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 "Complex Analysis for Mathematics and Engineering" by John H. Mathews and Russel W. Howell (M&H) [Fifth Edition] ... ...
I am focused on Section 3.2 The Cauchy Riemann Equations ...
I need help in fully understanding the Proof of Theorem 3.4 ...The start of Theorem 3.4 and its...
Reduced graph states are characterized as follows from page 46 of this paper:
Proposition: Let ##A \subseteq V## be a subset of vertices for a graph ##G = (V,E)## and ##B = V\setminus A## the corresponding complement in ##V##. The reduced state ##\rho_{G}^{A}:= tr_{B}(|G\rangle\langle G|)## is...
Brahmagupta's theorem:
A cyclic quadrilateral is orthodiagonal (diagonals are perpendicular) if and only if the perpendicular to a side from the point of intersection of the diagonals bisects the opposite side.
But I don't understand the first step of the proof for the necessary condition...
Hello,
This term in university I'm taking a second year intro to astrophysics course and my professor talks a lot about different situations and then solves a problem using the virial theorem. The reason I'm confused is because the range of topics that he applies this theorem to vary in many...
I am reading the book: "Theory of Functions of a Complex Variable" by A. I. Markushevich (Part 1) ...
I need some help with an aspect of the proof of Theorem 7.1 ...The statement of Theorem 7.1 reads as follows:
At the start of the above proof by Markushevich we read the following:
"If f(z)...
Show that if ##f## is a shrinking map ##d(f(x),f(y)) < d(x,y)## and ##X## is compact, then ##f## has a unique fixed point.
Hint. Let ##A_n=f^n(X)## and ##A=\cap A_n##. Given ##x\in A##, choose ##x_n## so that ##x=f^{n+1}(x_n)##. If ##a## is the limit of some subsequence of the sequence...
In the chemical engineering text of Smith, VanNess, and Abbott, there is a section on partial molar volume. It states that Gibbs theorem applies to any partial molar property with the exception of volume. Why is volume different? In other words, when evaluating the partial molar volume of a...
first state whether it can be solved using the Master Theorem, and if it can then use that. Otherwise, use the Akra-Bazzi formula.
1. T(n) = 3T([n/3])+n
2. T(n) = T([n/4])+T([n/3])+n
3. T(n) = 2T([n/4])+√n
I read on the Internet that the work done by a (rigid) body = the change in Kinetic energy.
What if I lift a rigid body slowly and vertically by 1 meter above the Earth's surface so that the initial velocity = final velocity =0?
According to the Work Energy theorem as stated on many sites on...
Hello good folks!
I'm stuck trying to solve the problem b). In the theory book examples they are skipping steps and shortly states 'use algebra' and parsevals theorem to rewrite the Fourier series into the answer that is given.
So I've tried to use parsevals theorem but I still can't rewrite...
Verify that $\dfrac{d}{dx}\ln(x)=\dfrac{1}{x}$ <br>
by applying Theorem 7<br>
Theorem 7 states that: If f is a one-to-one differentiable function with inverse $f^{-1}$ and $f'(f^{-1}(a))$ then the inverse function is differtiable at a and <br>
$$\dfrac{dy}{dx}=\dfrac{1}{\dfrac{dx}{dy}}$$<br>...
Veritasium posted a video, featuring a visualization of an "intuitive" explanation of the Intermediate Axis Theorem by Terry Tao, based on centrifugal forces in a rotating frame of reference:
Unfortunately, the animation is just as incomplete, as Tao's original explanation from 2011, and...
Hey guys,
Sorry that it's been a decent amount of time since my last posting on here. Just want to say upfront that I am extremely appreciative of all the support that you all have given me over my last three or four posts. Words cannot express it and I am more than grateful for it all. But, in...
A recent paper analyzing LIGO data suggests - though not conclusively - that the no-hair theorem likely holds, with implication for the information paradox and I was wondering where a confirmed no-hair outcome leaves us with regards going beyond the Standard Model...
