Forms Definition and 448 Threads

The question comes out of a corollary of this theorem: Let B be a symmetric bilinear form on a vector space, V, over a field \mathbb{F}= \mathbb{R} or \mathbb{F}= \mathbb{C}. Then there exists a basis v_{1},\dots, v_{n} such that B(v_{i},v_{j}) = 0 for i\neq j and such that for all...

I've been trying to get a meaningful understanding of the benefits of using differential forms. I've seen examples of physics formulas that are reduced to a very simple declarative form relative to their tensor counterparts. However to me it just seems like a notation change to implied tensor...

Homework Statement question concerning part c. Homework Equations The question is pretty simple if there is no zero of order ##N## at infinity, such that it does not cancel the pole of ##f(t)## at infinity of order ##N##. In this case it follows that ## f(t) g(t) \in M^{!}_2 ## and so we...

Hello there, I had some questions regarding k-forms. I was looking in the wiki page of differential forms(https://en.wikipedia.org/wiki/Differential_form) and noticed that it was was introduced to perform integration independent of the co-ordinates. I am not clear how? Is this because given a...

In undergraduate dynamics, they do things like this: -------------------- v = ds/dt a = dv/dt Then, from this, they construct: a ds = v dv And they use that to solve some problems. -------------------- Now I have read that it is NOT wise to treat the derivative like a fraction: it obliterates...

Sorry, I'm not sure what is the more appropriate word to use: shape or form. Let's to the question: How do we know what the shape of a given black hole is? I mean, how do we know whether it is spherical or whatever other form it has? Specifically, where do we look on the equations to get this...

Hello, Are the 20 amino acids that are usually referenced when building genetically coded proteins in all of life, and no other amino acids or are these only in humans and animals? I found the sentence below on this website and I wasn't sure what to make of it, is it true that there are...

Homework Statement I am trying to follow the attached solution to show that ##T_{p}f(\tau+1)=T_pf(\tau)## Where ##T_p f(\tau) p^{k-1} f(p\tau) + \frac{1}{p} \sum\limits^{p-1}_{j=0}f(\frac{\tau+j}{p})## Where ##M_k(\Gamma) ## denotes the space of modular forms of weight ##k## (So we know that...

I'm having trouble understanding a step in a proof about bilinear forms Let ## \mathbb{F}:\,\mathbb{R}^{n}\times\mathbb{R}^{n}\to \mathbb{R}## be a bilinear functional. ##x,y\in\mathbb{R}^{n}## ##x=\sum\limits^{n}_{i=0}\,x_{i}e_{i}## ##y=\sum\limits^{n}_{j=0}\;y_{j}e_{j}##...

Well, i came across the so-called both the forms of the uncertainty principle of Quantum Mechanics: the position-momentum form and the energy-time form; but i am not satisfied in one way. Here the trio: position, momentum and energy, all of them have their own operators, but time does not have...

Homework Statement Show that to a first approximation the equation of state of a gas that dimerizes to a small extent is given by, ##\dfrac{PV}{RT} = 1 - \dfrac{K_c}{V}## Where ##K_c## is equilibrium constant for ##A + A \iff A_2## Homework EquationsThe Attempt at a Solution Using virial...

Hi, My teacher tasked me with a complex waveform question, i have looked for some time to find out how to tackle these, but i still do not know where to begin. Any help would be greatly appreciated, not look for an answer just a method. $$i=12sin(40*\pi t) + 4sin(120* \pi t - /3\pi) + 2sin(200...

In a text I am reading (that I unfortunately can't find online) it says: "[...] differential forms should be thought of as the basis of the vector space of totally antisymmetric covariant tensors. Changing the usual basis dx^{\mu_1} \otimes ... \otimes dx^{\mu_n} with dx^{\mu_1} \wedge ...

Homework Statement What is the dimension of ##M_{24}##? Homework Equations attached The Attempt at a Solution [/B] I am confused what the (mod 12) is referring to- is it referring to the ##[k/12]## where the square brackets denote an equivalent class and the ## k \equiv 2## / ##k \notequiv...

I know, this question probably have asked many times, but I unable to post in that closed threads, so... We have a Planet, (let's say of size of the Earth, and that have a population of the Intelligent Creature - IC, on level of evolution and progress something near a mankind current level, so...

Hi was reading about differential forms, when I tried to solve the example given in this pdf https://www.rose-hulman.edu/~bryan/lottamath/difform.pdf. According to it, the answer is that on the image above. But when I tried to solve this same example by following the expression for ##w## given...

Yes, I know that I have already created another thread on this subject before. But, in this one, I would like to ask specifically why should we change from ##M## to ##\phi (M)## in the integral below? $$ \int_M (\partial_\nu w_\mu - \partial_\mu w_\nu) \ dx^\nu \wedge dx^\mu = \int_{\phi (M)}...

