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.
Homework Statement
No actual problem, thinking about the telescoping series theorem and Grandi's series
For reference Grandi's series S = 1 - 1 + 1 - 1...
Homework Equations
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The telescoping series theorem in my book states that a telescoping series of the form (b1 - b2) + ... + (bn -...
Hello! (Wave)
We say that the space $\Omega$ satisfies the exterior sphere condition at the point $x_0 \in \partial{\Omega}$ if there is a $y \notin \overline{\Omega}$ and a number $R>0$ such that $\overline{\Omega} \cap \overline{B_y(R)}=\{ x_0 \}$.
Let the function $\phi \in...
Homework Statement
Show that Dx∫f(u)du = f(x) Where the integral is evaluated from a to x. (Hint: Do Taylor expansion of f(u) around x).
Homework Equations
None
The Attempt at a Solution
I have
... = Dx(F(u)+C) = Dx(F(x-a)+C) = dxF(x) - dxF(a) = f(x)-f(a). My problem is that it should be...
Homework Statement
Prove that
$$ \lim_{x\to 0} \sqrt{x^3+x^2}\; \sin\left(\frac{\pi}{x}\right) = 0 $$
using Sandwich theorem
Homework Equations
Sandwich Theorem
The Attempt at a Solution
Now we know that sine function takes values between -1 and 1. ## -1 \leqslant...
Hi, this is a newbee question. Does the Fundamental Theorem of Calculus supply a visual (graphical) way of linking a function (F(x)) with its derivative (f(x))? That is, the two-dimensional area under a curve in [a,b] for f(x) is always equals to the one-dimensional distance F(b)-F(a)? If...
Hola, I tried to give a proof of this theorem and then check it against the one given by my book(Fasano, Marmi - Analytical Mechanics); I feel like mine seems reasonable and pretty intuitive, but the one on the book is a bit different and I don't really understand it completely, so I'd like to...
Homework Statement
In the first and second photo , it's stated earlier that the C is the boundary of surface on xy plane , but in the question in the 3rd picture , it's not stated that the C is on which surface , so , how to do this question ?
For ∫F.dr , i am not sure how to get r , coz i am...
Hey guys, I am supposed to solve for voltage and current on R4 using Thevenin's theorem here. Values are following:
U = 100V
R1 = 310
R2 = 610
R3 = 220
R4 = 570
R5 = 200
Now, I know I need to solve for Rth and Vth, so I put the circuit into simulation and figured out voltage on R3 is equal to...
Hello everyone! I am reviewing the derivation of the Virial Theorem from an introductory Astrophysics book (Carroll and Ostlie's) and found a step I couldn't follow. I've attached a photo of the step.
Can anyone explain how Newton's Third Law brings about eqn 2.41? I don't see how that first...
Homework Statement
I understand the proof of the implicit function theorem up to the point in which I have included a photo. This portion serves to prove the familiar equation for the implicit solution f(x,y) of F(x,y,z)=c. My confusion arises between equations 8.1-4 and 8.1-5 when it is stated...
Homework Statement
i can't find the normal vector here . In my book , outwards vector is . (Refer to photo 1 )
The question is in photo 2 , i am aksed to use stoke's theorem to evalutae line integral of vector filed
But , now the problem is i can't express z in terms of y and x . Can anyone...
I learned in Analytical Mechanics: "Emmy Noether's theorem shows that every conserved quantity is due to a symmetry".
The examples I learned where conservation of energy as symmetry in time and conservation of momentum as symmetries in space.
Now I wonder, do universal constants are also due to...
Homework Statement
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##C_\rho## is a semicircle of radius ##\rho## in the upper-half plane.
What is
$$\lim_{\rho\rightarrow 0} \int_{C_{\rho}} \frac{e^{iaz}-e^{ibz}}{z^2} \,dz$$Homework Equations
If ##C## is a closed loop and ##z_1, z_2 ... z_n## are the singular points inside ##C##...
Consider the lagrangian of the real scalar field given by $$\mathcal L = \frac{1}{2} (\partial \phi)^2 - \frac{1}{2} m^2 \phi^2 - \frac{\lambda}{4!} \phi^4$$
Disregarding snail contributions, the only diagram contributing to ## \langle p_4 p_3 | T (\phi(y)^4 \phi(x)^4) | p_1 p_2 \rangle## at...
Homework Statement
Consider the beam shown in (Figure 1) . Suppose that a = 15 in. , b = 8 in. , c = 1 in., and d = 4 in.
Determine the moment of inertia for the beam's cross-sectional area about the x axis...
Homework Statement
Find the solution of the following integral
Homework Equations
The Attempt at a Solution
I applied the above relations getting that
Then I was able to factor the function inside the integral getting that
From here I should be able to get a solution by simply finding the...
