Forms Definition and 448 Threads

Homework Statement Okay, I'm reopening this question because I didn't understand it as well as I thought I did. "a) Write r-hatc and phi-hat in hybrid component form ( )i + ( )j where the parentheses represent polar coordinate expressions. b) Now use these hybrid expressions to take...

Hi, All: I am trying to see how to classify all symmetric bilinear forms B on R^3 as a V.Space over the reals. My idea is to use the standard basis for R^3 , then use the matrix representation M =x^T.M.y . Then, since M is, by assumption, symmetric, we can diagonalize M...

------------------------------------------------------------------------- SUBJECT There are various forms of energy; some are potential while others are kinetic. An object or particle can have its energy change when work is performed upon it. Likewise, it may loss some of its energy when...

Hi guys! I am reading a paper which uses closed forms \omega on a p-dimensional closed submanifold \Sigma of a larger manifold M. When we integrate \omega we get a number Q(\Sigma) =\int _{\Sigma}\omega which, in principle, depends on the choice of \Sigma but because \omega is closed...

I have a matrix D (it happens to be in R^(nxm) where n>>m, but I don't think that is relevant at this point). I also have a vector t in R^n. I am interested in rewriting the set {x | (Dx-t)'(Dx-t) <= c} in standard ellipsoid form: --> {x | (x-z)'E(x-z)<=b} where E is an mxm positive...

Hello, i just came accros: Sum(i) , from i=1 to i=n which apparently equals n(n+1)/2 -Is there a way to derive this from the sum, or you just have to use your intuition and think through what exactly is being summed and the range of summation? -Do you have any resources to offer, that...

Hi, All: Given a quadratic form Q(x,y) over a field of characteristic different from 2, we can find the bilinear form B(x,y) associated with Q by using the formula: (0.5)[Q(x+y)-Q(x)-Q(y)]=B(x,y). I know there is a whole theory about what happens when we work over fields...

Even though life on Earth spans such diverse creatures from bacteria to humans, at its core, all life on Earth is pretty much the same. Life as we know it uses the same four bases to store information in DNA, the same 20 amino acids to build proteins, and the same genetic code to convert DNA...

What I answered is in brackets...what did I answer incorrectly? It is impossible to construct a device that, operating in a cycle, will produce no effect other than the extraction of heat from a reservoir and the performance of an equivalent amount of work. [True] For all naturally...

I have been going through my equations and writing them up on the computer so I can refer to that when needed and have go to angular acceleration, velocity and displacement equations yet I don't have very many equations for those topics and I wondered if anyone had some equations for finding...

I'm reading Flanders' Differential Forms with Applications to the Physical Sciences and I have some issues with problems 2 and 3 in chapter 3, which appear to ask the reader to compute the pullback a mapping from X to Y applied to a form over X, and I'm not sure how to interpret such a thing...

It is a well known fact that a symmetric bilinear form B on a finite-dimensional vector space V over any field F of characteristic not 2 is diagonalisable, i.e. there exists a basis \{e_i\} such that B(e_i,e_j)=0 for i\neq j. Does the same hold over an infinite dimensional vector space...

Lee 2003: Introduction to Smooth Manifolds ( http://books.google.co.uk/books?id=eqfgZtjQceYC&printsec=frontcover#v=onepage&q&f=false ) (search eg. for "computational"), Lemma 12.10 (b), p. 304: where I is an increasing multi-index: (i_1,...i_k) with each value less than or equal to all those...

LIMx\rightarrowinfinite (bn+an)1/n it is given that 0<a<b in this case why do we take bn or an common to make the inside element 0...and how do u guys take to solve this kind of situation how do u think?

*Bit of reading involved here, worth it if you have any interest in, or knowledge of, differential forms*. It took me quite a while to find a good explanation of differential forms & I finally found something that made sense, in a sense. Most of what I've written below is just asking you...

Homework Statement For the quadratic form x2-2xy+2yz+z2: a) Find a symmetric matrix that allows the quadratic form to be written as xTAx. b) Determine if the critical point at the origin is a minimum, maximum, or neither. c) Find the points for which the quadratic form achieves its...

