Forms Definition and 482 Threads

I'm studying physics at university, but there has always been one subject in pure maths which always interested me- that is modular forms. Is there an 'accessible' textbook on this topic? Can anyone recommend one? Is the GTM Springer 'A first course in modular forms' any good?

Zero does not have an inverse. And y=0 does not have an inverse. Does the wave form y=0 for all x have an inverse?

Partition each closed interval ##[a_i,b_i]## in the Cartesian product, ##A##. Denote the partition for the i-th closed interval as ##\{x_i^1,\ldots,x_i^{k_i}\}##. The Cartesian product of the partitions forms a partition of ##A## (think: a lattice of points that coincide with the points of each...

The last formula is what I was going for, since it arises as the momentum flux in fluid dynamics, but in the process I came across the rest of these formulas which I’m not sure about. The second equation is missing a minus sign (I meant to put [dA X grad(f)]). Are they correct? Do they have...

The equality is implied in the move from equation 15.43 line 2 to line 3. I do find Dray's book is admirably clear and absolutely says something I wish to understand, but my 78 year old brain has difficulty. However, in this case I can be precise about where I fail to follow. Oh! I find...

Any suggestions? Online courses or videos?

Why are n-forms called differential forms? What is differential about them? And why was the dx notation adopted for them? It must have something to do with the differential dx in calculus. But dx in calculus is an infinitesimal quantity. I don't see what n-forms have to do with infinitesimal...

Attempt of a solution. By the Rank–nullity theorem, $$ \dim V=\dim Im_{F}+\dim\ker\left(F\right) \Rightarrow n=1+\dim\ker\left(F\right) \Rightarrow \dim\ker\left(F\right)=n-1. $$ Similarly, it follows that $$\dim\ker\left(G\right)=n-1.$$ This first part, for obvious reasons, is very clear. The...

i) Let ##\pi : E \rightarrow M## be a vector bundle with a connection ##D## and let ##D'## be the gauge transform of ##D## given by ##D_v's = gD_v(g^{-1}s)##. Show that the exterior covariant derivative of ##E##-valued forms ##\eta## transforms like ##d_{D'} \eta = gd_D(g^{-1}\eta)##. ii) Show...

Hi this is my first message in this forum , I have this problem in my linear algebra course and I have never seen this type. Let $T : \mathbb{Q}^3 → \mathbb{Q}^3 $ a linear application s.t $(T^7 + 2I)(T^2 + 3T + 2I)^2 = 0$ Find all possible Jordan forms and the relative characteristic...

Hi guys i have this problem in my linear algebra curse . let $T:\mathbb{Q}^3→\mathbb{Q}^3$ a linear application s.t $(T^7+2I)(T^2+3T+2I)^2=0$ can you find all possible Jordan forms of T and related characteristic polynomial ? I am totally lost and that is the first time i see this type of problem

My book never references the Biot-Savart law with in terms of 1/r^3, only 1/r^2. How is it that it can be expressed as 1/r^3 as well

I've been noodling around with derivations of the relativistic energy and momentum, and I almost got it down to just a few lines. But not quite. I'm going to work in one spatial dimension, for simplicity (even though some derivations require a second spatial dimension) Let's assume that there...

First time looking at differential forms. What is the difference of the forms over R^n and on manifolds? Does the exterior product and derivative have different properties? (Is it possible to exaplain this difference without using the tangent space?)

from flask import Flask, render_template from forms import RegistrationForm @app.route("/register", methods = ['POST', 'GET']) def register(): forms = RegistrationForm() return render_template('register.html',form=forms) from wtforms import Form, BooleanField, StringField...

I had always known about Pangaea, but that always begged the question of what was before; since there didn't seem to be anything before, I just presumed that the crust was monolithic (i.e., as a general adjective, not "one rock", even though that it was it is, LOL), and under the sea, and a...

Going out to my car this morning, I noticed that frost preferentially formed on non-vertical windows. I can come up with three explanations 1) non-vertical windows are dirtier, providing more nucleation points 2) horizontal vs vertical boundary layer effect 3) gravity driven vertical diffusion...

