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.
NO TEMPLATE BECAUSE MOVED FROM ANOTHER FORUM
Hello,
I've been trying to figure out how it works for complicated problems, I know how to use long division, but I'm not understanding how this process is done for a problem like I have.
Instructions: Write the function in the form ƒ(x) = (x -...
Hello
I am struggling with proving theorem 1, pages 98-99, in Spivak's Calculus book: "A function f cannot approach two different limits near a."
I understand the fact that this theorem is correct. I can easily convince myself by drawing a function in a coordinate system and trying to find two...
Homework Statement
Consider the Klein-Gordon equation ##(\partial_\mu \partial^{\mu}+m^2)\varphi(x)=0##. Using Noether's theorem, find a continuity equation of the form ##\partial_\mu j^{\mu}=0##.
Homework Equations
##(\partial_\mu \partial^{\mu}+m^2)\varphi(x)=0##
The Attempt at a Solution...
Hi,
One silly thing is bothering me. As per one lemma, If a, b, and c are positive integers such that gcd(a, b) = 1 and a | bc, then a | c. This is intuitively obvious. i.e.
Since GCD is 1 'a' does not divide 'b'. Now, 'a' divides 'bc' so, 'a' divides 'c'. Proved.
What is bothering me is ...
Can someone please explain the progression of topics I would need to study in order to tackle Noether's Theorem? I keep hearing how important it is and am setting a self-study goal for myself to eventually understand it with enough rigour that I can appreciate it's beauty.
I have a feeling I...
Homework Statement
F(x) = (integral from 1 to x^3) (t^2 - 10)/(t + 1) dt
Evaluate F'(x)
Homework Equations
Using the chain rule
The Attempt at a Solution
Let u = x^3
Then:
[((x^3)^2 - 10) / (x^3 + 1)] ⋅ 3x^2
*step cancelling powers of x from fraction*
= (x^3 - 10)(3x^2)
= 3x^5 - 30x^2
I am...
Hi everybody! I'm currently studying Noether's theorem, but I'm a bit stuck around a stupid line of calculation for the variation of the symmetry. The script of my teacher says (roughly translated from German, equations left as he wrote them):
"V.2. Noether Theorem
How does the action change...
I am reading the book, Basic Abstract Algebra by P.B. Bhattacharya, S.K. Jain, and S.R. Nagpaul ... ... and am currently focused on Chapter 2: Integers, Real Numbers and Complex Numbers ...
I need help with an aspect of the proof of the Fundamental Theorem of Arithmetic in Section 1.3 ... ...
I am reading the book, Basic Abstract Algebra by P.B. Bhattacharya, S.K. Jain, and S.R. Nagpaul ... ... and am currently focused on Chapter 2: Integers, Real Numbers and Complex Numbers ...I need help with an aspect of the proof of the Fundamental Theorem of Arithmetic in Section 1.3 ... ...The...
I am reading Joseph J. Rotman's book: Advanced Modern Algebra (AMA) and I am currently focused on Section 5.1 Prime Ideals and Maximal Ideals ...
I need some help with understanding the proof of Theorem 5.12 ... ...Theorem 5.12 reads as follows:
In the above text Rotman writes the following:"...
I am reading Joseph J. Rotman's book: Advanced Modern Algebra (AMA) and I am currently focused on Section 5.1 Prime Ideals and Maximal Ideals ...
I need some help with understanding the proof of Theorem 5.12 ... ...Theorem 5.12 reads as follows:
In the above text Rotman writes the following:"...
I am reading Joseph J. Rotman's book: Advanced Modern Algebra (AMA) and I am currently focused on Section 5.1 Prime Ideals and Maximal Ideals ...
I need some help with understanding the proof of Proposition 5.9 ... ...Proposition 5.9 reads as follows:
In the proof of Proposition 5.9, Rotman...
I am reading Joseph J. Rotman's book: Advanced Modern Algebra (AMA) and I am currently focused on Section 5.1 Prime Ideals and Maximal Ideals ...
I need some help with understanding the proof of Proposition 5.9 ... ...Proposition 5.9 reads as follows: In the proof of Proposition 5.9, Rotman...
Bell's theorem states that super-luminal communication exists between particles that are separated by space-like separation viz. faster than light transmission of information. There is spontaneity in this. Relativistically this would amount to going back in time. The state of creation of...
Homework Statement
attempt to derive the equation of centripetal acceleration using work energy theorem
Homework Equations
work done = Change in kinetic energy The Attempt at a Solution
consider diametrically opposed points occurring in uniform circular motion - displacement = 2*R and let...
Moment of inertia is supposed to be defined with respect to a rotational axis such that for a system of point masses, I=∑miri2 where ri 's are the perpendicular distances of the particles from the axis.