Hi everyone. In reading some popular textbooks I noticed that in (maybe) most of GR and SR we don't encounter situations where we can use wedge-product and differential forms. However, these things are presented to us in most of the textbooks. But... if most of the books present them, it means...

In my ignorance, when first learning, I just assumed that one pushed a vector forward to where a form lived and then they ate each other. And I assumed one pulled a form back to where a vector lived (for the same reason). But I see now this is idiotic: for one does the pullback and pushforward...

Hey I was hoping someone could be me a succinct method of knowing what form of the Ideal gas law I need to use and in particular the different R's associated with each form. Form my Thermodynamics class we use PV = nRT Pv = RT PV = mRT Little v being the specific volume (which changes the R...

I'm trying to think of the curvature form of a connection on a tangent frame pricipal bundle as an alternative description of the Riemannian curvature of the connection(see i.e. https://en.wikipedia.org/wiki/Curvature_form) One thing I want to confirm is does a non-vanishing curvature-form...

(I am a mechanical engineer, trying to make up for a poor math education)' I understand that: A CLOSED form is a differential form whose exterior derivative is 0.0. An EXACT form is the exterior derivative of another form. And it stops right there. I am teaching myself differential forms...

So I understand that the integral of a differential form ω over the boundary of some orientable manifold Ω is equal to the integral of its exterior derivative dω over the whole of Ω. And I understand that one can pull back the integral of a 1-form over a line to the line integral between the...

Hello, does anyone have reference to(or care to write out) fully rigorous proof of Stokes theorem which does not reference Differential Forms? I'm reviewing some physics stuff and I want to relearn it. I honestly will never use the higher dimensional version but I still want to see a full proof...

From the second Friedmann equation, $$H^2 = \frac{1}{3M_p^2} \rho \quad (k=0, flat)$$ In warm inflation, radiation is present all the way therefore not requiring proper reheating process, so $$\rho = \rho_\phi + \rho_r \, ; \quad \rho_\phi = inflaton, \, \rho_r = radiation$$ But, $$\rho =...

Homework Statement ## r_{A} (n) = ## number of solutions of ## { \vec{x} \in Z^{m} ; A[\vec{x}] =n} ## where ##A[x]= x^t A x ##, is the associated quadratic from to the matrix ##A##, where here ##A## is positive definite, of rank ##m## and even. (and I think symmetric?) I am solving for the...

1) Defintion : A congruence subgroup of level ##N## is one that contains the principal subgroup at level ##N## which is defined as ## (a b c d) \in SL_2(Z) : a,d\equiv 1 (mod N), b,c \equiv 0 (mod N) ## (apologies ## ( a b c d)## is a 2x2 matrix.) The Hecke group is one such example given by...

I have in my lecture notes that ##E_{k}(t=0) =1 ##, ##E_k (t)## the Eisenstein series given by: ##E_k (t) = 1 - \frac{2k}{B_k} \sum\limits^{\infty}_1 \sigma_{k-1}(n) q^{n} ## ##B_k## Bermouli number ##q^n = e^{ 2 \pi i n t} ## context modular forms. Also have set ##lim t \to i\infty = 0## ...

##\theta(\tau, A) = \sum\limits_{\vec{x}\in Z^{m}} e^{\pi i A[x] \tau } ## ##=\sum\limits^{\infty}_{n=0} r_{A}(n)q^{n} ##, where ## r_{A} = No. [ \vec{x} \in Z^{m} ; A[\vec{x}] =n]## where ##A[x]= x^t A x ##, is the associated quadratic from to the matrix ##A##, where here ##A## is positive...

Homework Statement Express the quadratic equation ##x^2-6x+20## in the different form hence find,## 1. α+β, αβ , α^2+β^2## Homework EquationsThe Attempt at a Solution ## -(α+β)= -6 ⇒α+β= 6, αβ=20## [/B] now where my problem is finding ##α^2+β^2## , i don't have my reference notes here...

Hi, Excuse me this is probably a really stupid question but I ask because I thought that the definition of the dimension of a space is the number of elements in the basis. Now I have a theorem that tells me that ## dim M_{k} = [k/12] + 1 if k\neq 2 (mod 12) =[k/12] if k=2 (mod 12) ## for ## k...

Homework Statement I am wanting to show that ##\Delta (t) = 1/q (\sum\limits^{\infty}_{n=0} p(n)q^{n})^{24} ## where ##\Delta (q) = q \Pi^{\infty}_{n=1} (1-q^{n})^{24} ## is the discriminant function and ##p(n)## is the partition function, Homework Equations Euler's result that : ##...

