Forms Definition and 448 Threads

Hi, I had a silly idea that probably doesn't work, but I thought I'd ask about it anyway. I understand that vectors can be thought of as derivative operators, e.g. \frac{d}{d\lambda} = \frac{dx^\mu}{d\lambda} \partial_\mu, where lambda parametrizes some curve. I also gather that one-forms...

So I am looking for a program that can compile language similar to Latex, metlab, etc. I have tried the latex website, but there are many things to install and download and I am hoping there is something that doesn't require installing 4 different programs to get a compiler... Any help is...

How can I found out in which p-adic fields a quadratic form represent 0? For example in which p-adic fields does the form 3x2+7y2-15z2 represent zero?

Hi I'm not a philosophy student, so please keep your reply simple. I was reading Wikipedia article on Renaissance where I wasn't able to understand some of the statements. Q1: Renaissance philosophy was the period of the history of philosophy in Europe that falls roughly between the...

Not so much a homework problem as a curiosity on my part. I chose to give a presentation recently on undefined numbers. With that, indeterminate's unsurprisingly found their way into my presentation. After reading up on the list of indeterminate forms, I stumbled upon the form 0^\infty and...

I am having a hard time understanding the theory that a collapsed hypergiant forms a gamma ray pulsating black hole. Can someone explain how the em radiation can travel so fast with such energy as to not only escape the event horizon but also do so with such intensity?

In math, differential forms are alternating: dx^dy=-dy^dx. But in physics, we seem to exchange the order freely: dxdy=dydx. What's going on? I am comfortable with an answer that involves tensors, differential geometry, physics, volume forms, etc. In fact, this is really something I should...

Homework Statement Make a change of variable that transforms the quadratic form with no cross-product term: 9x1^2 - 8x1x2 = 3x2^2 Homework Equations A = PDP^-1 Q = y^TDy The Attempt at a Solution I know the answer. This is a question regarding concept. The eigenvalues for...

Do You Experience "Number Forms"? A "number form" is an involuntary chart, of sorts, that pops into some people's minds when they consider things like calendars (months, days), times of day, the alphabet, or even just numbers from 1 to infinity. These "charts" have their elements grouped...

I am having trouble visualizing when a 2 form is exact and have a specific case that I am struggling with at the moment. Any help is welcome. Take an oriented 2 torus and divide it ,using parallel circles, into an even number of tube shaped regions. In each tube, assign a 2-form that fades...

I thought they were unique, but given a fraction a/b, couldn't you always write it as the (decimal representation of a/b without the decimal place)/ 10^(n) ? thanks

In English which if the following, if any, is incorrect and if so, why: having had; having had had; had had; had having had; has had had; having have had had. What tense to you call the last one? Can you say "He had had having had had" ?

P.S. I'm not sure where to post this question, in particular I can't find a number theory forum on the coursework section for textbook problems. Please move this thread to the appropriate forum if this is not where it should belong to. Thanks!

Homework Statement Find the matrix of f relative to Alpha' and Beta'. Alpha' = [(1,0,0), (1,1,0), (2,-1,1)] Beta' = [(-13,9), (10,-7)] The question originally reads that f is a bilinear form. I've found a (correct according to answer key) matrix A that is 3 -4 4 -5 -1 2...

Hi PF, I am currently trying to teach myself the rudiments of differential forms, in particular their application to physics, and there's something I'd like to ask. It seems like diff forms can be used to express all kinds of physics, but the area I haven't been able to figure out is stuff...

Homework Statement Given a real symmetric matrix A, prove that: a) A is positive definite if and only if A = (B^T)B for some real invertible matrix B b) A is positive semidefinite if and only if there exists a (possibly singular) real matrix Q such that A = (Q^T)Q Homework Equations...

Hi all, I posted this awhile back in the homework sections of the forums and received only one reply, which suggested that I post it here instead, though I understand that it belongs in the homework section. The fundamental problem is not this particular exercise but about integration of...