Good Morning To cut the chase, what is the dx in an integral? I understand that d/dx is an "operator" on a function; and that one should never split, say, df, from dx in df/dx That said, I have seen it in an integral, specifically for calculating work. I do understand the idea of...

ok I am having a problem with getting permission to open up this 307 homework assignment i am logged into my Google account but can't seem to open the form here im sure here are using Google forms with the Google classroom app the class actually starts on jan 11 at UH west mahalo much

I am confused as to how exactly we integrate differential forms. I know how to integrate them in the sense that I can perform the computations and I can prove statements, but I don't understand how it makes sense. Let's integrate a 1-form over a curve for example: Let ##M## be a smooth...

I am test my knowledge of differential forms and obviously I am missing something because I can't figure out where I am going wrong here: Let ##C## denote the positively oriented half-circle of radius ##r## parametrized by ##(x,y) = (r \cos t, r \sin t)## for ##t \in (0, \pi)##. The value of...

Hello all, Apologies if this has already been asked before, but I tried researching this question for a while with no results. I was reading Grainger's Power System Analysis' derivation of the inductance of a single wire and got confused by his definition of magnetic flux linkage. He seems to...

In Resnick halliday book during finding capacitance of isolated sphere they used equation of spherical capacitor[4πε₀(ab)/b-a,where a is inner radius and b is outer radius.] And took b common and equation becames 4πε₀(a)(1-a/b) and then they put radius of outer sphere infinity and then a/b...

There are a few different textbooks out there on differential geometry geared towards physics applications and also theoretical physics books which use a geometric approach. Yet they use different approaches sometimes. For example kip thrones book “modern classical physics” uses a tensor...

Supposing the total mass of a stationary, non rotating Neutron Star is just one Kg below the mass required to form a black hole. Based on the wiki reference below the Schwarzschild radius must be just beneath the surface of the Neutron Star sphere. Now supposing an object with a mass of one Kg...

In college I learned Maxwell's equations in the integral form, and I've never been perfectly clear on where the differential forms came from. For example, using \int _{S} and \int _{V} as surface and volume integrals respectively and \Sigma q as the total charge enclosed in the given...

I am self-studying GR, using principally Carroll’s textbook and Alex Maloney’s online lectures, and nice book by a guy called Herbert Roseman. I am a bit confused by alternative ways of expressing the metric and it would be most helpful if someone could clarify J Basically, I am perplexed by...

I've been reading up on how nucleobases have both keto and enol forms, and how their enol forms can lead to point mutations in DNA replication. 1. What would cause a nucleobase to switch between keto and enol forms? 2. What happens to enol forms of nucleobases in the cell, are they used...

Hello! I was not quite sure about posting in this category, but I think my question fits here. I am wondering about Maxwell equations in vacuum written with differential forms, namely: \begin{equation} \label{pippo} dF = 0 \qquad d \star F = 0 \end{equation} I know ##F## is a 2-form, and It...

So I heard a k-form is an object (function of k vectors) integrated over a k-dimensional region to yield a number. Well what about integrals like pressure (0-form?)over a surface to yield a vector? Or the integral of gradient (1-form) over a volume to yield a vector? In particular I’m...

I have the following molecule: benzotriazole. It has pka=12. At pH=6 which form prevails? a) neutral b) anionic c) cationic d) none of the above. I thought the correct answer was c) cationic: pH<pka, benzotriazole has basic functional groups (amines).

I've been interested in relativistic spacecraft since news of the Breakthrough Starshot project announcement a few years ago. Breakthrough Starshot's method of laser propulsion still has many technical hurdles needed to be crossed. So I'm wondering what you guys think the first forms of...

Hey! :o I am looking at the following: Use the Quine-McCluskey method to determine the respective prime implicants for the following boolean functions and find a disjunctive minimal form. If available, also give all others disjunctive minimal forms. \begin{equation*}f(x_1, x_2, x_3...

nmh{1000} Suppose that A is a matrix whose characteristic polynomial is $$(\lambda-2)^2(\lambda+1)^2$$ find all possible Jordan Normal Forms of A (up to permutation of the Jordan blocks).ok i have been looking at examples so pretty fuzzy on this for the roots are 2 and -1so my first stab at...

I am reading Andrew Browder's book: "Mathematical Analysis: An Introduction" ... ... I am reading Chapter 13: Differential Forms ... ... and am currently focused on Section 13.1 Tensor Fields ... I need some help in order to fully understand some statements by Browder in Section 13.1 ... ...

