Questions tagged [exterior-algebra]
For questions on the exterior algebra, and related concepts such as the wedge product, the tensor algebra and differential forms.
1,268
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Equivalence of the two functors $Alt^{k}(-*)$ and $(Alt^{k})^{*}$
I am finishing Vector Analysis of Klaus Jänich. I am stuck at chapter $12$ because I am confused about a notation. I hope some of you could untangle it for me.
Lemma
We can interpret each $\varphi \in ...
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Hodge star decomposition in non-diagonal manifold product
I'm studying differential forms and I came across the following problem. From what I learnt in another question, when a manifold can be decomposed as $X \times Y$, then the formula found there works ...
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Can SAGE help me simplify wedge products of wedge products?
In particular, I have a vector space $V$ with basis $\{e_1, ..., e_n \}$ and I want to "foil" elements of $\bigwedge^k \bigwedge^m V$. For example, given a vector $((e_1 + e_2) \wedge e_3) \...
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Question about moving Hodge $\star$ to the argument of a 1-form
Let $\alpha$ be a 1-form on an $n$-dimensional vector space $V$ and $v_1,...,v_{n-1}$ (1-)vectors in $V$. Is it true that
$$
\star \alpha(v_1,...,v_{n-1}) = \alpha\big(\star(v_1 \wedge ... \wedge v_{n-...
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3
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Two definitions of antisymmetrization of a tensor?
I am currently learning about tensors and the exterior product, and I have found some contradictory information. I have seen some sources define the antisymmetrization of a tensor as the following:
...
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Confusion about differential forms and integration
I'm self-studying general relativity using Sean M. Carroll's textbook. I recently made it to sections 2.9 and 2.10, which talk about differential forms and integration of functions on manifolds. I ...
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Why is interior multiplication by $v$ an antiderivation implying $v\lrcorner\ \omega ^n = n(v\lrcorner\ \omega)\wedge\omega^{n-1}$?
In Proposition 22.8 of Lee's Introduction to Smooth Manifolds it is written that
[For $V$ a $2n$-dimensional vector space, $v$ a vector in $V$ and $\omega$ a degenerate $2$-covector of $V$] interior ...
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Confusion over tensor definition of exterior power of a vector space and exterior algebra
I am new to and currently learning about Tensor Algebra and Exterior Algebra. I am confused about the definition of the exterior power of a vector space $V$, $\textstyle \bigwedge^k (V)$, and the ...
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Wedge Product and Differential Forms, example
Let $x=id_{\mathbb{R}^4}$, $\alpha=dx^1+x_2dx^2\in \Omega^1\mathbb{R}^4$, $\beta=\sin(x_2)dx^1\wedge dx^3+\cos(x_3)dx^2\wedge dx^4\in \Omega^2\mathbb{R}^4$, $h(x_1, x_2, x_3, x_4)=(x_1, x_2, x_3x_4, ...
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A Question Regarding Cartan’s Absorption Method
I want to ask a question from the book named “ Cartan for Beginners : Differential Geometry via Moving Frames and Exterior Differential Systems” as to how one can absorb an apparent torsion. Suppose ...
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Nice proof that $\text{Alt}$ is natural.
There is a pair of functors $T:\text{kVect}\rightarrow \text{kAlg}$ and $\Lambda:\text{kVect}\rightarrow \text{kAlg}^-$ which are left adjoints to the forgetful functors $U$ (forget the multiplication ...
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How to show that a differential form is not exact?
I want to show that the differential form $w = x^2\sin(y) dx \wedge dy + 2x \sqrt{1+y^4} dx \wedge dz \in \Omega(\mathbb{R}^3)$ is not exact.
Would it be enough to show that $w$ is not closed? Or does ...
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Can the wedge product acting on continuous functions be written differently?
I have
$M=\mathbb{R}^2$, $N=\mathbb{R}^3$
$$F(\theta,\phi)=\big((\cos\phi+2)\cos\theta,(\cos\phi+2)\sin\theta,\sin\phi\big)$$
$$\omega=y\text{d}z\wedge\text{d}x$$
Calculating:
$$F^*\omega=F^*(y\text{d}...
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Compute d$\omega$ in Cartestian coordinates for a given $\omega$
Define a $2$-form $\omega$ on $\mathbb{R}^3$ by
$$\omega=x\text{d}y\wedge\text{d}z+y\text{d}z\wedge\text{d}x+z\text{d}x\wedge\text{d}y$$
Compute d$\omega$.
Using the formula for
$$\text{d}\omega=\text{...
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Do functions distribute over the wedge product?
I am wrapping my head around the arithmetic properties of the wedge product. I understand that constants do distribute over the wedge product, i.e. for $c_1,c_2\in\mathbb{R}$, it holds
\begin{equation}...