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 $(T^*_xM)^k \to \mathbb{R}$.

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Differential forms defined by integration

Let $\omega_1,\omega_2$ be two n-forms on a $n$-dimensional manifold $M$. Now, imagine we have for every open $N \subset M$ that $$\int_{N}\omega_1 = \int_N \omega_2.$$ Can anybody show me how to prove that both forms are equal?- I suspect that…
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Can every 2 form be represented as a linear combination of these specific two forms?

This question is Question 2 from Ilka's book on page 8. The first part is to prove that every $\omega^2\in \Lambda^2(V^{\ast})$ can be represented as \begin{equation*}\tag{1} \omega^2=\sigma_1\wedge\sigma_2+\dots…
Nirav
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Exercise about differential forms and pull-back: $\omega_p=0$ if and only if $(\omega|_U)_p=0$

Let $M$ be a manifold, $\omega\in\Omega^q(M)$. Let $U\subset M$ be an open subset embedded as manifold in $M$. Let $p\in U$. Show that $\omega_p=0$ if and only if $(\omega|_U)_p=0$. Notation: Let $F:U\to M$ be the inclusion map which is an…
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Under what conditions can a general 2-form be written as a wedge product of two 1-form

Assume we have a 2-form $\omega \in \Lambda^2\mathbb{R}^n$. It is usually stated one can write $$\omega = \alpha \wedge \beta,$$ with $\alpha, \beta \in \Lambda^1\mathbb{R}^n$ only for $n < 4$. How to prove this statement?
user54031
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Line integrals in differential form

I'm a bit confused as to the format of line integrals in differential form (i.e. the form in which Green's theorem is often presented). For example: $$ \oint\limits_\mathcal{C} \left( y^2 \mathrm{d}x + x^2 \mathrm{d}y \right)$$ where $\mathcal{C}$…
Anti Earth
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Is there a treatment/development of the Stokes' Theorem using differential forms and the Henstock-Kurzweil integral i.e. the gauge integral?

I'm working through an analysis text independently to prepare for grad school, and the author has discussed the limitations of both the Riemann and Lebesgue integrals and only hinted at the power of generalized Riemann integral which I had to look…
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Finding a $1$-form on a hyperboloid of one sheet which is closed but not exact

I want to find a closed 1-form $w$ on $$M=\{(x,y,z):x^2-y^2-z^2=-1\}\subset \mathbb R^3$$ which is not exact. I think that $$\frac{x\,\mathrm{d}x+y\,\mathrm{d}y+z\,\mathrm{d}z}{(x^2+y^2+z^2)^{3/2}}$$ is a such one, but how can I prove that?
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Show that $f^*\omega = \det(df) \, dx_1\wedge\cdots\wedge dx_n$

Let $f:\Bbb{R}^n\to \Bbb{R}^n$ be a differentiable map given my $f(x_1,\ldots,x_n)=(y_1,\ldots,y_n)$, and let $$\omega=dy_1\wedge\cdots\wedge dy_n.$$ Show that $$f^*\omega = \det(df) \, dx_1\wedge\cdots\wedge dx_n.$$ We can simplify the left hand…
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Linear Maps from $V$ to $V^*$ defined by a 2-form

I came across this idea at the very beginning of a book and I don't quite seem to grasp it. It states given $\omega \in \bigwedge ^2 (V) $ you can define a linear map $ \omega^\#: V \to V^* $ by $\omega(v, \cdot)$ . My first question is just a…
kmeis
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How do you compute the pull-back of a complex differential (1,1)-form given its potential?

Let $F: X \to Y$ be a holomorphic map. Let $\omega$ be a complex differential $(1,1)$-form on $Y$, $\omega=\partial \bar \partial f$, where $f$ is a pluri-subharmonic function. How would one explicitly compute the pull-back of this form under a…
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Symplectic form on 2-sphere?

I am trying to put the symplectic form of the 2-sphere defined by $\omega_u(v,w) := \langle u,v\times w\rangle,$ where $u \in \mathbb{S}^2$ and $v, w \in T_u\mathbb{S}^2$. I just can't do this in stereographic coordinates. Would you help me?
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exact and closed differential forms

This exercise is taken from the Meyer-Hall-Offin book on Hamiltonian systems. Let $Q(p,q)$ and $P(p,q)$ be smooth functions defined on an open set in $\mathbb{R}^2$. Consider the four differential forms $\Omega_1=PdQ-pdq,\ \Omega_2=PdQ+qdp,\…
Marc
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$e^{xy}dx \wedge dy$: determine the $1$-form that it induces on $S^1$ and check if the obtained $1$-form respects or not the induced orientation

Consider the $2$-form $e^{xy}dx \wedge dy$ on $\mathbb{R}^2$. Determine the $1$-form that it induces on $S^1$, viewed as the boundary of $B_2$. Check if the obtained $1$-form respects or not the induced orientation. Honestly I don't know where to…
Leafar
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Show $H_{dR}^1(S^n)=0$ for $n>1$ without de-Rham Theorem, and some similar questions.

Question Without using de Rham's theorem, prove: (1) Show $H_{dR}^1(S^n)=0$ for $n>1$. (2) Use (1) to show $H_{dR}^1(RP^n)=0$ (3) a n-form $\Omega$ is exact on $S^n$ if and only if $$ \int_{S^n}\,\Omega =0$$ (4) Use (3) to show that every smooth…
Hang
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The adjoint of left exterior multiplication by $\xi$ for hodge star operator

As we know, for $V$ vectoral space and a orientation $\mathcal{O}$ on $V$ and $e_{1},...,e_{n}$, the hodge star operator $\ast:\wedge V^*\rightarrow\wedge V^*$ is defined for $\ast(e_{1}\wedge...\wedge e_{n})=\pm 1$(acoording to the orientation…
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