# 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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### Solve the differential equation with the corresponding replacement

So I have this problem from the Elementary Differential Equations by Kells: Solve this differential equation with the corresponding replacement: $$(st+1)t\cdot ds+(2st-1)s \cdot dt=0$$ with $$z=st$$ ...
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### Why is $\Omega$ used to denote the space of differential forms?

A simple curious question about notation. The space of $k$-differential forms over a manifold $M$ is typically denoted by $\Omega^k(M)$. Does anybody know why, or where this notation originated? It ...
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### Is my understanding of dX accurate?

When writing $df$, for a differentiable function $f:\mathbb{R}^{n} \to \mathbb{R}$, I know rigorously speaking, it is a one form, such that $df(p)(v_{p}) = Df(p)(v)$, where $Df(p)$ is the linear ...
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### How to prove the existence of differential forms on a manifold using de Rham cohomology?

Let $S^3$ be the 3-sphere, and $\Sigma$ be a 2-dimension manifold. Let $\omega$ be a 2-form on $\Sigma$. $f:S^3\rightarrow \Sigma$ is a $C^{\infty}$ map.Then there is a 1-form $\alpha$ on $S^3$ such ...
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### If $\alpha\wedge d\alpha$ is a volume form, there exists a vector field $X$ such that $i_X\alpha\equiv1$ and $i_X (d\alpha)\equiv0$.

I'm currently stuck on the following problem: Let $\alpha$ be a 1-form on a connected 3-manifold $M$ such that $\alpha\wedge d\alpha$ is a volume form. Show that there exists a vector field $X$ on $M$...
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### Intuition: Topological notion of genus and Genus as the dimension of the complex vector space of holomorphic differentials

In my self-study of Lec.7 of Belyi Maps and Dessins d'Enfants, I came across the following statement Definition 3: Genus can be defined as the dimension of the complex vector space of holomorphic ...
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### Generalize the metric $g_s(r)$ to the class of surfaces $S$

Let $M=(0,1)^n$ and take $(M,g_n)$ where $g_n$ is the usual Euclidean metric. Let's assume that in dim. $n$ we want a smooth codimension one foliation of $M$ (can have exactly one singular leaf) whose ...
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### How is the fundamental theorem of calculus dependent on orientation?

Using standard Lebesgue integration, we can write: \begin{equation} \int_{(a,b)}f'(x) d\lambda = f(b) - f(a) \end{equation} There's no orientation on the left hand side of the equation, yet on the ...
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### Proving isotropy of a groupoid multiplication graph in a quasi-presymplectic groupoid

I'm studying quasi-presymplectic groupoids and I've come across the following proposition which I'm finding challenging to prove. Any help or guidance would be greatly appreciated. Given a Lie ...
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### On Marsden's 'Introduction to Mechanics and Symmetry' Exercise 5.2-3. (symplectic map is immersion)

I'm either confused with the definition of symplectic forms / immersions or the way exercise 5.2-3 in Marsden's 'Introduction to Mechanics and Symmetry' was stated. It reads as follows Exercise 5.2-3....
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### Finding a holomorphic 1-form in one chart that extends to a nearby coordinate domain.

I know that for a compact Riemann surface of genus $g$, there should be $g$ different globally defined holomorphic 1-forms. However, I am having trouble finding any in my case. Here is the set up: Let ...
Given two differentiable manifolds $\mathcal M$ and $\mathcal N$ I needed to show that $$\mathcal M, \; \mathcal N \mbox{ orientable } \Rightarrow \mathcal M \times \mathcal N \mbox{ orientable.}$$ ...