# Questions tagged [exterior-algebra]

For questions on the exterior algebra, and related concepts such as the wedge product, the tensor algebra and differential forms.

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### Determinant Formula for Wedge Product via Universal Property of Exterior Powers

I'm currently learning about differential forms in my analysis class, and I thought I'd dig a bit more into the linear algebra of exterior powers. I've seen the universal property of $\bigwedge^k(V)$: ...
1 vote
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### How to compute exterior derivative of generalized angular form in $\mathbb R^n$

On $\mathbb R^n$, with $m\in\mathbb Z$, I want to calculate the exterior derivative of the angular form \begin{align} w=\sum_{i=1}^n(-1)^{i-1}\frac{x_i}{\|x\|^m}dx_1\cdots dx_{i-1}\hspace{0.03cm}\...
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### exterior derivative of wedge of two one-forms

Let $\alpha, \beta$ be $k$-forms, then, where $d$ is the exterior derivative, we have $$d (\alpha \wedge \beta) = d \alpha \wedge \beta + (-1)^k (\alpha \wedge d \beta)$$ However, in many cases, e.g. ...
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### Linear dependence and nonzero p-vectors

If a, b, and c are linearly dependent, then it's easy to show that $a\wedge b\wedge c = 0$. However, if a, b, and c are linearly independent, how do you show that $a\wedge b\wedge c \ne 0$?
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1 vote
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### trace of wedge product and cyclic property [closed]

Surprisingly, while their are similar but more advanced questions on this site, I don't see any answers to the basic version I am asking herein. If I am taking the trace of a wedge product of matrices,...
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### The dot product is the norm of what vector (product)?

Let $a$, $b$, $c$ be vectors in $\mathbb{R}^3$ that form a triangle. (How this even makes sense formally, I don't know, since in a vector space all vectors are "glued" to the origin. In an ...
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### Wedge product defined on not alternating tensors?

I am currently reading Calculus on Manifolds by Spivak. In there, it defined wedge product as follows To determine the dimensions of $\Lambda^k(V)$, we would like a theorem analogous to Theorem 4-1. ...
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### Direct sum of a vector space V and a field R

What does it mean, that the exterior algebra of a vector space V over R is a superset of direct sum of real numbers and real vector space V? I thought that given vector spaces must have trivial ...
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### Integral involving Hadamard product with volume element [closed]

Consider the following integral: $$\int_{\mathbb{R}^{n\times m}}f(X)(H^T(A\odot dX)V)^{\wedge},$$ where $f:\mathbb{R}^{n\times m}\to \mathbb{R},$ $H\in O(n)$, $V\in O(m),$ $A$ is an $n\times m$ matrix ...
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Let us consider vector space $$V = (\Lambda^2 (\mathbb C(t))^\times)\otimes_\mathbb Z\mathbb Q.$$ Here I consider the group $A=(\mathbb C(𝑡))^\times$ of non-zero rational functions under ...