In mathematics and logic, a theorem is a non-self-evident statement that has been proven to be true, either on the basis of generally accepted statements such as axioms or on the basis of previously established statements such as other theorems. A theorem is hence a logical consequence of the axioms, with a proof of the theorem being a logical argument which establishes its truth through the inference rules of a deductive system. As a result, the proof of a theorem is often interpreted as justification of the truth of the theorem statement. In light of the requirement that theorems be proved, the concept of a theorem is fundamentally deductive, in contrast to the notion of a scientific law, which is experimental.Many mathematical theorems are conditional statements, whose proofs deduce conclusions from conditions known as hypotheses or premises. In light of the interpretation of proof as justification of truth, the conclusion is often viewed as a necessary consequence of the hypotheses. Namely, that the conclusion is true in case the hypotheses are true—without any further assumptions. However, the conditional could also be interpreted differently in certain deductive systems, depending on the meanings assigned to the derivation rules and the conditional symbol (e.g., non-classical logic).
Although theorems can be written in a completely symbolic form (e.g., as propositions in propositional calculus), they are often expressed informally in a natural language such as English for better readability. The same is true of proofs, which are often expressed as logically organized and clearly worded informal arguments, intended to convince readers of the truth of the statement of the theorem beyond any doubt, and from which a formal symbolic proof can in principle be constructed.
In addition to the better readability, informal arguments are typically easier to check than purely symbolic ones—indeed, many mathematicians would express a preference for a proof that not only demonstrates the validity of a theorem, but also explains in some way why it is obviously true. In some cases, one might even be able to substantiate a theorem by using a picture as its proof.
Because theorems lie at the core of mathematics, they are also central to its aesthetics. Theorems are often described as being "trivial", or "difficult", or "deep", or even "beautiful". These subjective judgments vary not only from person to person, but also with time and culture: for example, as a proof is obtained, simplified or better understood, a theorem that was once difficult may become trivial. On the other hand, a deep theorem may be stated simply, but its proof may involve surprising and subtle connections between disparate areas of mathematics. Fermat's Last Theorem is a particularly well-known example of such a theorem.
The last formula is what I was going for, since it arises as the momentum flux in fluid dynamics, but in the process I came across the rest of these formulas which I’m not sure about.
The second equation is missing a minus sign (I meant to put [dA X grad(f)]).
Are they correct? Do they have...
If correct, as non-physicist, I wonder why the vast jump to "spooky action" is seen as more plausible as some new type of particle faster than the speed of light. Consider the time long before the discovery of radio communication, how weird it must have been to theorize about that. The speed of...
Basically surface B is a cylinder, stretching in the y direction.
Surface C is a plane, going 45 degrees across the x-y plane.
Drawing this visually it's self evident that the normal vector is
$$(1, 1, 0)/\sqrt 2$$
Using stokes we can integrate over the surface instead of the line.
$$\int A(r)...
Solution Write the Taylor formula for ##e^x## at ##x=0##, with ##n## replaced by ##2n+1##, and then rewrite that with ##x## replaced with ##-x##. We get:
$$e^x=1+x+\dfrac{x^2}{2!}+\dfrac{x^3}{3!}+\cdots+\dfrac{x^{2n}}{(2n)!}+\dfrac{x^{2n+1}}{(2n+1)!}+O(x^{2n+2})$$...
I would like to check my understanding here to see if it is correct as I am currently stuck at the moment.
From the question, I can gather that:
P(Rain | Dec) = 9/30
P(Cloudy | Rain) = 0.6?
P(Cloudy | Rain) = 0.4
To answer the question:
P(Rain | <Cloudy, Morning, December> ) = P(Rain) *...
I parameterize surface A as:
$$A = (2cos t, 0, 2sin t), t: 0 \rightarrow 2pi$$
Then I get y from surface B:
$$y = 2 - x = 2 - 2cos t$$
$$r(t) = (2cost t, 2 - 2cos t, 2sin t)$$
Now I'm asked to integral over the surface, not solve the line integral.
So I create a new function to cover the...
I read about Noether's theorem that says how for every symmetry there is a conserved quantity. Seems kind of obvious. Does anyone understand it well enough that they can explain precisely why that notion is profound?
Hello,
I am currently working on the proof of Minkowski's convex body theorem. The statement of the corollary here is the following:
Now in the proof the following is done:
My questions are as follows: First, why does the equality ##vol(S/2) = 2^{-m} vol(S)## hold here and second what...
I am trying to solve the given question based on energy conservation,but am stuck with the analysis of the equations.
The question says find the velocity of the bigger block when the smaller block initially given a velocity v and sliding on the horizontal part of the bigger block reaches the...
Consider a certain integer between ## 1 ## and ## 1200 ##.
Then ## x\equiv 1\pmod {9}, x\equiv 10\pmod {11} ## and ## x\equiv 0\pmod {13} ##.