However, in some derivations of the virial theorem (like the one on wiki), the so-called...
Homework Statement
Let S be the portion of the paraboloid ##z = 4 - x^2 - y^2 ## that lies above the plane ##z = 0## and let ##\vec F = < z-y, x+z, -e^{ xyz }cos y >##. Use Stoke's Theorem to find the surface integral ##\iint_S (\nabla × \vec F) ⋅ \vec n \,dS##.
Homework Equations
##\iint_S...
Let me set up the question briefly. Emmy Noether's theorem relates symmetry to conserved quantities, e.g. invariance under translations in time => conservation of energy. A fundamental truth revealed.
Massive gauge bosons, leptons and quarks all appear to acquire mass through the spontaneous...
So, I know that the gauss law states that the Flux of the electric field through a closed surface is Q/ε , but does the gauss theorem works also for non inverse square law Fields?
I think not because in order to not have a Flux depending on distance but a constant one we need that r^2 of the...
Suppose we have equal and opposite charge densities on a parallel plate capacitor. Let the plates be separated some small distance d (small when compared with the plate size). Now slowly separate the plates so that their separation is now doubled to 2d. We have done work and the electrostatic...
Homework Statement
[/B]
Suppose we have a 2\pi-periodic, integrable function f: \mathbb{R} \rightarrow \mathbb{C} whose Fourier coefficients are known. Parseval's theorem tells us that:
\sum_{n = -\infty}^{\infty}|\widehat{f(n)}|^2 = \frac{1}{2\pi}\int_{-\pi}^{\pi}|f(x)|^{2}dx,
where...
As I understand it Blochs theorem says that the solutions to the one electron Schrödinger equation in a periodic potential has the form:
ψ(r) = exp(i k⋅r)un(r)
, where un(r) has the same periodicity as the lattice and n labels the band number.
Now a detail that confuses me: In a book I am...
I am trying to prove this function theorem:
Let F:X→Y and G:Y→Z be functions. Then
a. If F and G are both 1 – 1 then G∘F is 1 – 1.
b. If F and G are both onto then G∘F is onto.
c. If F and G are both 1 – 1 correspondences then G∘F is a 1 – 1 correspondence.
Part a has already been...
Hello, I know a theorem that say that if ##F : \mathbb{R} \times \Omega \rightarrow E## is continuous and local lispchitziann in is seconde set value(where ##\Omega## is an open of a Banach space E.). we have that the maximum solution ##(\phi, J)##(where J is an open intervall and ##\phi : J...
The proofs of the Fundamental Theorem of Calculus in the textbook I'm reading and those that I have found online, basically show us:
1) That when we apply the definition of the derivative to the integral of f (say F) below, we get f back.
F(x) = \int_a^x f(t) dt
2) That any definite integral...
I am reading T. S. Blyth's book "Module Theory: An Approach to Linear Algebra" ... ... and am currently focussed on Chapter 1: Modules, Vector Spaces and Algebras ... ...
I need help with a basic and possibly simple aspect of Theorem 2.3 ...
Since the answer to my question may depend on...
71. Use the Comparison Theorem to determine weather the integral
$$\displaystyle
I=\int\frac{{x}^{2}}{{x}^{5}+2} \, dx$$
is convergent or divergent.
Comparison Theorem Suppose that $f$ and $g$ are continuous with
$f(x) \ge \, g(x) \ge 0 $ for $x\ge a$
(a) if $\displaystyle \int_{a}^{\infty}...
Homework Statement
Let be ##f \in C^{1}(\mathbb{R}^{n}, \mathbb{R}^{n})## and ##a \in \mathbb{R}^{n}## with ##f(a) = a##. I'm looking for a suffisent and necessar condition on f that for all ##(x_{n})## define by ##f(x_{n}) = x_{n+1}##, then ##(x_{n})## converge.
Homework Equations
##f(a) =...
I am reading T. S. Blyth's book "Module Theory: An Approach to Linear Algebra" ... ... and am currently focussed on Chapter 1: Modules, Vector Spaces and Algebras ... ...
I need help with an aspect of Theorem 1.1 part 4 ...
Theorem 1.1 in Blyth reads as follows:
In the above text, in part 4...
I am reading T. S. Blyth's book "Module Theory: An Approach to Linear Algebra" ... ... and am currently focussed on Chapter 1: Modules, Vector Spaces and Algebras ... ...
I need help with an aspect of Theorem 1.1 part 4 ...
Theorem 1.1 in Blyth reads as follows:In the above text, in part 4 of...
