Questions tagged [differential-forms]

For questions about differential forms, a class of objects in differential geometry and multivariable calculus that can be integrated.

A degree $k$ differential form on a smooth manifold $M$ is a quantity that can be integrated on $k$-dimensional submanifolds of $M$.

Formally, a degree $k$ differential form is an element of $\Omega^k(M) = \Gamma(M, \bigwedge^kT^*M)$ which is the vector space of smooth sections of the vector bundle $\pi: \bigwedge^kT^*M \to M$; a section is a map $\alpha : M \to \bigwedge^kT^*M$ such that $\pi\circ\alpha = \operatorname{id}_M$. In particular, if $\alpha \in \Omega^k(M)$, for each $x \in M$, $\alpha(x) \in \bigwedge^kT^*_xM$; that is, $\alpha(x)$ is an alternating map $(T^*_xM)^k \to \mathbb{R}$.

3182 questions
213
votes
10 answers

Exterior Derivative vs. Covariant Derivative vs. Lie Derivative

In differential geometry, there are several notions of differentiation, namely: Exterior Derivative, $d$ Covariant Derivative/Connection, $\nabla$ Lie Derivative, $\mathcal{L}$. I have listed them in order of appearance in my education/in…
102
votes
3 answers

Direct proof that the wedge product preserves integral cohomology classes?

Let $H^k(M,\mathbb R)$ be the De Rham cohomology of a manifold $M$. There is a canonical map $H^k(M;\mathbb Z) \to H^k(M;\mathbb R)$ from the integral cohomology to the cohomology with coefficients in $\mathbb R$, which is isomorphic to the De Rham…
85
votes
1 answer

Lebesgue measure theory vs differential forms?

I am currently reading various differential geometry books. From what I understand differential forms allow us to generalize calculus to manifolds and thus perform integration on manifolds. I gather that it is, in general, completely distinct from…
74
votes
5 answers

What is a covector and what is it used for?

From what I understand, a covector is an object that takes a vector and returns a number. So given a vector $v \in V$ and a covector $\phi \in V^*$, you can act on $v$ with $\phi$ to get a real number $\phi(v)$. Is that "it" or is there more to it?…
Mike Flynn
  • 1,687
  • 2
  • 13
  • 21
62
votes
6 answers

Geometric understanding of differential forms.

I would like to understand differential forms more intuitively. I have yet to find a book which explains how the use of the exterior product in differential forms ties into the geometrical significance of it. Most books briefly introduce the…
60
votes
6 answers

Why do differential forms have a much richer structure than vector fields?

I apologize in advance because this question might be a bit philosophical, but I do think it is probably a genuine question with non-vacuous content. We know as a fact that differential forms have a much richer structure than vector fields, to name…
Jia Yiyang
  • 1,013
  • 9
  • 20
56
votes
3 answers

How to calculate the pullback of a $k$-form explicitly

I'm having trouble doing actual computations of the pullback of a $k$-form. I know that a given differentiable map $\alpha: \mathbb{R}^{m} \rightarrow \mathbb{R}^{n}$ induces a map $\alpha^{*}: \Omega^{k}(\mathbb{R}^{n}) \rightarrow…
Tony Burbano
  • 561
  • 1
  • 5
  • 3
56
votes
3 answers

Is Stokes' Theorem natural in the sense of category theory?

Stokes' Theorem asserts that for a compactly-supported differential form $\omega$ of degree $n-1$ on a smooth oriented $n$-dimensional manifold $M$ we have the marvellous equation $$\int_M d\omega = \int_{\partial M} \omega.$$ Doesn't that look like…
39
votes
2 answers

What's the geometrical intuition behind differential forms?

This question can look like a duplicate of this one, but it's kind of different. I'm trying to relate some geometrical meanings I've seem in some books to the definition of differential forms in $\mathbb{R}^n$ as mappings $p \mapsto \omega(p)\in…
Gold
  • 24,123
  • 13
  • 78
  • 179
38
votes
1 answer

Wanted: A purely algebraic proof of the Frobenius theorem on distributions

Is there a purely algebraic proof of the Frobenius theorem? Here's a rough sketch of what i'm looking for: Let $Der(R)$ denote the $R$-module of ($R$-valued) derivations of the algebra $R$ endowed with the lie bracket given by the commutator.…
35
votes
3 answers

Symmetric and wedge product in algebra and differential geometry

Which is the correct identity? $dx \, dy = dx \otimes dy + dy \otimes dx$ $~~~$or$~~~$ $dx \, dy = \dfrac{dx \otimes dy + dy \otimes dx}{2}~$? $dx \wedge dy=dx \otimes dy - dy \otimes dx$ $~~~$or$~~~$ $dx \wedge dy=\dfrac{dx \otimes dy - dy \otimes…
35
votes
3 answers

What does it mean to multiply differentials?

Often times in multi-variable calculus you would have expressions for the differentials of area and volume like this $dA =dxdy$ or $dV = dxdydz$ which we are supposed to just accept because it makes sense in that if you take a tiny piece of…
Ziad Fakhoury
  • 2,547
  • 1
  • 11
  • 29
32
votes
2 answers

Characterization of the exterior derivative $d$

In the paper Natural Operations on Differential Forms, the author R. Palais shows that the exterior derivative $d$ is characterized as the unique "natural" linear map from $\Phi^p$ to $\Phi^{p+1}$ (Palais' $\Phi^p$ is what is perhaps more commonly…
echinodermata
  • 3,434
  • 1
  • 15
  • 36
31
votes
3 answers

Geometric interpretation of connection forms, torsion forms, curvature forms, etc

I have just begun learning about the connection 1-forms, torsion 2-forms, and curvature 2-forms in the context of Riemannian manifolds. However, I am finding it hard to relate these notions to any sort of geometric intuition. How can one interpret…
Jesse Madnick
  • 29,652
  • 7
  • 90
  • 147
31
votes
1 answer

Proving that the pullback map commutes with the exterior derivative

I'm trying to prove that the pullback map $\phi^{\ast}$ induced by a map $\phi:M\rightarrow N$ commutes with the exterior derivative. Here is my attempt so far: Let $\omega\;\in\Omega^{r}(N)$ and let $\phi :M\rightarrow N$. Also, let…
Will
  • 2,925
  • 1
  • 17
  • 41
1
2 3
99 100