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 potential inside the crystal is periodic ##U(\vec{r} + \vec{R}) = U(\vec{r})## for lattice vectors ##\vec{R} = n_i \vec{a}_i##, ##n_i \in \mathbb{Z}## (where the ##\vec{a}_i## are the crystal basis), and Hamiltonian for an electron in the crystal is ##\hat{H} = \left( -\frac{\hbar^2}{2m}...
So i derived the moment of inertia about the axis of symmetry (with height h) and I am confused about the perpendicular axis theorem.
The problem ask to find the moment of inertia perpendicular to axis of symmetry
So the axis about h, i labelled as z, the two axis that are perpendicular to z, i...
1. The factor theorem states that (x-a) is a factor of f(x) if f(a)=0
Therefore, suppose (x+1) is a factor:
f(-1)=3(-1)^3-4(-1)^2-5(-1)+2
f(-1)=0
So, (x+1) is a factor.
3x^3-4x^2-5x+2=(x+1)(3x^2+...)
Expand the RHS = 3x^3+3x^2
Leaving a remainder of -7x^2-5x+2
3x^3-4x^2-5x+2=(x+1)(3x^2-7x+...)...
I thought i understood the theorem below:
i) If A is a matrix in ##M_n(k)## and the minimal polynomial of A is irreducible, then ##K = \{p(A): p (x) \in k [x]\}## is a finite field
Then this example came up:
The polynomial ##q(x) = x^2 + 1## is irreducible over the real numbers and the matrix...
We work with Maxwell's equations in the frequency domain.
Let's consider a bounded open domain ## V ## with boundary ## \partial V ##.
1. The equivalence theorem tells me that if the field sources in ## V ## are assigned and if the fields in the points of ## \partial V ## are assigned, then I...
Hi,
I just had a quick question about conventions in potential flow theory:
Question: What is the convention for ## \Gamma ## for the streamline ## \Psi = \frac{\Gamma}{2\pi} ln(\frac{r}{a} ) ## and how can we interpret the Kutta-Jukowski Theorem ## Lift = - \rho U \Gamma ##?
Approach:
For the...
Homework Statement:: What is the convention for ## \Gamma ## for the streamline ## \Psi = \frac{\Gamma}{2\pi} ln(\frac{r}{a} ) ## and how can we interpret the Kutta-Jukowski Theorem?
Relevant Equations:: ## v_{\theta} = - \frac{\partial \Psi}{\partial \theta} ##
[Mentor Note -- moved from the...
Hi,
I just had a quick question about a step in the method of calculating the surface integral and why it is valid. I have already done the divergence step and it yields the correct result.
Method:
Let us calculate the normal: ## \nabla (z + x^2 + y^2 - 3) = (2x, 2y, 1) ##. Just to double...
Hi,
I was trying to gain an understanding of a proof of the divergence theorem in curvilinear coordinates. I have found these online notes here and am looking at the proof on pages 4-5. The method intuitively makes sense to me as opposed to other proofs which fiddle around with vector...
Hi,
I've a doubt about the applicability of the substitution theorem in circuit theory.
Consider the following picture (sorry for the Italian inside it :frown: )
As far I can understand the substitution theorem can be applied to a given one-port element attached to a port (a port consists of...
Hello, there. A friend asked me a problem last night.
Suppose that a system consists of a rod of length ##l## and mass ##m##, and a disk of radius ##R##. The mass of the disk is negligible. Now the system is rotating around an axis in the center of the disk and perpendicular to the plane where...
Summary:: I'm reading Adkins' book "Algebra. An approach via Module Theory" and I'm trying to prove theorem 3.15
In theorem 3.15 of Adkins' book says:
Let ##N \triangleleft G##. The 1-1 correspondence ##H \mapsto H/N## has the property
$$H_1 \subseteq H_2 \Longleftrightarrow H_1/N \subseteq...
Good day all
my question is the following
Is it correct to (after calculation the new field which is the curl of the old one)to use the divergence theroem on the volume shown on that picture?