Let's say we want to write down the metric in a uniform field. I see two ways of going about this. Method 1: Straightforward arguments using the equivalence principle and photons in elevators show that if a photon with initial energy E rises or falls by dy, then its energy shift is given...

On page 7 it gives two conditions for a linear function on the space of p-vectors built from a linear function on the underlying L space. I do not understand! Does anybody ? Then it continues by saying that the two properties are an axiomatic characterization on the space of p-vectors. So, if...

Homework Statement Let x_1,...,x_n: M \rightarrow R be functions on a manifold which form a local coordinate system on some region. Show that every differential form on this region can be written uniquely in the form w^k = \sum_{i_1<...<i_k} a_{i_1,...i_k}(\bf{x})dx_{1_i} \wedge .. \wedge...

Homework Statement find all Jordan forms of 8x8 matrices given the minimal polynomial x^2*(x-1)^3 Homework Equations The Attempt at a Solution The roots are clearly 0,1 and 0 has degree 2 while 1 has degree 3. The forms would be made up of the blocks [0,0;1,0] corresponding to 0...

So I am trying to compute all possible Jordan forms over the complex numbers given a minimal polynomial. My question is this: If the roots of the minimal polynomial are both real, should I proceed as if all of the possible forms are over real numbers?

Introduction to smooth manifolds, by John Lee, page 304. The right-hand side of (c) near the top of the page has a factor \omega_I\circ F. I've been doing the calculation over and over for hours now and I keep getting just \omega_I. Is that F supposed to be there? Edit: I should add that...

I read a problem a while ago which was to find a differential form on the circle which is not the differential of any function. Being a hapless physicist, this puzzled me for a while. I've found an answer in Spivak's Calculus on Manifolds, but I need a little help in following his reasoning...

Homework Statement Hi all I can find a differential form defined on R2\{0,0}, which is closed but not exact, but is it possible to find a differential form defined on all R2, which is closed but not exact?

Homework Statement show that \left(\begin{array}{cc}2 & -1\\-1 & 1\end{array}\right) forms a basis for R^2Homework Equations The Attempt at a Solution ok...my instructor said he wants me to show that they are linearly independant and to show that they span to form a basis...not just by a...

Which book/books are a good intro into manifolds? Maybe a book that is both oriented towards a physicist but also includes rigor. How is this book An Introduction to Manifolds by Loring W. Tu In the preface it says one year of real analysis and a semester of abstract algebra would suffice as a...

Homework Statement 48. Lim (cscx- cotx) x-0 52. lim (xe^1/x -x) x-∞ Homework Equations The Attempt at a Solution

Say I'm given a random 3-dimensional vector, pointing from the origin. How can I find the angle it forms with a plane defined by two other vectors?

Let Q: R3 \rightarrow R be the quadratic form given by Q(x) = 2x1x2 + 2x1x3 + 2x2x3 where x = (x1x2x3)t How do I write down the matrix A of the quadratic form Q in the standard matrix E. and how do I find the numeric values for \lambda

Homework Statement Claim: The solution space of a linear homogeneous PDE Lu=0 (where L is a linear operator) forms a "vector space". Proof: Assume Lu=0 and Lv=0 (i.e. have two solutions) (i) By linearity, L(u+v)=Lu+Lv=0 (ii) By linearity, L(au)=a(Lu)=(a)(0)=0 => any linear...

I'm just curious, why, when solving limits, is 1^\infty considered an indeterminate form? Isn't 1 raised to any power equal to 1? Why isn't it so simple?

I'm just wondering: in what field of mathematics are differential forms frequently used by professional mathematicians?

What is the benefit of expressing Maxwell's equation in the language of differential forms? Differential forms seem to be inferior to the language of tensors. Sure you can do fancy things with the exterior derivative and hodge star, but with tensors you can derive those same identities with...

Prove the following result: let G be a compact Lie group, H its closed subgroup and X = G/H. Let T(X) denote the space of G-invariant differential forms on X (e.g. \omega \in T(X) \Leftrightarrow \forall g \in G g^{*}\omega = \omega). Then T(X) is isomorphic to H^{*}(X), de Rham cohomology...

