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Questions tagged [exterior-derivative]

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Pullbacks commute with the exterior derivative

I am trying to show by induction that pullbacks and the exterior derivative commute. I know that this question has been discussed on this site, eg. here and here. However, none of these questions ...
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Significance of function $I$ such that $\omega = I(d\omega) + d(I\omega)$ for differential forms

In Calculus on Manifolds, Spivak proves the Poincaré Lemma by introducing a function $I$ which maps $p$-forms to $(p-1)$-forms (for each $p$). Suppose there is a differential form $\omega\in \Omega^p (...
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Exterior derivative of a 1-form on a surface for non-regular mappings

I would appreciate some help for this problem. I have no idea how to start. Let $M\subset \mathbb{R}^3$ be a smooth surface. Let $\phi$ be a $1$-form on $M$. By definition, the exterior derivative of ...
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Prove the exterior derivative is a linear transformation of vector spaces.

Let $\omega_1$ be a $k$-form and $\omega_2$ be an $l$-form, both defined in an open subset $U\subset \mathbf R^3$. Let $d:\wedge^k(U)\rightarrow\wedge^{k+1}(U)$ be the exterior derivative of ...
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Exterior derivative of a vector valued function

I'm trying to understand the exterior derivative in the simplest context that I can. I feel like I understand how an exterior derivative should behave for a function. For example, let's take a simple ...
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What's the intuition behind the shear not being possible to be obtained with exterior differentiation?

Allow me to present some context, first. If we have a $K$-vector space $E$ and a non-degenerate symmetric bilinear form (a pseudo-riemannian metric) $g$, then we can project any 2nd order tensor (the ...
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How can there be a non-covariant exterior derivative?

So, if I'm not wrong, the definition of the exterior derivative of a differential $k$-form is the differential $k\text{+}1$-form $$\text{d}\omega = \omega_{i_1...i_k,i_{k+1}} \text{d}x^{i_{k\text{+}1}...
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Given a differential form $\omega$, is there a differential form $\phi$ such that $\omega\wedge\phi$ is closed?

Let $M$ be a differential manifold and $\Omega^p(M)$ the vector bundle of $p$-forms. My question is: Given a differential $p$-form $\omega$, is there a differential $q$-form $\phi$ such that $d(\...
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1answer
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Why is the exterior differential independent of the chosen basis?

Let M be as smooth manifold, W a k-dimensional vector space with basis $(e_1,...,e_k)$ and $\Omega^r(M,W)$ be the differential r-form on M with values in W. Let $\omega \in \Omega^r$ and $v_1,..., v_r ...
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Properties of forms and pullbacks

Let $j: \mathbb{S}^2 \rightarrow \mathbb{R}^3\setminus\{0\}$ be the canonical injection and $\alpha$ a k-form over $\mathbb{R}^3\setminus\{0\}$. If $\alpha$ is closed or exact, is it the same for $j^*...
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find a vector field in $\mathbb{R^3}$ with specific properties

Let $α = dz - ydx \in Ω^1 (\mathbb{R}^3)$ Find a vector field Z in $\mathbb{R^3}$ such that $α(Z) = 1$ and $dα(Z,.) = 0$ What I did: I computed $d\alpha = dx \wedge dy $. Then let $Z=a\partial_x + ...
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Exterior Derivative over Quaternions

I was wondering if it is possible to define the exterior derivative of a quaternionic valued function. I am doing the quaternionic analogue of a previously complex valued computation, namely something ...
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Product manifolds and exterior derivative with interior product

While studying differential geometry, I read this part of a proof and I didn't understand it. Given a $2$-manifold $\Omega$ and an interval $I=(-\epsilon, \epsilon)$, consider the cartesian product $M=...
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Confusion over an exterior derivative product

I have an elementary question regarding the exterior derivative which confuses me. I am reading Theorem 2.1.13 of the book Aspects of Multivariate Statistical Theory by Robb J Muirhead. The part that ...
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The wedge of an exact form with a closed form is exact.

I'm trying to prove that the wedge of a closed form $\xi$ with an exact form $\omega$ is exact. We already have that half of it is exact. Maybe we can use the equation of $\xi$ being closed to rewrite ...
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1answer
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Exterior derivative calculation in Chern and Hamilton paper

Chern and Hamilton in their paper "On Riemannian metrics adapted to three-dimensional contact manifolds" constructed an structure on 3-sphere as follows (Example 3.2 of paper): Let $$\omega_1=...
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1answer
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existence of de Rham complexes

I have a very basic question about the exterior derivative of differential forms and de Rham complexes. It is very basic, I know that the exterior derivative satisfies $d^2=0$. Knowing that, how is a ...
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1answer
66 views

Show that $d(\omega ∧ \tau ) = (d\omega)∧\tau +(−1)^{\text{deg}\omega} w ∧d\tau.$

The exterior differentiation $d : \Omega^∗(U)→\Omega^∗(U)$ is an antiderivation of degree $1$: $$d(\omega ∧ \tau ) = (d\omega)∧\tau +(−1)^{\text{deg}\omega} w ∧d\tau.$$ Suppose $\omega=\sum a_Idx^...
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What is the physical OR geometrical interpretation of the exterior derivative?

The exterior derivative of a $C^{\infty} 0-$ forms on an open set $U$ of $\mathbb R^n$ look like the total derivative of $f$ in the calculus. My doubt- Why did an exterior derivative of $C^{\infty} ...
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1answer
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Express covariant derivative in terms of exterior derivative

I know there is an intimate relation between covariant, Lie and exterior derivative. I know that the covariant derivative requires more structure than the exterior, so it would be possible. How do I ...
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1answer
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Exterior derivative of alternating form

Let $V$ be a real vector space of dimension $n>2$ and $\omega\in \wedge^2V^*$ an alternating bilinear form on $V$. I'm wondering if there is a notion of exterior derivative $d:\wedge^2V^*\...
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1answer
217 views

Unique exterior derivative

While reading that article wiki I was confused by the note saing that there is another formula for exterior derivative differing by a constant. On the other hand from what I have checked in my ...
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Why is the exterior derivative called exterior derivative

I am studying exterior calculus, and I think I have some grasp of what is the exterior derivative. However its name still eludes me - why is it called a derivative? Is it just because the operator $d$ ...
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Function and its exterior derivative

Is there an example of a function $f:M\to \mathbb R$, where $M$ is a differentiable manifold, such that $f$ is constant on the hypersurface $\Sigma$ and its exterior derivative $df\neq 0$ on $\Sigma$?...