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.
I am reading Houshang H. Sohrab's book: "Basic Real Analysis" (Second Edition).
I am focused on Chapter 4: Topology of R and Continuity ... ...
I need help in order to fully understand the proof of Theorem 4.3.4 ... ... Theorem 4.3.4 and its proof read as follows:
In the above proof by...
Stokes' Theorem states that:
$$\int (\nabla \times \mathbf v) \cdot d \mathbf a = \oint \mathbf v \cdot d \mathbf l$$ Now, if for a specific situation, I can work out the RHS and it's equal to zero, does it necessarily mean that ##\nabla \times \mathbf v = 0##? I mean all that tells me is that...
##I_{AB} = I_{GXX} + A.(y^{2})##
Same applies to CD;
##I_{CD} = I_{GYY} + A.(x^{2})##
In the above statement, "any axis in its plane" where does the plane exist in this sketch?
I am reading Houshang H. Sohrab's book: "Basic Real Analysis" (Second Edition).
I am focused on Chapter 4: Topology of R and Continuity ... ...
I need help in order to fully understand the proof of Theorem 4.1.10 ... ... Theorem 4.1.10 and its proof read as follows:
In the above proof by...
To be honest i don't know from where to start. I know how i can test the stokes theorem if i have a cylindrical shape and a cylindrical vector or spherical vector and a spherical shape but here I am out of ideals.
The first thing i tried was to compute the left part of the stokes theorem but i...
State where in the ty-plane the hypotheses of Theorem 2.4.2 are satisfied
$\displaystyle y^\prime= \frac{t-y}{2t+5y}$
ok I don't see how this book answer was derived since not sure how to separate varibles
$2t+5y>0 \textit{ or }2t+5y<0$
I am reading Tom M Apostol's book "Mathematical Analysis" (Second Edition) ...
I am focused on Chapter 3: Elements of Point Set Topology ... ...
I need help in order to fully understand Theorem 3.28 (Lindelof Covering Theorem ... ) .Theorem 3.28 (including its proof) reads as follows:
In the...
I am reading Tom M Apostol's book "Mathematical Analysis" (Second Edition) ...
I am focused on Chapter 3: Elements of Point Set Topology ... ...
I need help in order to fully understand Theorem 3.28 (Lindelof Covering Theorem ... ) .Theorem 3.28 (including its proof) reads as follows:
In...
Hello guys, is it possible to "see" the mean value theorem when one is only thinking of numerical values without visualizing a graph? Perhaps through a real world problem?
This theorem is from the stewart calculus book 11.1.6
If $$ \lim_{n\to\infty} |a_n| = 0$$, then $$\lim_{n\to\infty} a_n = 0$$
I wonder whether converse of this theorem true or not
I am reading Houshang H. Sohrab's book: "Basic Real Analysis" (Second Edition).
I am focused on Chapter 2: Sequences and Series of Real Numbers ... ...
I need help with the proof of Theorem 2.3.9 (a) ...
Theorem 2.3.9 reads as follows:
Now, we can prove Theorem 2.3.9 (a) using the Cauchy...
In my textbook when proving continuity implies uniform continuity (which is very similar to the proof given here), BWT is used to find a converging subsequence. I cannot see why this is needed. Referring to the linked proof, if we open up the inequality ##|x_n-y_n|<\frac{1}{n}##, isn't by the...
I am reading D. J. H. Garling's book: "A Course in Mathematical Analysis: Volume I: Foundations and Elementary Real Analysis ... ...
I am focused on Chapter 3: Convergent Sequences
I need some help to fully understand some remarks by Garling made after the proof of Theorem 3.1.1...
I am reading D. J. H. Garling's book: "A Course in Mathematical Analysis: Volume I: Foundations and Elementary Real Analysis ... ...
I am focused on Chapter 3: Convergent Sequences
I need some help to fully understand some remarks by Garling made after the proof of Theorem 3.1.1...
I am reading D. J. H. Garling's book: "A Course in Mathematical Analysis: Volume I: Foundations and Elementary Real Analysis ... ...
I am focused on Chapter 3: Convergent Sequences
I need some help to fully understand the proof of Theorem 3.1.1 ...Garling's statement and proof of...
Section ##3.8## talks about the gradient and smooth surfaces, defining when the directional derivative ##(\partial f/\partial\mathbf{u})(\mathbf{p})## takes maximum value and that when it equals ##0##, then ##\mathbf{u}## is a unit vector orthogonal to ##(grad\ f)(\mathbf{p})##.It also says that...