Applying the Chinese Remainder Theorem produces:
## n=9\cdot 11\cdot 13=1287 ##.
This means ## N_{1}=\frac{1287}{9}=143, N_{2}=\frac{1287}{11}=117 ## and...
prove:
The 2nd axiom of mathematical logic
2) $((P\implies(Q\implies R))\implies((P\implies Q)\implies(P\implies R))$
By using only the deduction theorem
do these proposals violate the Coleman-Mandula theorem since they combine space-time and internal symmetries via SU(2)r and SU(2)l
how plausible are these proposals as actual physical theories that unify gravity with SU(2) weak force?
Graviweak Unification
F. Nesti, R. Percacci...
My question is, if the determinism theorem is a good explanation, which covers all holes of the entanglement experiment. why are people still concluding its a 'spooky' superposition which is only determined by a measure and then somehow affects the other measurement.
What am I missing? Why is...
Hello physics researchers, teachers and enthusiasts.
I notice one little thing is confusing me in the derivation of Bernoulli's equation in the article, they write:$$dW = dK + dU$$where dW is the work done to the fluid, dK is the change in kinetic energy of the fluid, and dU is the change in...
According to Helmholtz’s theorem, if electric charge density goes to to zero as r goes to infinity faster than 1/r^2 I'm able to construct an electrostatic potential function using the usual integral over the source, yet I don't understand how this applies to a chunk of charge in some region of...
I am a little confused with the Poynting theorem https://en.wikipedia.org/wiki/Poynting%27s_theorem .
When we use this equation, the energy density that enters in $$\partial u / \partial t$$ is the one due only to the fields generated by charges/source itself? That is, if we have a magnetic...
Hi,
I am looking for a formal proof of Thevenin theorem. Actually the first point to clarify is why any linear network seen from a port is equivalent to a linear bipole.
In other words look at the following picture: each of the two parts are networks of bipoles themselves.
Why the part 1 -- as...
This video demonstrates the Dzhanibekov effect (instability when spinning arround the intermediate axis).
In order to achieve the best results, is it better for the three MoI's to be close together, or for them to have widely differing values?
(OBS: Don't take the index positions too literal...)
Generally it is easy to deal with these type of exercises for discrete system. But since we need to evaluate it for continuous, i am a little confused on how to do it.
Goldstein/Nivaldo gives these formulas:
I am trying to understand how...
Greetings
the solution is the following which I understand
I do understand why the current orientation of the Path is positive regarding to stocks (the surface should remain to the left) but I don´t understand why the current N vector of the surface is positive regarding stockes theorem...
My attempt;
##4x^3+kx^2+px+2=(x^2+λ^2)(4x+b)##
##4x^3+kx^2+px+2=4x^3+bx^2+4λ^2x+bλ^2##
##⇒k=b, p=4λ^2 , bλ^2=2##
##\dfrac{4λ^2}{bλ^2}=\dfrac{p}{2}##
##\dfrac{4}{b}=\dfrac{p}{2}##
##⇒8=pb## but ##b=k##
##⇒8=kp##
Any other approach appreciated...
My first approach;
##(a-b)^3+(b-c)^3+(c-a)^3=a^3-3a^2b+3ab^2-b^3+b^3-3b^2c+3bc^2-c^3+c^3-3c^2a+3ca^2-a^3##
##=-3a^2b+3ab^2-3b^2c+3bc^2-3c^2a+3ca^2##
what i did next was to add and subtract ##3abc## ...just by checking the terms ( I did not use...
Does the second uniqueness theorem just say that if there is an electric field that satisfies Gauss's law for a surface surrounding each conductor + a surface enclosing all the conductors, it is indeed the true electric field, and no other electric field will satisfy those conditions?
in this example in Griffiths' electrodynamics, he says the following :(Figure 3.7 shows
a simple electrostatic configuration, consisting of four conductors with charges
±Q, situated so that the plusses are near the minuses. It all looks very comfort-
able. Now, what happens if we join them in...
From my research I have found that since Fermat proved his last theorem for the n=4 case, one only needs to prove his theorem for the case where n=odd prime where c^n = a^n + b^n. But I am not clear on some points relating to this. For example, what if we have the term (c^x)^p, where c is an...
Hi, PF
Taylor's formula provides a formula for the error in a Taylor approximation ##f(x)\approx{P_{n}(x)}## similar to that provided for linear approximation.
Observe that the case ##n=0## of Taylor's formula, namely,
##f(x)=P_{0}(x)+E_{0}(x)=f(a)+\dfrac{f'(s)}{1!}(x-a)##,
is just the...
The k-th Pontrjagin class of a real vector bundle is defined as the 2k-Chern class of the complexified bundle. Therefor, a Pontrjagin class lives in cohomology with integer coefficients. But then the statement of Theorem 15.9 is that if the coefficient ring is taken to be a PID \Lambda...
