Forms Definition and 482 Threads

Say I start with a quadratic form: x^2 - y^2 - 2z^2 + 2xz - 4yz. I complete the square to get: (x+z)^2 - (y+2z)^2 + z^2. (So the rank=3, signature=1) The symmetric matrix representing the quadratic form wrt the standard basis for \mathbb{R}^3 is A =\begin{bmatrix} 1 & 0 & 1 \\...

Hello, Everyone: My apologies for not including a descriptive title; I was just very distracted: In the page: http://en.wikipedia.org/wiki/Closed_...erential_forms there is a reference to the form dw= (xdx/(x^2+y^2) -ydy/(x^2+y^2) ) , next to which there is the graph of " the...

Homework Statement Show that exterior differentiation of a 0-form f on R3 is essentially the same as calculating the gradient of f. The Attempt at a SolutionLet U be a differentiable 0-form on R3. I think dU = \sum _{j=1} ^n \frac{δF_I}{δx_j}dx_j dx_IHowever, since U is a 0-form, I can...

Homework Statement Nullity(B-5I)=2 and Nullity(B-5I)^2=5 Characteristic poly is: (λ-5)^12 Find the possible jordan forms of B and the minimal polynomials for each of these JFs. The Attempt at a Solution JFs: Jn1(5) or ... or Jni(5). Not sure how to find these jordan forms and minimal polynomials.

I'm having a bit of a brain fart here. Given a positive definite quadratic form \sum \alpha_{i,j} x_i x_j is it possible to re-write this as \sum k_i x_i^2 + \left( \sum \beta_i x_i \right)^2 with all the ki positive? I feel like the answer should be obvious

In VB6 this was very easy. you just used form1.hide form2.show In c# you have to do this form2 openForm2 = new form2(); //create a new instance form2.show(); this.hide(); When I put this in a button it worked, but every time I changed forms it reset the form values to default...

Homework Statement See attachment. The Attempt at a Solution In a) ii); The use of a chain diagram is required but I have no idea how to produce one. As for i); I have no idea how to do this. In b); (B+5I)v=(1,2,-1) and (B+5I)^2v=0. The eigenvalues are 5 (with multiplicity 2) and 2 (w/...

Homework Statement See Attachment. The Attempt at a Solution I need an efficient way to find the det(A) so I can find the eigenvalues together with the trace or will use Cayley-Hamilton Thm. I can find the algebraic multiplicities but cannot find the geometric ones. If a matrix is...

So, I've been thinking about this for a while...and I can't seem to resolve it in my head. In this thread I will use a tilde when referring to one forms and a vector sign when referring to vectors and boldface for tensors. It seems to me that if we require the basis vectors and one forms to obey...

Homework Statement A 20 × 20 matrix C has characteristic polynomial (λ^2 − 4)^10. It is given that ker(C−2I), ker (C − 2I)^2, ker (C −2I)^3 and ker (C −2I)^4 have dimensions 3,6,8,10 respectively. It is given that ker (C + 2I), ker (C +2I)^2, ker (C +2I)^3 and ker (C +2I)^4 have di- mensions...

Homework Statement Are chain and ring forms of glucose isomers? They aren't, because they have the same structure, right? Homework Equations The Attempt at a Solution

The professor for my symbolic logic course requires us to be extremely precise with our explanations. Given the subject, I understand his reasoning and appreciate his rigor. I am studying for our first exam by doing some of the exercises at the end of the sections on which we're going to be...

I know this may sounds silly but I am confused consider this two form for example, by substitution, I get \omega = dx \wedge dy = d(rCos\theta)\wedge d(rSin\theta) = r dr \wedge d\theta also consider this smooth map F(x,y)=(rCos\theta,rSin\theta) then F^{*}\omega = rdr \wedge...

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...

I am reading a derivation of the Euler Lagrange equations. They defined coordinate charts φ: U -> R^n where x -> φ(x) = (x1, x2, ... xn). then they said for the one form dφ: TU -> TR^n ~= R^n X R^n where (x,y) -> dφ(x,y) = (x1, ..., xn, y1, ..., yn). what I am confused about is that when i...

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...

Homework Statement Here is my problem http://i51.tinypic.com/34dihx5.jpg However my teacher had some suggestions of solving this problem in a nice mathematical way. Here is the plan of solving which I would like to follow 1) find the Hodge dual to f 2) compute df 3) is f exact...

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).

Homework Statement Given the radial function F: R_n without zero vector -----> R_n F(x) = phi(||x||)*x ---- the function value is depends on it's distance from the origin. Phi is the function : phi: (0,infinity) ---> Real Numbers How to prove that the diff. form defined by this...

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...

I am looking for any help I can get, I have been working on this for hours and cannot figure it out. Should I include more info in my post? I am getting an error stating: The forms of these values must match This value has the form: Unitless but others have the form: f(any1, [unitless]) =>...

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...