the residuals are normally distributed, as is a requirement for the regression model, can I assume so due the Central limit theorem (given the size is sufficient)? Or does it not apply when there is a time...

Homework Statement Homework EquationsThe Attempt at a Solution The correct answer is 0.25V. I'm getting 0.3335V[/B]

im trying to use angles on the same arc theorem

Homework Statement Verify the Divergence Theorem for F=(2xz,y,−z^2) and D is the wedge cut from the first octant by the plane z =y and the elliptical cylinder x^2+4y^2=16 Homework Equations \int \int F\cdot n dS=\int \int \int divF dv The Attempt at a Solution For the RHS...