Read about differential forms | 17 Discussions | Page 1

  1. K

    A Can we always rewrite a Tensor as a differential form?

    I read in the book Gravitation by Wheeler that "Any tensor can be completely symmetrized or antisymmetrized with an appropriate linear combination of itself and it's transpose (see page 83; also this is an exercise on page 86 Exercise 3.12). And in Topology, Geometry and Physics by Michio...
  2. K

    A Differential Forms or Tensors for Theoretical Physics Today

    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...
  3. Abhishek11235

    Differential 1 form on line

    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...
  4. K

    Integral of a differential form

    Homework Statement Suppose that a smooth differential ##n-1##-form ##\omega## on ##\mathbb{R}^n## is ##0## outside of a ball of radius ##R##. Show that $$ \int_{\mathbb{R}^n} d\omega = 0. $$ Homework Equations [/B] $$\oint_{\partial K} \omega = \int_K d\omega$$ The Attempt at a Solution...
  5. K

    I Differential forms and bases

    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...
  6. K

    A Diff. forms: M_a = {u /\ a=0 | u in L}

    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...
  7. K

    A The Hodge star operator

    I'm reading section 2.7 of Flanders' book about differential forms, but I have some doubts. Let ##\lambda## be a ##p##-vector in ##\bigwedge^p V## and let ##\sigma^1,\ldots,\sigma^n## be a basis of ##V##. There's a unique ##*\lambda## such that, for all ##\mu\in \bigwedge^{n-p}##,$$ \lambda...
  8. beefbrisket

    I Sign mistake when computing integral with differential forms

    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...
  9. P

    A Exterior forms in wiki page

    Hello there, I had some questions regarding k-forms. I was looking in the wiki page of differential forms(https://en.wikipedia.org/wiki/Differential_form) and noticed that it was was introduced to perform integration independent of the co-ordinates. I am not clear how? Is this because given a...
  10. P

    I Differential forms as a basis for covariant antisym. tensors

    In a text I am reading (that I unfortunately can't find online) it says: "[...] differential forms should be thought of as the basis of the vector space of totally antisymmetric covariant tensors. Changing the usual basis dx^{\mu_1} \otimes ... \otimes dx^{\mu_n} with dx^{\mu_1} \wedge ...
  11. M

    I Difference between 1-form and gradient

    I have seen and gone through this thread over and over again but still it is not clear. https://www.physicsforums.com/threads/vectors-one-forms-and-gradients.82943/ The gradient in different coordinate systems is dependent on a metric But the 1-form is not dependent on a metric. It is a...
  12. O

    A Exact vs Closed forms

    (I am a mechanical engineer, trying to make up for a poor math education)' I understand that: A CLOSED form is a differential form whose exterior derivative is 0.0. An EXACT form is the exterior derivative of another form. And it stops right there. I am teaching myself differential forms...
  13. O

    A The meaning of an integral of a one-form

    So I understand that the integral of a differential form ω over the boundary of some orientable manifold Ω is equal to the integral of its exterior derivative dω over the whole of Ω. And I understand that one can pull back the integral of a 1-form over a line to the line integral between the...
  14. V

    A How to switch from tensor products to wedge product

    Suppose we are given this definition of the wedge product for two one-forms in the component notation: $$(A \wedge B)_{\mu\nu}=2A_{[\mu}B_{\nu]}=A_{\mu}B_{\nu}-A_{\nu}B_{\mu}$$ Now how can we show the switch from tensor products to wedge product below...
  15. O

    A Why the terms - exterior, closed, exact?

    Hi all, (Thank you for the continuing responses to my other questions...) I am gaining more and more understanding of differential forms and differential geometry. But now I must ask... Why the words? I understand the exterior derivative, but why is it called "exterior?" Ditto for CLOSED and...
  16. F

    I How to interpret the differential of a function

    In elementary calculus (and often in courses beyond) we are taught that a differential of a function, ##df## quantifies an infinitesimal change in that function. However, the notion of an infinitesimal is not well-defined and is nonsensical (I mean, one cannot define it in terms of a limit, and...
  17. O

    A Integrating the topics of forms, manifolds, and algebra

    Hello, As you might discern from previous posts, I have been teaching myself: Calculus on manifolds Differential forms Lie Algebra, Group Push forward, pull back. I am an engineer approaching this late in life and with a deficient background in math. It is all coming together and I almost...
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