Hi!
I am given the lagrangian:
## L = \dot q_1 \dot q_2 - \omega q_1 q_2 ##
(Which corresponds to a 2D harmonic oscillator) And I am given two transformations and I am asked to say if there is a constant of motion associated to each transformation and to find it (if that's the case).
I am...
In the derivation of energy conservation, there is the transformation ##q(t)\rightarrow q'(t)=q(t+\epsilon)##, whose end points are kind of fuzzy. The original path ##q(t)## is only defined from ##t_1## to ##t_2##. If this transformation rule is imposed, ##q'(t_2-\epsilon)=q(t_2)## to...
Hi.
Is the binomial theorem ##(1+x)^n = 1+nx+(n(n-1)/2)x^2 + ….## valid for x replaced by an infinite series such as ##x+x^2+x^3+...## with every x in the formula replaced by the infinite series ?
If so , does the modulus of the infinite series have to be less than one for the series to...
I am reading Charles G. Denlinger's book: "Elements of Real Analysis".
I am focused on Chapter 2: Sequences ... ...
I need help with the proof of Theorem 2.9.6 (a)Theorem 2.9.6 reads as follows:
In the above proof of part (a) we read the following:
" ... \forall \ m, n \in \mathbb{N}, \...
A question about Bell's theorem :
Consider the ##CHSH=AB-AB'+A'B+A'B'##
Then the theorem states : ##-2\leq CHSH\leq 2##
Implying ##|<CHSH>|\leq 2##.
We could repeat the average : ##\langle |\langle CHSH\rangle|\rangle\leq 2##
Now Bell's theorem deals with large numbers average, in order to...
I am reading Tom M Apostol's book "Mathematical Analysis" (Second Edition) ...
I am focused on Chapter 4: Limits and Continuity ... ...
I need help in order to fully understand the example given after Theorem 4.29 ... ... Theorem 4.29 (including its proof) and the following example read as...
I am reading Tom M Apostol's book "Mathematical Analysis" (Second Edition) ...
I am focused on Chapter 4: Limits and Continuity ... ...
I need help in order to fully understand the proof of Theorem 4.25 ... ... Theorem 4.25 (including its proof) reads as follows:
In the above proof by...
Summary: Rudin theorem 1.21
He has said that as t=X/(X+1) then t^n<t<1 then maximum value of t is 1. then in the next part he has given that t^n<t<x. as maximum value of t is less than 1 why has he given that t<x ?
Suppose a wave function is a linear combination of 2 stationary states: ##\psi(x)=c_1\psi_1(x)+c_2\psi_2(x)##.
By [5.52] and [5.53], we have ##\psi(x+a)=e^{iK_1a}c_1\psi_1(x)+e^{iK_2a}c_2\psi_2(x)##. But to prove [5.49], we need ##K_1=K_2##. That means all the eigenvalues of the "displacement"...
I am reading the book: Complex Analysis: A First Course with Applications (Third Edition) by Dennis G. Zill and Patrick D. Shanahan ...
I need some help with an aspect of the proof of Theorem 3.1.1 (also named Theorem A1 and proved in Appendix 1) ...
The statement of Theorem 3.1.1 (A1) reads...
At the end of appendix C (concerning the non-interaction theorem of classical relativistic hamiltonian systems) of the book "Classical Relativistic Many-Body Dynamics" by Trump and Schieve it is stated that"It follows that Currie's equation (C.21), and subsequently the assertion of vanishing...
count(ℝ) > count(ℚ) ; count(ℚ) == count(ℕ)
But still in-between any members of ℝ, we are quarantine to find element of ℚ
Can someone help me understand: were are these members of ℝ we cannot account for?
For reference: https://en.wikipedia.org/wiki/Rational_number
"The rationals are a dense...
i know its pretty basic but please give some insight for
triangle law of vector addition and pythgoras theorem.
becuase ofcourse if you use traingle law to find resultant it will be different from what is pythagoras theorem
I am reading Karl R. Stromberg's book: "An Introduction to Classical Real Analysis". ... ...