Andrew Browder in his book: "Mathematical Analysis: An Introduction" ... ... defines a differential form in Section 13.1 which reads as follows: In the above text from Browder we read the following: " ... ... A differential form of degree ##r## (or briefly an ##r##-form) in ##U## is a map...

I am reading the book: Multivariable Mathematics by Theodore Shifrin ... and am focused on Chapter 8, Section 2, Differential Forms ... I need some help in order to fully understand some statements of Shifrin at the start of Chapter 8, Section 2 on the dual space ... The relevant text from...

Homework Statement This problem is from V.I Arnold's book Mathematics of Classical Mechanics. Q) Show that every differential 1-form on line is differential of some function Homework Equations The differential of any function is $$df_{x}(\psi): TM_{x} \rightarrow R$$ The Attempt at a Solution...

Hi there all, I'm currently taking a course in Multivariable Calculus at my University and would appreciate any recommendations for a textbook to supplement the lectures with. Thus far the relevant material we've covered in a Single Variable course at around the level of Spivak and some Linear...

In the exercises on differential forms I often find expressions such as $$ \omega = 3xz\;dx - 7y^2z\;dy + 2x^2y\;dz $$ but this is only correct if we're in "flat" space, right? In general, a differential ##1##-form associates a covector with each point of ##M##. If we use some coordinates...

Homework Statement Given the following quadric surfaces: 1. Classify the quadric surface. 2. Find its reduced equation. 3. Find the equation of the axes on which it takes its reduced form. Homework Equations The quadric surfaces are: (1) ##3x^2 + 3y^2 + 3z^2 - 2xz + 2\sqrt{2}(x+z)-2 = 0 ##...

Homework Statement A switching network has 4 inputs and a single output (Z) as shown in the figure below. The output Z is 1 iff the binary number represented by ABCD ( A is the MSB) is an even number greater than 5. Find : a) The standard POS of Z (abbreviated form). b) The standard SOP of...

Here's exercise 1 of chapter 2 in Flanders' book. Let ##L## be an ##n##-dimensional space. For each ##p##-vector ##\alpha\neq0## we let ##M_\alpha## be the subspace of ##L## consisting of all vectors ##\sigma## satisfying ##\alpha\wedge\sigma=0##. Prove that ##\dim(M_\alpha)\leq p##. Prove also...

this is probably a stupid question but for the fundamental domain for SL2(Z), we say the cusp is only at infinity. Compare say to hecke subgroups which are congruence subgroups where we say the equivalence classes are given by the points where the fundamental domain intercepts the real axis as...

nmh{2000} Let A be a matrix with characteristic polynomial $ p_A(t) = (t − 1)^3(t − 5)^2(t − 6)$ (a) List the possible Jordan Canonical forms for A. (b) Suppose all eigenspaces are one dimensional. What is the Jordan form for A in this case?

The question provides the vector field (xy, 2yz, 3zx) and asks me to confirm Stokes' theorem (the vector calc version) but I am trying to use the generalized differential forms version. So, I am trying to integrate \omega = xy\,dx + 2yz\,dy + 3zx\,dz along the following triangular boundary...

Alright, so this might be a stupid question, but nevertheless, I ask. I am to consider whether the quadratic form ## P(x,y) = a x + b y + d xy ## can map the integers onto the integers. So through a change of basis, I re-express this as ## P'(u,v) = Au^2 + Bv^2 ## for rational A and B...

I ran across exercise 2.8.4 in Oneill's Elementary Differential Geometry. It says "Given a frame field ##E_1## and ##E_2## on ##R^2## there is an angle function ##\psi## such that ##E_1=\cos(\psi)U_1+\sin(\psi)U_2##, ##E_2=-\sin(\psi)U_1+\cos(\psi)U2## (where ##U_1##, ##U_2##, ##U_3## are the...

Hello I was playing with maths for magnetic Skyrmions. There is very prominent mathematical construct in there that I would like to understand, but I do not know where to look. It is easiest to state it for simple 2d space. We can define a 1-form: ##\omega=\sqrt{\left| g \right|}...

Homework Statement Problem- Determine if the set of all function y(t) which have period 2pi forms a vector space under operations of function addition and multiplication of a function by a constant. What I know- So I know this involves sin, cos, sec, and csc. Also I know that a vector space...