A problem on geometry proof
Hi (Smile),
When considering the \triangle ABM E is the midpoint of AB
& EO //OM (given).I think this is the way to tell AO=OM , Help .Many Thanks (Smile)
According to Kirchoff
$$e=J(T,f)A$$
##e## is the power emitted and ##A## is the power absorbed
If ##E## is the power supplied, can I say that
$$e=E-A$$
Hello. I have a question about a step in the factorization theorem demonstration.
1. Homework Statement
Here is the theorem (begins end of page 1), it is not my course but I have almost the same demonstration : http://math.arizona.edu/~jwatkins/sufficiency.pdf
Screenshot of it:
Homework...
Homework Statement
Homework Equations
Formula for Area of a retangle : A = L x W
Pythagorean theorem: A2 + b2 = c2
The Attempt at a Solution
So I am pretty sure I did it correct but I just want to be 100% certain I will get this right, By the way its a picture cause I found it easier to...
Hi,I have been stuck on this problem
The midpoints of the sides AB and AC of the triangle ABC are P and Q respectively. BQ produced
and the straight line through A drawn parallel to PQ meet at R. Draw a figure with this information
marked on it and prove that, area of ABCR = 8 x area of APQ.
I...
Homework Statement
[/B]
(a) Calculate the load current using Thevenin's Theorem
(b) Calculate the load current using Superposition
Homework Equations
N/A
The Attempt at a Solution
There have already been a couple of historical posts of this question but those threads don't give me any...
This relates to a homework question which I have spent considerable time on and although I understand the concepts, the process of getting to the answer is difficult because of several different 'versions' of the right answer I see.
The relevant threads are...
look figure (b)
suppose that baseball deliver F through horizontal motion.
imagine that the O point of the system is same line of F (+x is F direction)
then before percussion, the angular momentum of the system is "0" because r and v of baseball are same direction (L = r x mv = 0)
so after...
Homework Statement
"Under mild continuity restrictions, it is true that if ##F(x)=\int_a^b g(t,x)dt##,
then ##F'(x)=\int_a^b g_x(t,x)dt##.
Using this fact and the Chain Rule, we can find the derivative of
##F(x)=\int_{a}^{f(x)} g(t,x)dt##
by letting
##G(u,x)=\int_a^u g(t,x)dt##,
where...
(Sorry for my bad English.) I was reading about the Green's theorem and I notice that the book only shows for the case where the function is a vector function. When learning about line integrals, I saw that we can do line integrals using "ordinary" functions. For example, the line integral of...
Let's say I have multiple spin systems (atoms in a protein) in a solution of water and the spin systems are all producing a magnetic field \mathrm{B_{loc}} that affects nearby spin systems.
Will the fluctuation-dispersion theorem apply to the force generated by a spin's magnetic field...
When you have single parameter transformations like this in Noether's Theorem
\begin{array}{l}
{\rm{ }}t' = t + \varepsilon \tau + ...{\rm{ }}\\
{\rm{ }}{q^\mu }^\prime = {q^\mu } + \varepsilon {\psi ^\mu } + ...
\end{array}
The applicable form of the Rund-Trautman Identity is
{\rm{...
The math theorem to be proven
We want to join three given points using any number of straight lines of any length while minimising the total length of the straight lines. Show that this is achieved by using three lines that are 120##^\circ## apart as shown above.
The following is the answer to...
hi everyone, I would like say that there are lots of proofs related to pythagoras theorem in a flat space, but When I searched it's general form I have not found something worthwhile. Besides, I also involved myself to have a nice proof of it, as a result I have not any valuable or very close...
Homework Statement
Find \int_{0}^{\infty} \frac{\cos(\pi x)}{1-4x^2} dx
Homework Equations
The residue theorem
The Attempt at a Solution
The residue of this function at $$x=\pm\frac{1}{2}$$ is zero. Therefore shouldn't the integral be zero, if you take a closed path as a hemisphere in the...
Hello, I have a question about Heine Borel Theorem.
First, I am not sure why we have to show
"gamma=Beta"
gamma is the supremum of F(which is equivalent to H_squiggly_bar in the text ), and it has to be greater than beta. Otherwise, S contains H_squiggly_barSecond, for the case 1, why...
Use the squeeze theorem to show that
$\displaystyle
\lim_{{n}\to{\infty}} \frac{n!}{{n}^{x}}=0 \\
\text{have never used the squeeze theorem } \\
\text{but by observation the denominator is increasing faster}$
"Write out a series of three or more different whole numbers in geometric progression, starting from one, so that the numbers should add up to a square. So like, 1 + 2 + 4 + 8 + 16 + 32 = 63 (one short of a square)"(can't find an actual real life example)
I can't seem to find an answer for this?