Homework Statement Show that d\omega_{ij}+\sum_{k=1}^{n} \omega_{ik}\wedge\omega_{kj} =0 Homework Equations Let G be the group of invertible nxn matrices. This is an open set in the vector space M=Mat(n\times n, R) and our formalism of differential forms applies there with the...

I am told that the set of positive definite quadratic forms on R^2 has a metric that turns it into H x R where H is the hyperbolic plane. Can you describe this metric? * As a space the forms are viewed as GL(2,R)/O(2).

Hi, Everyone: I have a quadratic form q, defined on Z⁴ , and I know the value of q on each of the four basis vectors ( I know q is not linear, and there is a sort of "correction" for non-bilinearity between basis elements , whose values --on all pairs (a,b) of...

In my reference books differential forms are integrated by means of pullbacks. Actually, integrals of differential forms are DEFINED by means of pullbacks. In other words, the integration domain must have a parametrization. Since differential forms and their integrals are under regularity...

I am asked to find all the modular forms with weight k which don't have zeros on the upper half plane. I know that a modular form with weight k is composed of an Eisenstein series with index k and a cusp form with weight k, and I have at my disposal the zeros formula for modular forms. So...

Which quantities are "naturally" forms, and which are (multi)vectors? I'm undertaking a self-study of geometric algebra and differential forms. It is very enlightening, but I find I'm getting a bit confused by which kind of beast is most "natural" for a particular physical quantity. Where...

This isn't homework. I asked my professor for help on figuring out a way to know the possible combinations of reduced row echelon forms of nxn matrices, or mxn matrices. He only could show me why it was really hard to find this out, not how to actually do it. His method was to use...

Mathcad help: "The forms of these values must match" error Homework Statement I am getting an error, when I assign a value to 'num', stating: The forms of these values must match This value has the form: Unitless, but others have the form: f(any1, [unitless]) => [unitless] I cannot...

Homework Statement Determine whether the following set forms a vector space: {(x1, x2, x3) E R^3 | x1 + 2x2 - x3 = 0 and x1x2 = 0} Homework Equations The axioms! The Attempt at a Solution I know that the first equation in the set fulfills the axioms for a vector space, since...

I made a post titled the same thing but it didnt seem to show up for some reason so if i am just reposting this over again i apologize. I recently got the book A geometrical approach to differential forms by David Bachman. At the moment the biggest issue i am having is just visualizing what...

I was wondering if it is possible to extract such information from the toric data. It will be very useful if you even have rough reference that might discuss related topics. Thanks!

Hi everyone, Homework Statement I've been studying a paper in which there is a connection given by, A = f(r)\sigma_1 dx+g(r)\sigma_2 dy, where \sigma's are half the Pauli matrices. I need to calculate the field strength, F = dA +[A,A]. Homework Equations A = f(r)\sigma_1...

Hi everyone, I've been studying a paper in which there is a connection given by, A = f(r)\sigma_1 dx+g(r)\sigma_2 dy, where \sigma's are half the Pauli matrices. I need to calculate the field strength, F = dA +[A,A]. I have computed it, but a factor is given me problems. I would say, dA =...

I'm just wondering whether magentic fields can used to do work on objects? For example, can an electromagnet create a large enough magnetic field within a magnetic bullet to deflect off it's original path? I know this isn't practical, I'm just wondering whether its possible.

Why is it a (for example) 3x3 matrix of linear forms cannot necessarily be written as the sum of at most 3 rank one matrices of linear forms but the statement is true if "linear forms" is replaced with scalars? Does it have something to do with the 2x2 minors being calculated differently when...

Homework Statement True or False and Why? "The sum of two quadratic forms in three variables must be a quadratic form as well." Homework Equations q(x_1,x_2,x_3)=x_1^2+x_2^2+x_3^2+x_1x_3+x_2x_3 The Attempt at a Solution I am definitely missing something. To me this is a...

I was reading about dual spaces and dual bases in the book Linear Algebra by Friedberg, Spence and Insel (FSI) and they give an example of a linear functional, f_i (x) = a_i where [x]_β = [a_1 a_2 ... a_n] denotes the matrix representation of x in terms of the basis β = {x_1, x_2, ..., x_n} of...