Some time ago I was looking around the web for the use of differential equations in General Relativity. Then I found a definition (below) of differential forms, but I noted that the definition on my book is different from this one. Could someone tell me if it is right?

Does gravity affect waves such as gamma, xray, radio etc. and how does it interact with other wave forms considering gravity is a wave itself. Respectfully, Pat Hagar

Hi, As part of showing that ##E^*_{2}(-1/t)=t^{2}E^*_{2}(t)## where ##E^*_{2}(t)= - \frac{3}{\pi I am (t) } + E_{2}(t) ## And since I have that ##t^{-2}E_{2}(-1/t)=E_{2}(t)+\frac{12}{2\pi i t} ## I conclude that I need to show that ##\frac{-1}{Im(-1/t)}+\frac{2t}{i} = \frac{-t^{2}}{Im(t)} ##...

The differential form of a function is \partial{f(x^1,...,x^n)}=\frac{\partial{f(x^1,...,x^n)}}{\partial{x^1}}dx^1+...+\frac{\partial{f(x^1,...,x^n)}}{\partial{x^n}}dx^nIs there (especially in General Relativity) differential of higher orders, like \partial^2{f(x^1,...,x^n)}? If so, how is it...

The electromagnetic action can be written in the language of differential forms as ##\displaystyle{S=-\frac{1}{4}\int F\wedge \star F.}## The electromagnetic action can also be written in the language of vector calculus as $$S = \int \frac{1}{2}(E^{2}+B^{2})$$ How can you show the...

If ##\bf{v}## is a vector and ##\alpha## is a ##p##-form, their interior product ##(p-1)##-form ##i_{\bf{v}}\alpha## is defined...

Let ##0##-form ##f =## function ##f## ##1##-form ##\alpha^{1} =## covariant expression for a vector ##\bf{A}## Then consider the following dictionary of symbolic identifications of expressions expressed in the language of differential forms on a manifold and expressions expressed in the...

Consider a curve ##C:{\bf{x}}={\bf{F}}(t)##, for ##a\leq t \leq b##, in ##\mathbb{R}^{3}## (with ##x## any coordinates). oriented so that ##\displaystyle{\frac{d}{dt}}## defines the positive orientation in ##U=\mathbb{R}^{1}##. If ##\alpha^{1}=a_{1}dx^{1}+a_{2}dx^{2}+a_{3}dx^{3}## is a...

Homework Statement T/F: If a finite set of vectors spans a vector space, then some subset of the vectors is a basis. Homework EquationsThe Attempt at a Solution It seems that the answer is true, due to the "Spanning Set Theorem," which says that we are allowed to remove vectors in a spanning...

Why does the definition of a differential form requires a totally antisymmetric tensor?

I was reading some textbooks on doppler echo for medicine and came across this version of the bernoulli equation for blood flow in the aorta. $$P_{1} - P_{2}= 1/2 \rho (v{_{2}}^{2}- v{_{1}}^{2}) + \rho \int_{1}^{2} \frac{\overrightarrow{dv}}{dt}\cdot \overrightarrow{ds} +...

Hi I remember reading an article some years back (5?) on a description of energy categorized into either potential or kinetic energy. I think it was an article in "The physics teacher" but can't find it... Anyone remember it? Martin

I am very confused about that in some literature the Maurer Cartan forms for a matrix group is written as ##{\omega _g} = {g^{ - 1}}dg## what is ##dg## here? can anyone give an example explicitly? My best guess is ## dg = \left( {\begin{array}{*{20}{c}} {d{x^{11}}}& \ldots &{d{x^{1m}}}\\...

Homework Statement Are the Gibbs and Boltzmann entropies always equivalent? Homework Equations $$ S=k_{B}ln\Omega $$ [Boltzmann entropy, where ##\Omega## is the number of available microstates $$ S=-k_{B}\sum_{i}p_{i} ln(p_{i}) $$ [Gibbs entropy, where ##p_{i}## is the probability of a...

Hello, I would like to know if anybody here has used the book "Differential Forms and the Geometry of General Relativity" by Tevian Dray and how they found it. Thanks!

I have a certain set of problems (i.e. https://www.math.ucdavis.edu/~kouba/CalcOneDIRECTORY/limcondirectory/LimitConstant.html), where many problems are in an indeterminate form ($\frac{0}{0}$) but if we apply L'Hopital's rule it yields an incorrect answer. Instead, I have to simplify the...

If there is enough gravity and the universe starts to collapse in on itself, would life forms that live during this phase of the universe get younger as time passes and they would need to expend energy to stay old?

so the solar wind is the result of resonance of orbiting planets, and solar cycles too, after this paper [Link to crackpot paper deleted]