The divergence theorem should be applied on a closed surface , can I consider this as closed?
Thanks...
Here is the video in question:
In the video at 4:23, Michael says, "Now Thales's Theorem tells us that the two other points, where these rays contact the circumference, are diametrically opposed. They are on opposite sides of the circle and a line passing through them will pass through the...
My main issue with this question is the manipulation of the two arbitrary fields into a single one which can then be substituted into the divergence theorem and worked through to the given algebraic forms.
My attempt:
$$ ∇(ab) = a∇b + b∇a $$
Subsituting into the Eq. gives $$ \int dS ·...
Hi,
I was just working on a homework problem where the first part is about proving some formula related to Stokes' Theorem. If we have a vector \vec a = U \vec b , where \vec b is a constant vector, then we can get from Stokes' theorem to the following:
\iint_S U \vec{dS} = \iiint_V \nabla...
I've tried a few ways of solving this, both directly and by using Stokes' Theorem. I may be messing up what the surface is in the first place
F= r x (i + j+ k) = (y-z, z-x, x-y)
Idea 1: Solve directly. So ∇ x F = (-2,-2,-2). I was unsure on which surface I could use for the normal vector...
If we consider a system of fixed mass as well as a control volume which is free to move and deform, then Reynolds transport theorem says that for any extensive property ##B_{S}## of that system (e.g. momentum, angular momentum, energy, etc.) then$$\frac{dB_{S}}{dt} = \frac{d}{dt} \int_{CV} \beta...
Dear all,
While simulating a coupled harmonic oscillator system, I encountered some puzzling results which I haven't been able to resolve. I was wondering if there is bug in my simulation or if I am interpreting results incorrectly.
1) In first case, take a simple harmonic oscillator system...
fig one:
I just want to know if i am right in attack this problem by this integral:
*pi
Anyway, i saw this solution:
In which it cut beta, don't know why.
So i don't know.
$f: \mathbb{R^2} \rightarrow \mathbb{R}$, $f(x,y) = x^2+y^2-1$
$X:= f^{-1} (\{0\})=\{(x,y) \in \mathbb{R^2} | f(x,y)=0\}$
1. Show that $f$ is continuous differentiable.
2. For which $(x,y) \in \mathbb{R^2}$ is the implicit function theorem usable to express $y$ under the condition $f(x,y)=0$...
Hello,
Within Griffith's text - chap 12 section 12.2 page 423 - this is a brief summary of Bell's Theorem and description of Bell's 1964 work.
There is a table on page 423 showing the spin of the electron and positron (from pi meson decay) - these would be in the singlet state, one would be...
https://www.feynmanlectures.caltech.edu/I_19.html
"Suppose we have an object, and we want to find its moment of inertia around some axis. That means we want the inertia needed to carry it by rotation about that axis. Now if we support the object on pivots at the center of mass, so that the...
This is problem 18.3 from QFT for the gifted amateur. I must admit I'm struggling to interpret what this question is asking. Chapter 18 has applied Wick's theorem to calculate vacuum expectation values etc. But, there is nothing to suggest how it applies to a product of operators.
Does the...
I have worked out a FRW domain wall cosmological model in f(G) theory of gravitation. I have received one comment that this model violets Lovelock theorem.
Are there any constraints to cut massive gravitation modes with higher derivative models in gravitational wave GW170817?.
I've been discussing Newton's Shell Theorem re: gravity with someone, and thought of the analogy to charge.
1. I think the net effect on a negative charge inside a hollow sphere of positive charge will be zero. i.e. No net attraction. Yes?
2. But what would happen to the magnetic field if the...
First i tried proving Newton shell theorem directly for r=R and solved the integral as above but still got the wrong solution.
Here i tried using general case:
Here r' is the distance of a small ring from the point particle of mass m
So my doubt is when we take r=R and then evaluate this...
Now I realize this is not the simplest way to do this problem, I get that, so please don't answer me with the "Try doing it this way..." posts. I would like to see if we can please make this solution come to life. The first kink in the proof is the functional equations, I know it should work...