Hello guys, I am new here.
I was wondering whether I could get some help about the highlighted part. What I don't understand is why we are able to ignore the 5-ohms resistor when we have short circuited terminals a-b.
Thanks in advance.
A friend of mine shared a YouTube video with me, saying he was sure I would love it. He described it as very strange with a rotating wingnut in the space station flipping over on its rotation axis, over and over, while it spun rapidly.
After watching the video, I verified I was taught the...
Going through Axler's awful book on linear algebra. The complex spectral theorem (for operator T on vector space V) states that the following are equivalent: 1) T is normal 2) V has an orthonormal basis consisting of eigenvectors of T and 3) the matrix representation of T is diagonal with...
From Stokes' theorem: ##\int_{C}^{}\vec F\cdot d\vec r=\iint_{S}^{}curl\vec F\cdot d\vec S=\iint_{D}^{}curl\vec F\cdot(\vec r_u \times \vec r_v)dA ##
To get to the latter surface integral, I started by parametrizing the triangular surface in ##uv## coordinates as:
$$\vec r=<1-u-v,u,v>, 0\leq...
From Stokes we know that ##\iint_{\textbf{S}}^{}curl \textbf{F}\cdot d\textbf{S}=\int_{C}^{}\textbf{F}\cdot d\textbf{r}##.
Now, we can calculate the surface integral of the curl of F by calculating the line integral of F over the curve C.
The latter ends up being 0(I calculated it parametrizing...
If I have a many-body Hamiltonian, and I choose a coordinate x with canonical momentum p, I can say that by the generalized equipartition theorem that
<p(dH/dx)> = -<p(dp/dt)> = 0
Since p and x are distinct phase space variables, and since by the Hamiltonian equations of motion the force...
This is not really a homework problem (it could be made to be though). I kind of made it up, inspired by a youtube math challenge problem involving parabolas, a water fountain where A = 1, R = 3, and H = 3. The solution given (h = 9/4) was based off simple math utilizing vertex form of a...
There is a nice uniqueness theorem of electrostatics, which I have found only after googling hours, and deep inside some academic site, in the lecture notes of Dr Vadim Kaplunovsky:
Notice that the important thing here is that only the NET charges on the conductors are specified, not their...
Moderator's note: Spin-off from previous thread due to topic change.
Not in the sense in which it is used in the no communication theorem. That sense is basically the information theoretic sense, which in no way requires humans to process or even be aware of the information.
Can we have two tangents (two turning points) within the given two end points just asking? I know the theorem holds when there is a tangent to a point ##c## and a secant line joining the two end points.
Or Theorem only holds for one tangent point. Cheers
I have to prove three equations above.
For first two equations, I've been thought and made reasonable answer by using a definition of the electricfield.
However, for third, I can't use a definition of a magnetic field due to the cross product
Like J_2 X J_1 X (r_2 - r_1).
I think three of 'em...
I have been tasked with calculating amplitudes of a B meson decaying to a photon and lepton/lepton anti-neutrino pair ,upto one loop and have pretty much never seen this thing before. I will ask my questions along the way as I describe what I am doing.
This factorization theorem seems to go thus...
Let me first list the four axioms that a determinant function follows:
1. ## d (A_1, \cdots, t_kA_k, \cdots, A_n)=t_kd(A_1, \cdots A_k, \cdots, A_n)## for any ##A_k## and ##t_k##
2. ##d(A_1, \cdots A_k + C , \cdots A_n)= d(A_1, \cdots A_k, \cdots A_n) + d(A_1, \cdots C, \cdots A_n)## for any...
Its Good to be Back!
From Resnik, Fundamentals of physics: Consider a particle of mass m, moving along an x-axis and acted on by a net force F(x) that is directed along that axis. The work done on the particle by this force as the particle moves from position ##x_i## to position ##x_f## is given...
I don't need an answer (although I don't have sadly, it's from a test).
I need just a tip on how to start it...
i cannot use Taylor in here (##\ln(x)## is not Taylor function), therefore, its only MVT, but I don't know which point I should try... since I must get the annoying ##\ln(x)##...
I have completed a formal proof of D&K Theorem 6.2.8 Part (ii) ... but I am unsure of whether the proof is correct ... so I would be most grateful if someone could check the proof and point out any errors or shortcomings ...
Theorem 6.2.8 reads as follows:
Attempted Proof of Theorem 6.2.8...
I am reading J. J. Duistermaat and J. A. C. Kolk: Multidimensional Analysis Vol.II Chapter 6: Integration ...
I need help with the proof of Theorem 6.2.8 Part (iii) ...The Definition of Riemann integrable functions with compact support and Theorem 6.2.8 and a brief indication of its proof...
Greetings!
My question is: is it possible to use the green theorem to compute the circulation while in presence of a scalar function ? I know how to solve by parametrising each part but just in case we can go faster? thank you!