I am working with a simulation which generates an arbitrary number ##n## of identical curves with different phases and calculates their (normalized) sum. As expected, the fluctuation depth of the curves decreases as we increase ##n##. Here is an example of my simulation (when ##n>1##, the...
My question is: What is wrong with my working/ method (in the attached pictures) to find i_{sc} ? I can get the Norton equivalent from there, but seem to get the same answer as the solution scheme.
Context: we are given the circuit depicted in the picture (initially with no connection between...
Picture of the circuit is posted below. Apologies, the voltage source on the left should read 24 V. My question is: What is wrong with this method? [Edit: Sorry if it wasn't clear- the method in the picture yields the wrong answer]
When I originally did the question, I just turned the LHS into...
I am reading Andrew McInerney's book: First Steps in Differential Geometry: Riemannian, Contact, Symplectic ... and I am focused on Chapter 3: Advanced Calculus ... and in particular on Section 3.3: Geometric Sets and Subspaces of ##T_p ( \mathbb{R}^n )## ... ...
I need help with...
I am reading Andrew McInerney's book: First Steps in Differential Geometry: Riemannian, Contact, Symplectic ... and I am focused on Chapter 3: Advanced Calculus ... and in particular on Section 3.3: Geometric Sets and Subspaces of ##T_p ( \mathbb{R}^n )## ... ...
I need help with an aspect...
I want to check Stokes' theorem for the following exercise:
Consider the vector field ##\vec F = ye^x \hat i + (x^2 + e^x) \hat j + z^2e^z \hat k##.
A closed curve ##C## lies in the plane ##x + y + z = 3##, oriented counterclockwise. The parametric representation of this curve is defined as...
As far as I can tell the divergence theorem might be one of the most used theorems in physics. I have found it in electrodynamics, fluid mechanics, reactor theory, just to name a few fields... it's literally everywhere. Usually the divergence theorem is used to change a law from integral form to...
I don't understand proof of uniqueness theorem for polynomial factorization, as described in Stewart's "Galois Theory", Theorem 3.16, p. 38.
"For any subfield K of C, factorization of polynomials over K into irreducible polynomials in unique up to constant factors and the order in which the...
Since it is known than the number N of primitive Pythagorean triples up to a given hypotenuse length A is given on average by N = Int(A/(2.pi)) and according to my calculations with primitive triples and A = B + C, I get on average N = Int(0.152 A^2) (for A = 10^3 I get N = 152,095 compare...
I am checking the divergence theorem for the vector field:
$$v = 9y\hat{i} + 9xy\hat{j} -6z\hat{k}$$
The region is inside the cylinder ##x^2 + y^2 = 4## and between ##z = 0## and ##z = x^2 + y^2##
This is my set up for the integral of the derivative (##\nabla \cdot v##) over the region...
From gauss divergence theorem it is known that ##\int_v(\nabla • u)dv=\int_s(u•ds)## but what will be then ##\int_v(\nabla ×u)dv##
Any hint??
The result is given as ##\int_s (ds×u)##
A monic polynomial of degree N has N number of coefficients. The product of N number of linear factors has N number of free terms. A complex number has 2 DOF. Therefore, both a monic polynomial and the product of free terms have 2N number of DOF of real values. Thus, it must be possible to...
Sorry for the misspelling, but this forum doesn't allow enough characters for the title. The title should be:
For the topological proof of the Fundamental Theorem of Algebra, what is the deal when the roots are at the same magnitude, either at different complex angles, or repeated roots?
I...
I am reading Gerard Walschap's book: "Multivariable Calculus and Differential Geometry" and am focused on Chapter 1: Euclidean Space ... ...
I need help with an aspect of the proof of Theorem 1.3.1 ...
The start of Theorem 1.3.1 and its proof read as follows:
I tried to understand how/why...
Homework Statement
Find all $$n \in Z$$, for which $$ (\sqrt 3+i)^n = 2^{n-1} (-1+\sqrt 3 i)$$
Homework Equations
$$ (a+b i)^n = |a+b i|^n e^{i n (\theta + 2 \pi k)} $$
The Attempt at a Solution
First I convert everything to it`s complex exponential form: $$ 2^n e^{i n (\frac {\pi}{3}+ 2\pi...
Hello everybody!
