Forms Definition and 482 Threads

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The band's name was inspired by the "Secret Chiefs" said to inspire and guide various esoteric and mystical groups of the previous two centuries. Spruance has expressed interest in, and drawn inspiration from, various mystical or occult systems such as Sufism, Kabbalah, Hermeticism and alchemy.

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  1. Joppy

    MHB Differential forms - Reference request

    Hi. Can anyone recommend a text introducing differential forms along with all the necessary pre-requisites for understanding them? For example, I'm not really familiar with tensor calculus but would like to shortcut studying it completely separately to learning differential forms. If that's too...
  2. binbagsss

    Modular Forms, Fundamental Domain Question

    Homework Statement My question is below it makes more sense there, after I have gave my interpretation of the definition of the fundamental domain to confirm my understanding Homework Equations The fundamental domain is defined as: 1) Every point in H is equivalent to a point in F. 2)...
  3. binbagsss

    Hecke Bound for Cusp - Modular Forms

    Homework Statement i have a questions on the piece of lecture notes attached: 2. Homework Equations The Attempt at a Solution [/B] I agree 2) of proposition 2.12 holds once we have 1). I thought I understood the general idea of 1), however, my reasoning would hold for ##M_k## it does not...
  4. Euler2718

    I Clarifying a corollary about Quadratic Forms

    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...
  5. jedishrfu

    I Some Questions on Differential Forms and Their Meaningfulness

    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...
  6. binbagsss

    Product of modular forms: poles, zeros expansion

    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...
  7. P

    A What Is the Purpose of Exterior Forms in Differential Geometry?

    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...
  8. JTC

    A Split the differential and differential forms

    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...
  9. davidge

    I Understanding Black Hole Shapes & Forms

    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...
  10. I

    Different amino acids in different life forms?

    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...
  11. binbagsss

    Modular forms, Hecke Operator, translation property

    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...
  12. nightingale123

    I Why is there a Matrix A that satisfies F(x,y)=<Ax,y>?

    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}##...
  13. Thejas15101998

    I Forms of the Uncertainty Principle

    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...
  14. B

    Equation of states for a gas that forms dimers

    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...
  15. J

    MHB Complex wave forms and fundamentals.... Very very stuck

    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...
  16. P

    I Differential forms as a basis for covariant antisym. tensors

    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 ...
  17. binbagsss

    Modular Forms, Eisenstein series, show it transforms with modular of weight 2

    Homework Statement I need to show that transforms with modular of weight ##2## for ##SL_2(Z)## We have the theorem that it is sufficient to check the generators S and T We have that E_2 is (whilst holomorphic) fails to transform with modular weight ##2## as it has this extra term...
  18. binbagsss

    Modular Forms, Dimension, Valence Formula

    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...
  19. frostysh

    How to destroy a Planet, or 100% of the life forms on it

    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...
  20. davidge

    I Problem when solving example with differential forms

    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...
  21. davidge

    I Integrals: Why Change from M to $\phi(M)$?

    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)}...
  22. davidge

    I Applications of Wedge-Product and Differential Forms

    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...
  23. O

    A Pullback and Pushforward in Manifolds: Why Do We Do It?

    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...
  24. J

    Understanding the Different Forms of the Ideal Gas Law and Their Applications

    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...
  25. R

    I Curvature forms and Riemannian curvatures of connections

    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...
  26. O

    A Understanding Exact vs. Closed Forms for Mechanical Engineers

    (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...
  27. O

    A The meaning of an integral of a one-form

    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...
  28. C

    I Proofs of Stokes Theorem without Differential Forms

    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...
  29. S

    I Different forms of energy density in inflation

    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 =...
  30. binbagsss

    Computing representation number quad forms

    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...
  31. binbagsss

    I Congruence Subgroups and Modular Forms Concept Questions

    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...
  32. binbagsss

    I Eistenstein series E_k(t=0) quick q? Modular forms

    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## ...
  33. binbagsss

    A Representation number via quad forms of theta quadratic form

    ##\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...
  34. chwala

    Expressing Quadratic Equations in Different Forms

    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...
  35. binbagsss

    A Modular forms, dimension and basis confusion, weight mod 1

    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...
  36. binbagsss

    Discriminant function and paritition function - modular forms - algebra really

    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 : ##...
  37. K

    I Differential Forms in General Relativity: Definition & Use

    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?
  38. H

    B Gravity & Wave Forms: Interactions Explained

    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
  39. binbagsss

    A Modular Forms: Non-holomorphic Eisenstein Series E2 identity

    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)} ##...
  40. K

    I Differential Forms in GR: Higher Order Derivatives

    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...
  41. S

    Helicity integral in differential forms

    Homework Statement Let ##V^{3}(t)## be a compact region moving with the fluid. Assume that at ##t=0## the vorticity ##2##-form ##\omega^{2}## vanishes when restricted to the boundary ##\partial V^{3}(0)##; that is, ##i^{*}\omega^{2}=0##, where ##i## is the inclusion of ##\partial V## in...
  42. S

    Euler's equations in differential forms

    Homework Statement Euler's equations can be written using vector calculus as ##\displaystyle{\frac{\partial v_{i}}{\partial t}+v^{j}\left(\frac{\partial v_{i}}{\partial x^{j}}\right) = -\left(\frac{1}{\rho}\right)\frac{\partial p}{\partial x^{i}}+f_{i}}.## Euler's equations can also be...
  43. S

    Electromagnetic action in differential forms

    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...
  44. S

    A Interior products, exterior derivatives and one forms

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

    A Differential forms and vector calculus

    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...
  46. S

    A Line integrals of differential forms

    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...
  47. Mr Davis 97

    T/F: Subset of a spanning set always forms a basis

    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...
  48. K

    I Differential Forms: Definition & Antisymmetric Tensor

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

    Different forms of Bernoulli's equation

    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} +...
  50. Z

    Article in "The physics teacher" "Only two forms of energy"

    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
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