I am focused on Chapter 3: Limits and Continuity ... ...
I need help in order to fully understand the proof of Theorem 3.55 on page 110 ... ...Theorem 3.55 and its proof read as follows:
At the start...
I am reading Karl R. Stromberg's book: "An Introduction to Classical Real Analysis". ... ...
I am focused on Chapter 3: Limits and Continuity ... ...
I need help in order to fully understand the proof of Theorem 3.55 on page 110 ... ... Theorem 3.55 and its proof read as follows:
At the...
Hi to all!
The ordinary Gauss theorem states that ##\Phi\left(\vec{E}\right)\,=\, \frac{\sum_{i=1}^{n}q_{i}}{\varepsilon_{0}}## where ##\sum_{i=1}^{n}q_{i}## is the sum of all charges internal of a closed surface and ##\varepsilon_{0}## is the dielectric constant in the empty. Now I ask to the...
I am reading Karl R. Stromberg's book: "An Introduction to Classical Real Analysis". ... ...
I am focused on Chapter 3: Limits and Continuity ... ...
I need further help in order to fully understand the proof of Theorem 3.47 on page 107 ... ... Theorem 3.47 and its proof read as follows:
In...
I am reading Karl R. Stromberg's book: "An Introduction to Classical Real Analysis". ... ...
I am focused on Chapter 3: Limits and Continuity ... ...
I need help in order to fully understand the proof of Theorem 3.47 on page 107 ... ... Theorem 3.47 and its proof read as follows:
In the...
I am reading Karl R. Stromberg's book: "An Introduction to Classical Real Analysis". ... ...
I am focused on Chapter 3: Limits and Continuity ... ...
I need further help in order to fully understand the proof of Theorem 3.43 on pages 105-106 ... ... Theorem 3.43 and its proof read as follows...
Equivalent Statements to Compactness ... Stromberg, Theorem 3.43 ... ...
I am reading Karl R. Stromberg's book: "An Introduction to Classical Real Analysis". ... ...
I am focused on Chapter 3: Limits and Continuity ... ...
I need help in order to fully understand the proof of Theorem 3.43 on...
I am reading Karl R. Stromberg's book: "An Introduction to Classical Real Analysis". ... ...
I am focused on Chapter 3: Limits and Continuity ... ...
I need help in order to fully understand the proof of Theorem 3.40 on page 104 ... ... Theorem 3.40 and its proof read as follows:
In the...
I am reading Karl R. Stromberg's book: "An Introduction to Classical Real Analysis". ... ...
I am focused on Chapter 3: Limits and Continuity ... ...
I need help in order to fully understand the proof of Theorem 3.36 on page 102 ... ... Theorem 3.36 and its proof read as follows:
In the...
I am reading Karl R. Stromberg's book: "An Introduction to Classical Real Analysis". ... ...
I am focused on Chapter 3: Limits and Continuity ... ...
I need help in order to fully understand the proof of Theorem 3.18 on pages 98-99 ... ... Theorem 3.18 and its proof read as follows:
In the...
I recently found the centre of mass of a semicircle using polar coordinates, by first finding the centre of mass of a sector, and then summing all the sectors from 0 to pi to get the centre of mass of the semicircle. However, being a beginner at integrals, I struggled for a long time getting the...
I am reading Karl R. Stromberg's book: "An Introduction to Classical Real Analysis". ... ...
I am focused on Chapter 3: Limits and Continuity ... ...
I need help in order to fully understand the proof of Theorem 3.6 on page 94 ... ... Theorem 3.6 and its proof read as follows:
In the above...
is there a rigorous version of this proof of fundamental theorem of calculus?if yes,what is it?and who came up with it?
i sort of knew this short proof of the fundamental theorem of calculus since a long while...but never actually saw it anywhere in books or any name associated with it.
i know...
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...
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?