I've been trying to create a program that takes a user supplied indefinite integral and integrates it using the methods I have been learning in my calculus class; substitution, integration by parts, trig substitution, partial fractions, elementary anti-derivatives, etc. I've been trying to use...

Homework Statement Construct the analytic mapping \phi(x,y) for the H^{2+} \times S^1 representation of SL(2;R) Homework Equations g(x) \circ g(y) = g(\phi(x,y)) The Attempt at a Solution So, all points in SL(2;R) lie on the manifold H^{2+} \times S^1. I also know that SL(2;R) is...

As in the title , I recently somehow want to learn differential form , but ,actually , I do not really know where I should start , or what books I should read .. So,can anyone recommend some useful books ?

As 'unnati D' stated, in the now locked thread, "i believe the concept of a separate life force, existing without d support of an atomic body is worth exploring." This caught my interest, and even if his views may seem misguided and somewhat chaotic, there is an interesting concept worth...

Benzene forms the electron shell configuration of two rings of electrons, parallel to the molecule, shown here http://upload.wikimedia.org/wikipedia/commons/thumb/9/90/Benzene_Orbitals.svg/750px-Benzene_Orbitals.svg.png so my question is, if an electron beam is run through the center of the...

Homework Statement http://d.imagehost.org/view/0659/Capture Link to wolfram alfa:http://www.wolframalpha.com/input/?i=integrate%28cos%28e^x%29*e^x What i don't understand is why whey do it like this and why i can't integrate by parts in this case? Thanks for any replies!

Hi, I'm trying to understand the use of Maurer-Cartan one-forms in physics. As far as I understand it's a Lie-algebra valued one-form which sends vectors at an arbitrary point g on the Lie-group to the identity e (the Lie algebra). But my question is: what is the use of these things in...

Can general relativity be constructed with differential forms?

Hey, A quick question. In the definition of a differential form, we normally require that they be sections of the k-th exterior power of the cotangent bundle. However, on page 14 of Jurdjevic's book on...

Hello, I am interested to know how many readers think that we will find life in our solar system, if so when?. I am confident that we will find basic life forms in a human lifetime. also if anyone thinks that there is more that one intelligent civilisation in our galaxy other than our own...

Homework Statement Give the two common forms of the mathematical equation for slope (involving X & Y)? Homework Equations ? The Attempt at a Solution Can you guys please give me the two common forms of the mathematical equation for slope? I don't know what it is but I am guessing...

Tensor densities are normally defined in terms of coordinate transformations. Could they also be defined as functions of p tangent vectors and q cotangent vectors, just as tensors are defined, except relaxing the condition of linearity? Can anyone suggest a good, basic introduction to...

limit proofs(indeterminate forms??) Homework Statement We work in the real numbers. Are the following true or false? Give a proof or counterexample. (a) If \sum a^4_n converges, then \sum a^5_n converges. (b) If \sum a^5_n converges, then \sum a^6_n converges. (c) If a_n \geq 0 for all...

Homework Statement Well, I've made a double limit using the polar forms. The thing is the limit is wrong, I've made a plot, and then I saw that the limit doesn't exist, and what I want to know is what I'm reasoning wrong, and some tips to get a deeper comprehension on this limits, and on what I...

Hi I'm just wondering how other life forms on different planets might be able to evolve. Some of my thoughts: 1) other life forms don't necessarily need oxygen do they? On this planet we use oxygen to generate energy through combustion. Other life forms could generate energy by i)...

Let R be a commutative semiring. That is a triple (R,+,.) such that (R,+) is a commutative monoid and (R,.) is a commutative semigroup. Let {\mathbf \alpha}_i = \alpha_1,\alpha_2,\ldots,\alpha_n . The n-variate indeterminate is just free monoid on n letters. However, it is common to...

Most places I've read say it's because the car forms a Faraday cage, but a few say that is incorrect and that it's actually from the skin effect. The notable case of the latter explanation is from the Boston Museum of Science: http://www.mos.org/sln/toe/cage.html This guy, Dr. Davis from the...