Forms Definition and 448 Threads

I have been working through Spivak's fine book, but the part about differential forms and tangent spaces has left me confused. In particular, Spivak defines the Tangent Space \mathbb R^n_p of \mathbb R^n at the point p as the set of tuples (p,x),x\in\mathbb R^n. Afterwards, Vector fields are...

Homework Statement Given the equivalent impedance of a circuit can be calculated by the expression Z= (Z_1 Z_2)/(Z_1+ Z_2 ) If Z1 = 4 + j10 and Z2 = 12 – j3, calculate the impedance Z in both rectangular and polar forms. Homework Equations j2=-1 The Attempt at a Solution Z=...

I read recently that all energy is either kinetic or potential. In high energy physics, it is easy to understand the kinetic bit, but potential energy eludes me. What are some examples of potential energy at the high energy physics/elementary particle level? Also, if strings exist, how is...

Hi, All: There is a standard method to construct a nowhere-zero form to show embedded (in R^n ) manifolds are orientable ( well, actually, we know they're orientable and we then construct the form). Say M is embedded in R^n, with codimension -1. Then we can construct a nowhere-...

Hi, Again: I'm trying to show that, given a 3-manifold M, and a plane field ρ (i.e., a distribution on TM) on M, there exists an open set U in M, so that ρ can be represented as the kernel of a differential form w , for W defined on U. The idea is that the kernel of a linear map...

Hi, All: I need some help with some "technology" on differential forms, please: 1)Im trying to understand how the hyperplane field Tx\Sigma< TpM on M=\Sigma x S1 , where \Sigma is a surface, is defined as the kernel of the form dθ (the top form on S1). I know that...

Homework Statement Lim as x→∞ of ((2x+1)/(2x-1))^(sqrtx)Homework Equations The Attempt at a Solution When I initially plugged in ∞ for my x, I get (∞/∞)^∞, correct? If so, should I just let y=((2x+1)/(2x-1))^(sqrtx) and take the limit of both sides using ln? That's what I attempted to do...

So far the best general physics book for EM I've found has been Alonso and Finn. The problem is that I just spent too much time trying to understand electric displacement using the hand-wave magic mathematical definitions they give. The rest of the book seems fine (it gives vector forms for...

Hi, Everyone: A way of defining an orientation form when given a codimension-1 , orientable n-manifold N embedded in R^{n+1} , in which the gradient ( of the parametrized image ) is non-zero (I think n(x) being nonzero is equivalent to N being orientable), is to consider the nowhere-zero...

Homework Statement Imagine that we discover a new particle, called P particle by a head-on collision of two photons of energies 500 Mev and 200 MeV. The photons are annihilated in the process. a) what is the mass of the newly discovered particle P? b) what is the kinetic energy of the P...

I was wondering that since most forms of energy can either be in wave or particle form, (example; photons and electromagnetic wave) and also since mass is also a form of energy, could mass energy also be in wave form, and if so, what would its characteristics be?

I'm having problems in my differential geometry class. Does anyone know of a good tutorial or set of notes? The 1-form, 2-form, 3-form stuff is confusing and so is the wedge product. Anyways, here's the problem. After some algebra I arrived at my answer, but I'm unsure of how to incorporate...

Homework Statement the first friedmann equation is: (\frac{\dot{a}}{a})^2=\frac{8\pi G\rho}{3}-\frac{kc^2}{a^2}+\frac{\Lambda}{3} In the case of a closed Universe (k > 0) containing only non-relativistic matter and no cosmological constant, write the Friedmann equation in terms of, H(a), H0...

As you can tell from the question, my science knowledge is lacking. :confused: I can see that mammals, reptiles, etc. must eat other living creatures (mammals, insects, plants, etc.) for energy. But do plants, microbes and other types of life forms all depend on eating other life forms? For...

Hi I was reading a book that introduced momentum and energy in integral forms and I had some confusion regarding what the terms meant. All integrals below are closed integrals For the momentum equation, the result was: F = d(mV)/dt = ∫∫ρ(V[dot]dS)V + ∫∫∫∂(ρV)/∂tdV From product rule...

Homework Statement there are 2 blocks along a frictionless track. block 1 has a mass of m1 and block 2 has a mass of m2. Block 1 is initially moving at a speed of v0. it collides and sticks to the initial stationary block 2. (m2=9m1) what fraction of the inital kinetic energ of the system is...

A is a square matrix. x, b are vectors. I know for Ax=b, that given b, there are an infinite number of pairs (A, x) which satisfy the equation. I'm wondering if the same is true for xAx=b. in particular, what if (x, A, b) are all stochastic vectors/matrices (i.e the entries of b and x add to...

