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...
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...
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...
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...
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...
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...
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}##,$$
\lambda...
The question provides the vector field (xy, 2yz, 3zx) and asks me to confirm Stokes' theorem (the vector calc version) but I am trying to use the generalized differential forms version. So, I am trying to integrate \omega = xy\,dx + 2yz\,dy + 3zx\,dz along the following triangular boundary...
Hello there,
I had some questions regarding k-forms. I was looking in the wiki page of differential forms(https://en.wikipedia.org/wiki/Differential_form) and noticed that it was was introduced to perform integration independent of the co-ordinates. I am not clear how? Is this because given a...
In a text I am reading (that I unfortunately can't find online) it says:
"[...] differential forms should be thought of as the basis of the vector space of totally antisymmetric covariant tensors. Changing the usual basis dx^{\mu_1} \otimes ... \otimes dx^{\mu_n} with dx^{\mu_1} \wedge ...
I have seen and gone through this thread over and over again but still it is not clear.
https://www.physicsforums.com/threads/vectors-one-forms-and-gradients.82943/
The gradient in different coordinate systems is dependent on a metric
But the 1-form is not dependent on a metric. It is a...
(I am a mechanical engineer, trying to make up for a poor math education)'
I understand that:
A CLOSED form is a differential form whose exterior derivative is 0.0.
An EXACT form is the exterior derivative of another form.
And it stops right there. I am teaching myself differential forms...
So I understand that the integral of a differential form ω over the boundary of some orientable manifold Ω is equal to the integral of its exterior derivative dω over the whole of Ω.
And I understand that one can pull back the integral of a 1-form over a line to the line integral between the...
Suppose we are given this definition of the wedge product for two one-forms in the component notation:
$$(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...
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...
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...
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...