I have a question regarding the first step of the quantistic proof of the Goldstone's theorem. Defining
$$a(t) = \lim_{V \rightarrow +\infty} {\langle \Omega|[Q_v(\vec{x},t),A(\vec{y})]| \Omega \rangle}$$
where ##|\Omega\rangle## is the vacuum state of the Fock space, ##Q_v##...
Hello.
In chapter 3 (Quantum Black Holes) of this book... https://www.amazon.com/dp/069116844X/?tag=pfamazon01-20 ...Stephen Hawking writes...
"The no-hair theorem, proved by the combined work of Israel, Carter, Robinson and myself, shows that the only stationary black holes in the absence of...
Following my instructor's notes the statement of the Uniqueness Theorem(s) are as follows
"If ##\rho_{inside}## and ##\phi_{boundary}## (OR ##\frac{d \phi_{boundary}}{dn}## ) are known then ##\phi_{inside}## is uniquely determined"
A few paragraphs later the notes state
"For the field inside...
Hello! I have been searching the web and textbooks for a certain theorem which generalizes the value of the integral around a infinitesimal contour in the real axis, or also called indented contour over a nth order pole.
It is easy to prove that if the pole is of simple order, the value of the...
Homework Statement
In comparison with the sampling sine wave, in order to reconstruct a square wave, do we need to increase or decrease sampling frequency?
Homework Equations
Aliasing effect
Leakage effect
The Attempt at a Solution
No matter square wave or sine wave, the experimental results...
I am reading Andrew Browder's book: "Mathematical Analysis: An Introduction" ... ...
I am currently reading Chapter 12: Multilinear Algebra and am specifically focused on Section 12.1: Vectors and Tensors ...
I need help in fully understanding Corollary 12.4 to Theorem 12.2 ... ...
Theorem...
How do you prove 1.85 is valid for all closed surface containing the origin? (i.e. the line integral = 4pi for any closed surface including the origin)
Homework Statement
Given: sin(Πx/a)e6Πix/Na
and e2Πi/a(7/N+4)x
can these equations be represented in Bloch form?[/B]
Homework Equations
Given that Bloch form can be represented as:
Ψ(x) = u(x) eikx[/B]
The Attempt at a Solution
sin(Πx/a)eikx w/n = 3
and...
This is the image of theorem from V.I Arnold's Mathematical method of mechanics. I understood the example given in text. But I want to know what is physical meaning of example? Can anybody help?
Hey! :o
We have the system \begin{align*}&x_1=\left (5+x_1^2+x_2^2\right )^{-1} \\ &x_2=\left (x_1+x_2\right )^{\frac{1}{4}}\end{align*} and the set $G=\{\vec{x}\in \mathbb{R}^2: \|\vec{x}-\vec{c}\|_{\infty}\leq 0.2\}$ where $\vec{c}=(0.2,1)^T$.
I want to show with the Banach fixed-point...
One version of Liouville’s Theorem for non-dissipative classical systems, governed by a conserved Hamiltonian, is that the volume in phase space (position-momentum space) of an ensemble of such systems (the volume is the Lebesgue measure of the set of points where the ensemble’s density is...
Homework Statement
Let a and b be integers, and let m be a positive integer. Then a ≡ b (mod m) if and only
if a mod m = b mod m.
Homework EquationsThe Attempt at a Solution
By definition a ≡ b (mod m) => m| (a-b)
mx = a -b => mx + b = a => b = a mod m
b = a - mx => b = m(-x) + a => a = b...
I am wondering if it existes some discret version of the Noether symmetry for potential with discrete symmetry (like $C_n$ ).
The purpose is to describe the possible evolution of the phase space over the time without having to solve equations numerically (since even if the potential may have...
In https://arxiv.org/ftp/arxiv/papers/1409/1409.5158.pdf, the author (Donald A. Graft) concludes that Bell tests cannot refute local realism, because they employ a wrong analysis. He says:
"The quantum joint prediction cannot be recovered in an experiment with separated (marginal) measurements...
Homework Statement
A man owns two old cars, A and B, and has trouble starting them on cold mornings. The probability both will start is 0.1; the probability B starts and A does not is 0.1; the probability that neither starts is 0.4
a) Find the probability that car A will start.
b) Find the...
I'm reading through Crooks's paper:
https://journals.aps.org/pre/abstract/10.1103/PhysRevE.60.2721
as well as a review paper by Mansour et al.:
https://aip.scitation.org/doi/10.1063/1.4986600
trying to figure out their derivation of the fluctuation theorem (section II of both papers). I had a...