So as I have read in Schutzs book on GR, and I'm finding his section on tensors and one forms very confusing. Schutz describes gradients of functions as a one form, I cannot quite grasp why. In calculus I was taught that the gradient was a vector pointing in the direction of the fastest...

why does (P^-1)AP form a triangular matrix?

Let r and s be positive integers. Show that {nr + ms | n,m ε Z} is a subgroup of Z Proof: ---- "SKETCH" ----- Let r , s be positive integers. Consider the set {nr + ms | n,m ε Z}. We wish to show that this set is a subgroup of Z. Closure Let a , b ε {nr + ms | n,m ε...

Is it true that for every standard formulation T of ZFC, T ⊢ the power set of {naturals}? After all, the empty set axiom and the pairing axiom are in T, and so we get N. Then by the power set axiom we get P(N).

Hey guys, in class today we learned about the life cycle of a star and at the very first stage gravity pulls the helium or Hydrogen nuclei at such a speed that they fuse (nuclear fusion). As I understand it there is little gravity in space so where is this extra gravity coming from? Thanks...

Hello everybody, This is my first time on Physics forums. I am a sophomore in high school who LOVES math. I have lots of free time this summer and would like to learn multivariable calculus and/or linear algebra (whichever is a prerequisite for the other, depending on the textbook I choose)...

State whether the sequence converges as n--> ##∞##, if it does find the limit i'm having trouble with these two: n!/2n and ∫ e-x2 dx now I know they're special forms so the ordinary tricks won't work. Any help or hints?

Homework Statement Express z=-1+4i in polar for then find z^4 converting to Cartesian form Homework Equations r = sqrt(x^2+y^2) theta = y/x z= r cos (theta) + i r sin (theta) The Attempt at a Solution r= sqrt(-1^2+4^2) = sqrt(17) theta = tan a = 4/1 a = tan^-1...

Let me preface by saying I am a physics major. So I am coming at differential forms from the perspective of physics, i.e. work, flows, em fields, etc. My question is this. My understanding is that a basic 1-form dx, dy, or dz takes a vector v = (v1,v2,v3) and gives back the corresponding...

So about a hundred years ago there was a live (sort of) differential forms thread hosted by someone named Lethe that was really helpful but short-lived. There have been some other diffl forms threads, too, such as the one centered on Bachman's book, but they all seem to peter out without any...

Hi, I’m currently doing research for a science fiction story and was wondering if anyone would be interested in helping me out. I’m trying to understand the various applications of different forms of energy. More specifically, here is what I want to know: If someone was equipped with a...

Can someone recommend a good textbook for learning differential forms for someone with an understanding of calculus at the level of Spivak? Thanks.

Homework Statement 1. Homework Statement [/b] Enumerate all possible Jordan forms for 3 x 3 systems where all the eigen-values have negative real parts. Do not use specific values. Instead, use possibilities like λ1; λ2; λ3, each with multiplicity 1, or λ (multiplicity 3). Homework...

Hi, I'm just wondering, a vector valued (or (n,0) tensor valued) p form is the same as a rank (n,p) tensor which is totally anti-symmetric bottom indices right? Is there a difference? A (n,m) valued p form is a (n,m+p) tensor which is anti-symmetric in the lower p indices?

Hello Everyone, I'm currently working through a differential geometry book that uses Clifford's algebra instead of differential forms. If anybody has knowledge of both, would you please explain what the differences between the approaches are, and what (if any) are the advantages of each...

Homework Statement *indeterminate* oops the limit of x^x as x goes to zero from the right Homework Equations Going to be using L'hopital, and related algebraic manipulations to convert to indefinite form 0/0, infinity/infinity The Attempt at a Solution My understanding is that this limit...

A sphere has charge density \rho=k\cdot r. Using the integral form of Gauss's Law, one easily finds that the electric field is E=\frac{k\cdot r^2}{4\epsilon} anywhere inside the sphere. However, \nabla\cdot E=\frac{k\cdot r}{2\epsilon}, which is half of what should be expected from the...

Hi, I have a complex number and understand that the rectangular form of the number is represented by s = σ + jω, where σ is the real part and jω is imaginary. I am having trouble locating them in the number below: I know that "2" is a real number, and the numerator is imaginary...

Source: http://en.wikipedia.org/wiki/Linear_equation General form:- It says (under the title General Form) "If B is nonzero, then the y-intercept, that is the y-coordinate of the point where the graph crosses the y-axis (where x is zero), is −C/B, and the slope of the line is −A/B."...

FYI this is a homework problem which I already have the answer to but would like to clarify some points on. Homework Statement Find the Smith Normal Form of the matrix \left[ \begin{array}{cccc} 6 & 0 & 4 \\ 0 & 6 & 8 \\ 0 & 3 & 0 \end{array} \right] over the ring of integers. Homework...

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...