Questions tagged [differential-forms]

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

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Explicit evaluation of symplectic form on cotangent bundle

Let $M$ be a manifold and denote by $T^*M$ its cotangent bundle. Let $(x,U)$ be a coordinate chart so that $x: U\to \mathbb{R}^{n}$. Let $p\in U$ and $v\in T^*_pM$, then we can write $v = v_i dx^i$ ...
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Can someone explain me the notation of k-forms (Differentialforms)?

I'm looking at K-forms (Differential forms) and I somehow struggle a bit to understand the notation and the meaning of it. To be more precise I have problems in understanding the indexing. Let me ...
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In which way do the differential forms give a different view on multidimensional integration?

I am reading up on differential forms calculus, so more specifically we started with pfaff-forms and curve integrals. The multidimensional forms come later. Our prof. said at the time that you can ...
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Is the existence of a continuous top form sufficient for orientability of a smooth manifold?

I'm currently working through Tu's Introduction to Manifolds. He first defines orientability in terms of the existence of orientations on the tangent spaces which can be locally represented by a ...
66 views

What is the definition of the integral of a differential form on a submanifold

Let $M$ be a smooth oriented manifold of dimension $n$. Let $\alpha = \alpha_{} + \alpha_{} + ...+ \alpha_{[n]}$ be smooth differential form on $M$ such that $\alpha_{[I]} \in \mathcal{A}^i(M)$....
42 views

Inverse exterior derivative

The exterior derivative is defined as the unique $\mathbb{R}$-linear mapping, such that $df$ is a differential one-form for a zero-form $f$, $d d\alpha = 0$ for any $\alpha$, and that it is an ...
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Volume form on the sphere $S^2$ [closed]

I know the volume form (in french "forme d'aire") on the sphere $S^2$ is given by, $$\nu_{S^2} = z(dx \wedge dy) + x(dy \wedge dz) + y(dz \wedge dx)$$ I'm wondering how can I compute it ...
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Showing a specific $p$-form on a star-shaped domain is the exterior derivative of a specific $(p-1)$-form.

The folloing is a problem from Chapter 17 of Lee's Introduction to Smooth Manifolds: Suppose $U\subseteq \mathbb{R}^n$ is open and star shaped with respect to $0$, and $\omega=\sum'\omega_Idx^I$ is a ...
60 views

2-Form on $\mathbb{R}^3$ That Restricts to Surface Area 2-Form on Torus?

I would like to find a 2-form on $\mathbb{R}^3$ that 1) restricts to the torus to give a generator of the top-dimensional de Rham cohomology of the torus and 2) restricts to the torus to give the ...
56 views

Curl of a Lie bracket of two vector fields

I am wondering if there is a nice (ideally coordinate free, something that holds in manifolds) identity of the form $\nabla \times [X,Y] = [\nabla \times X,Y] + [X, \nabla \times Y] + ...$, and if ...
71 views

Existence of $k$-form with nonzero integral

Say that $N$ is an oriented, compact, connected manifold without border. If $\operatorname{dim}(N) = k$, does it always exists some $k$-form $\omega$ such that $\int_N \omega \neq 0$? I know how to ...
23 views

Verification of this integration on forms by two different methods

Let $\Omega\subset\mathbb{R}^3$ be the upper half ball with radius $a$, i.e. the region bounded below by $z=0$ and above by $x^2+y^2+z^2=a^2$. Compute  \int_{\partial\Omega} xz\,dy\wedge dz + yz\,dz\...
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Interchange of limits of differential operators on differential forms

Let $M$ be a complex 2-dimensional manifold with the additional property that a smooth function $h:M \to \mathbb C$ is constant if and only if $\partial\overline\partial h = 0$ (which is equivalent (...
Integration of differential form inclusion of $S^2$
Take this example. We have the natural inclusion $i : S^2 \rightarrow R^3$ and the differential form: $\omega = x dy \wedge dx + y dz \wedge dx + z dx \wedge dy$. How can we say that that the ...