I need a book like Schaum's Outline of Differential Forms (which doesn't exist). One that sets out a few ideas, then beats them into your thick skull with a TONS of exercises and provides fully worked out solutions. Does anyone know of such a book?

I wasnt quite sure where to put this thread. This question occurred to me while I was looking at the group theory of standard model groups so I thought I'd put it here. Anyway, here is my question: One can define the Killing form for a group by taking the trace of two generators. One can...

Hello, I have a question related to the calculation of curvature using exterior differential forms (Misner, pp. 354-363). In all the examples given in the book (i.e. Friedmann, Schwarzschild, pulsating star metrics), the "guess and check" method used to find the connection forms (Eq. (14.31))...

Homework Statement Let the quadratic form F(x,y,z) be given as F(x,y,z) = 2x^2 + 3y^2 + 5z^2 - xy -xz - yz. Find the transitional matrix that would transform this form to a diagonal form. Homework Equations A quadratic form is a second degree polynomial equation in three...

Recently I discovered geometric algebra which looks very exciting. I was wondering if there is any connection between geometric algebra and differential forms? I see that different research groups recommend the use of differential forms (http://www.ee.byu.edu/forms/forms-home.html" ), and...

Homework Statement the set R^(2) with the usual vector addition forms an abelian group. For a belongs to R and x=(x1,x2) belongs to R^(2) we put a *x :=(ax1,0),this defines a scalar multiplication R*R^2 ---R^2 (a,x)---a*x. determine which of the axioms defining a vector space hold for the...

Homework Statement http://img16.imageshack.us/img16/6606/50381320.jpg Homework Equations Please see above picture The Attempt at a Solution I believe for question a) I just need to add up all the matrices and then row reduce to RREF, which gives me: [1,0,0] [0,1,0] or Do I...

(i) if \alpha=\sum_i \alpha_i(x) dx_i \in \Omega^1, \beta=\sum_j \beta_j(x) dx_j then\alpha \wedge \beta = \sum_{i,j} \alpha_i(x) \beta_j(x) dx_i \wdge dx_j \in \Omega^2 NOW THE STEP I DON'T FOLLOW - he jumps to this in the lecture notes: \alpha \wedge \beta = \sum_{i<j} (\alpha_i...

Ok i don't really get how everyone forms isomers, in my class people just move carbons and other things to turn them into branches Is there a specific way to determine the isomer of a compound without guesswork and counting to c if the hydrogens and carbons stay the same? For example how...

I'm trying to study geometric algebra using Artin's book and am having some difficulty with what degenerate symmetric bilinear forms would be like. Does someone know of an example and brief explanation. Also, the opposite being "nondegenerate nonsymmetric bilinear form" would help me out. If I...

I know \lim_{n \rightarrow \infty} (1 + 1/n)^n = \lim_{n \rightarrow \infty} 1^{\infty} , which is an indeterminate form, converging to e in this case. But what if the original sequence is a_n = 1^n . Then as n tends to infinity, the function converges to 1 (because it's constant and the limit...

Hi, I'm trying to solve a problem in David Bachman's Geometric Approach to Differential Forms (teaching myself.) The problem is to integrate the scalar function f(x,y,z) = z^2 over the top half of the unit sphere centered at the origin, parameterized by \phi(r,\theta) = (rcos\theta, rsin\theta...

Can a quadratic form always be diagonalised by a rotation?? Thx in advance

On an n-dimensional vector space an alternating n-form defines a measure. However a measure can be defined on its own right, without mentioning any alternating form. My question is that what condition must a measure satsfy that it can be originated from an alternating multilinear form. I mean an...

Can someone point me to a good comparison of the various forums of temperature measurement? (Thermocouples, Thermistors, IR, etc), I'm interested in finding out more about : relative cost, accuracy, size, output type, any other constraints on the technology.