Stokes theorem relates a closed line integral to surface integrals on any arbitrary surface bounded by the same curve. Gauss theorem relates a closed surface integral to the volume integral within a unique volume bounded by the same surface. What causes this asymmetry in these 2 theorems, in the...
I see this in my book but there is something I don't get!
If we consider a Carnot cycle where heat Qh enters and heat Ql leaves,
We know Qh/Ql=Th/Tl
And we define ΔQ_rev then :
∑(ΔQ_rev/T) = (Qh/Th) - (Ql/Tl) =0
I insert an image:
Which shows the heat dQi entering the reservoir at Ti from a...
In Stokes' theorem, the closed line integral of f=the surface integral of curl f on ANY surface bounded by the same curve. But in Gauss' theorem, the surface integral of f on a surface=the volume integral of div f on a unique volume bounded by the surface. A surface can only enclose 1 volume...
According to Basic proportionalit theorem
if a line is parallel to a side of a triangle which intersects the other sides into two distinct points, then the line divides those sides proportionaly.
I can't figure a way out how to prove it.
Here is an attempt.
we know that AE/EB = AD/DC.
Hello,
A generic vector field ##\bf {F} (r)## is fully specified over a finite region of space once we know both its divergence and the curl:
$$\nabla \times \bf{F}= A$$
$$\nabla \cdot \bf{F}= B$$
where ##B## is a scalar field and ##\bf{A}## is a divergence free vector field. The divergence...
I am reading Stephen Willard: General Topology ... ... and am studying Chapter 2: Topological Spaces and am currently focused on Section 3: Fundamental Concepts ... ...
I need help in order to prove Theorem 3.11 Part 1-a using the duality relations between closure and interior ... ..The...
Consider a point A outside of a line α. Α and α define a plane.Let us suppose that more than one lines parallels to α are passing through A. Then these lines are also parallels to each other; wrong because they all have common point A.
I've been slowly grinding away with what I can about quantum mechanics and QFT. I'm not sure how far I've gotten but I've come up against a bit of a roadblock concerning how the relativity of simultaneity applies in QFT with specific reference to the outcome of Bell tests.
My misunderstanding...
I am reading Stephen Willard: General Topology ... ... and am currently reading Chapter 2: Topological Spaces and am currently focused on Section 1: Fundamental Concepts ... ...
I need help in order to fully understand an aspect of the proof of Theorem 3.7 ... ..Theorem 3.7 and its proof...
hi guys
our solid state professor gave us a series of power point slides that contains the derivation of Bloch theorem , but some points is not clear to me , and when i asked him his answer was also not clear :
in the first part i understand the he represented both the potential energy and the...
Hi,
I just have a quick question about a problem involving Gauss' Theorem.
Question: Vector field F = \begin{pmatrix} x^2 \\ 2y^2 \\ 3z \end{pmatrix} has net out flux of 4 \pi for a unit sphere centred at the origin (calculated in earlier part of question). If we are now given a vector...
To my mind, there are two distinct approaches to energy problems that different authors tend to use, and I wondered whether either is more fundamental than the other. The first is variations on the work energy theorem, and the second consists of defining a system boundary and setting the change...
Let ##S_t## be a uniformly expanding hemisphere described by ##x^2+y^2+z^2=(vt)^2, (z\ge0)##
I assume by verify they just want me to calculate this for the surface. I guess that ##\textbf{v}=(x/t,y/t,z/t)## because ##v=\frac{\sqrt{x^2+y^2+z^2}}{t}##. The three terms in the parentheses evaluate...
In deriving the work-energy theorem, Griffiths does the following:
##\frac{d\mathbf{p}}{dt}\cdot\mathbf{u} = \frac{d}{dt}\bigg(\frac{m\mathbf{u}}{\sqrt{1-u^2/c^2}}\bigg)\cdot\mathbf{u}=\frac{m\mathbf{u}}{(1-u^2/c^2)^{3/2}}\cdot\frac{d\mathbf{u}}{dt}##
I may have forgotten something essential...