1. Homework Statement
i attached the problem statement as an image file
Homework Equations
p(x) = (x-c)q(x) + r
The Attempt at a Solution
i've simplified it down to ((x-1)^114) / (2^114)(x+1). is there a practical way to approach this besides long division? wolfram alpha gave an extremely...
I just suppose the Bell's Ansatz for the result of measurement to be $$A (\theta,\lambda) $$
Now the parameter lambda could be anything :
-a physical quantity like the polarization angle of the incoming photon
-the coordinate of a 'world'
- the whole wavefunction.
...
In the case of the...
I'm having a little trouble starting on this problem. Can someone help? I was trying to solve it out, but I just ended up telling how to draw the center. Attached is the problem:
Hello again, everyone. Have a multivariate calculus question this time around. If anyone can point me in the right direction and help me see where WebAssign finds me wrong, it would be greatly appreciated.
1. Homework Statement
Homework Equations
∫∫ScurlF ⋅ dS = ∫CF ⋅ dr
The Attempt at a...
Homework Statement
I am trying to express ##T(\phi(x1)\Phi(x2)\phi(x3)\Phi(x4)\Phi(x5)\Phi(x6))## in terms of the Feynman propagators ##G_F^{\phi}(x-y)## and ##G_F^{\Phi}(x-y)##
where ##G_F^{\phi}(x-y) =\int \frac{d^{4}k}{(2\pi)^{4}}e^{ik(x-y)} \frac{ih}{-k.k - m^2 -i\epsilon} ##
and...
<Moderator's note: Moved from a technical forum, so homework template missing>
Sorry
i have one question to ask
how to check the v.dl part in this problem
i cannot do this problem as it is too hard to integrate the equation
How did griffith get this long-horrible equation(see the orange...
I am reading Matej Bresar's book, "Introduction to Noncommutative Algebra" and am currently focussed on Chapter 1: Finite Dimensional Division Algebras ... ...
I need help with some aspects of the proof of Theorem 1.4 ... ...
Theorem 1.4 reads as follows:
Questions 1(a) and 1(b)
In...
I am reading Matej Bresar's book, "Introduction to Noncommutative Algebra" and am currently focussed on Chapter 1: Finite Dimensional Division Algebras ... ...
I need help with some aspects of the proof of Theorem 1.4 ... ...
Theorem 1.4 reads as follows:
Questions 1(a) and 1(b)
In the above...
The thread I wanted to post my question on got closed. Recapitulating:
The best (simplest) account I have found to date for the Bell inequality (SPOT stands for Single Photon Orientation Tester):
Imagine that each random sequence that comes out of the SPOT detectors is a coded message. When...
1. The problem statement, all variables and given/
You must push a box up an incline plane (the angle being constant : a), to a person waiting to receive it, who is a distance of h(constant) vertically above you. Though the slope is slippery, there is a small amount of friction with kinetic...
I don't know how to start proving this theorem, so can someone please help? I need to prove that the circumcircles all intersect at a point M. Thank you!
Miquel's Theorem: If triangleABC is any triangle, and points D, E, F are chosen in the interiors of the sides BC, AC, and AB, respectively...
Homework Statement
The moment of inertia for a perpendicular axis through the center of a uniform, thin, rectangular metal sheet with sides a and b is (1/12)M(a2 + b2). What is the moment of inertia if the axis is through a corner?
The answer is given as this was a powerpoint lecture and it...
Homework Statement
Find the remainder of ##4^{87}## in the division by ##17##.
Homework Equations
Fermat's Little Theorem:
If ##p## is prime and ##a## is an integer not divisible by ##p##, then
##a^{p-1} \equiv 1 (\mod \space p)##
or equivalently,
##a^p \equiv a (\mod \space p)##
The...
Hello! I am a bit confused about the first Sylow theorem. So it says that if you have a group of order ##p^mn##, with gcd(n,p)=1, you must have a subgroup H of G of order ##p^m##. So, if I have a group G of order ##p^k##, there is only one subgroup of G of order ##p^k## which is G itself. Does...
Hello! (Wave)
We consider the following problem.
$$Lu=f(x) \text{ in } \Omega \\ u|_{\partial{\Omega}}=0$$
I want to show that if $c(x) \leq -c_0<0$ in $\overline{\Omega}$, then it holds that $\min\{ 0, \frac{\min_{\Omega}f(x)}{-c_0}\}\leq u(x) \leq \max_{\Omega} \{ 0...
Liouville's theorem states that the total time-derivative of the distribution function is zero along a system trajectory in phase-space. Where the system follows a trajectory that satisfies the Hamilton's equations of motion.
I have a hard time getting an inuitive understanding of this...
Homework Statement
Homework Equations
Green's theorem
The Attempt at a Solution
DO I first parametrize? For 1st part, I have 3 parametrizations, which I can then find the normal vector, and use in the integrals?
Hawking area theorem says that area of black hole generally never decrease. Penrose process says that energy can be extracted from black hole. Energy extraction will decrease mass? if yes then if mass is decreased then will area also decrease?
I am confusing things here :(
According to DeMorgan’s theorem (break the bar and change the sign), the complement of ܽa⋅b+c⋅d is a'+b'⋅c'+d' Yet both functions are 1 for ܾܽܿ abcd 1110. How can both a function and its complement be 1 for the same input combination? What’s wrong here?I honestly have no idea. I mean, shouldn't...
I am reading D. J. H. Garling: "A Course in Mathematical Analysis: Volume I Foundations and Elementary Real Analysis ... ...
I am currently focused on Garling's Section 1.7 The Foundation Axiom and the Axiom of Infinity ... ...
I need some help with Theorem 1.7.4 ... and in particular with...
Homework Statement
(See attachment: if it doesn't work, see below for poorer formatting)[/B]
Theorem 6.1 (Centered Formula of Order O(h2)). Assume that f ∈ C^3[a, b] and that x − h, x, x + h ∈ [a, b]. Then (3) f (x) ≈ f (x + h) − f (x − h) 2h . Furthermore, there exists a number c = c(x) ∈ [a...
I am reading D. J. H. Garling: "A Course in Mathematical Analysis: Volume I Foundations and Elementary Real Analysis ... ...At present I am focused on Chapter 1: The Axioms of Set Theory and need some help with Theorem 1.2.2 and its relationship to the Separation Axiom ... ...
The...
I am reading D. J. H. Garling: "A Course in Mathematical Analysis: Volume I Foundations and Elementary Real Analysis" ... ...
At present I am focused on Chapter 1: The Axioms of Set Theory and need some help with Theorem 1.2.2 and its relationship to the Separation Axiom ... ...
The...
Obviously ##\mathbb{R^2}## is convex, that is, any points ##a,b\in\mathbb{R^2}## can be connected with a line segment. In addition, ##f## is differentiable as a composition of two differentiable functions. Thus, the conditions of mean value theorem for vector functions are satisfied. By applying...
The population mean and standard deviation are given below. Find the required probability and determine whether the given sample mean would be considered unusual.
μx̄ = μ = 12,749
σ = 1.2
n = 35
For the given sample n = 35, the probability of a sample mean being less than 12,749 or greater...
I am wondering is someone could comment on a question I have recently answered. I have attached the question and my answer. Apologies for not following the standard procedure of Latex but there are drawings associated with this question. I answered section A and my results are written on the...
Homework Statement
Using paper, pencil and the Virial theorem, calculate the position uncertainty (an estimate of the vibration amplitude) of the H atom in its ground state C-H stretching mode. In more precise language, calculate the bond length uncertainty in a C-H bond due to the C-H...
Homework Statement
Using the maximum power transfer theorem, find the value of R which will result in the maximum power being delivered to R.
Homework Equations
P_max= (V_oc)^2/4(R_th))
R_L=R_th
P_RL=(V_th/(R_th+R_L))^2*R_L
The Attempt at a Solution
I have no clue where to even begin with...
I am having trouble doing this problem from my textbook... and have
no idea how to doit.
1. Homework Statement
I am having trouble doing this problem from my textbook...
Show that the equation x + y - z + cos(xyz) = 0 can be solved for z = g(x,y) near the origin. Find dg/dx and dg/dy
(dg/dx...
I know that for rigid bodies only the work-energy theorem states that the net work done on the body equals the change in kinetic energy of the body since a rigid body has no internal degrees of freedom and hence no other forms of energy such as potential energy. Is there a most generalized form...
Homework Statement
Homework Equations
Parallel axis theorem: Ip = Icm + Md^2
Icm = I = ML²/12 + 2 * mr²
3. The attempt
Ip = Icm + Md^2 ==> wrong
I = Md^2 ==> right
Why don't I need to add "Icm"?
Thanks.
Hello everyone. I need help on proofs. I have to proof the Angle-Angle-Side theorem. Can someone help me with this?
The AAS states : If triangles ABC and DEF are two triangles such that angle ABC is congruent to angle DEF, angle BCA is congruent to angle EFD, and segment AC is congruent to DF...
Homework Statement
I need to accommodate a dashpot in an intentionally simple work-kinetic energy analysis method. For example, for a box being dragged up a ramp via a rope while attached to a spring, I can deal with the work done by gravity, rope tension, spring force, and friction via the...