It is a quotient - of the tensor algebra, obtained by taking graded sum over whole numbers $n$ of $n$-fold tensor products - by the ideal generated by elements of the form $a\otimes a$. We write the residue class of $a\otimes b$ in this algebra, as $a\wedge b$ and call it the wedge product.

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30 views

Does the Hodge dual (star) operator make the exterior algebra an involutive (*-) algebra?

Does the Hodge dual (star) operator make the exterior algebra an involutive (*-) algebra? https://en.m.wikipedia.org/wiki/Hodge_dual https://en.m.wikipedia.org/wiki/*-algebra This would seem to be a ...
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1answer
21 views

Since the variance matrix is the expected value of a dyadic tensor, why is it not singular? Which is the probabilistic property behind that?

I will try to explain better my annoying doubt. The variance matrix (or covariance matrix, according to an alternative notation) $\Sigma_v \in \mathbb{R}^{n\times n}$ of the vector random variable $v\...
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1answer
50 views

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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2answers
74 views

A possible generalized determinant?

This will likely seem a bit contrived, and admittedly it is, but I wanted to see just how "close" we could get to generalizing the concept of a determinant. In what follows, we will lose quite a few ...
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0answers
51 views

Simple properties of wedge product [closed]

How to prove a) $\omega \wedge \eta =(-1)^{kl}\eta\wedge\omega, \omega$ is $k$-tensor and $\eta$ is $l$-tensor. b)$f^*(\omega \wedge \eta)=f^*(\omega)\wedge f^*(\eta)$ where $f:V\rightarrow W$ ...
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28 views

Relation between inner product and wedge product

$M$ is an Riemannian n-manifold, $p\in M$. $x, y\in T_pM$. Is it true that $\langle x,x \rangle \langle y, y \rangle-\langle x, y \rangle^2=|x \wedge y|^2$? I'm reading Do Carmo's Riemannian ...
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1answer
22 views

Why this map implies the decomposition of the tensor product space?

Let $V$ be a vector space and consider the tensor product of $V$ with itself, that is $V\otimes V$. Define $\alpha' : V\times V\to V\otimes V$ by $$\alpha'(v,w)=w\otimes v.$$ In that case, $\alpha'$ ...
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19 views

A question about the definition of the exterior product

Let $T\in\wedge^p(V^*)$ and $S\in \wedge^q(V^*)$. Then $T\wedge S$ is defined to be $\text{Alt}(T\otimes S)$ How is $\text{Alt}(T\otimes S)$ defined? I know that $\text{Alt}(T)$ is defined as an ...
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1answer
48 views

Restriction of an $n$-form to a submanifold

I am currently studying exterior calculus on manifolds and am not sure if I understand things correctly. In my textbook (R.W.R. Darling, Differential Forms and Connections) there is an example of a $2$...
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1answer
56 views

Exterior derivative of complex differential form

I have this question, from several complex variables: Start with the differential form: $$\omega(z)=\sum_{\nu=1}^{n} \frac{(-1)^{\nu-1}\bar{z}_{\nu}}{|z|^{2n}} d\bar{z}[\nu] \wedge dz, $$ where $dz=...
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33 views

Do I understand the divergence theorem correctly?

Suppose the area, volume or hyper volume covered by a vector is $$ \mathrm{V}\left(\vec{u}\right) = u_x \times u_y \times \ldots $$ And the area, volume or hyper volume covered by a matrix is $$ \...
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1answer
76 views

Geometric Intuition about the relation between Clifford Algebra and Exterior Algebra

It is common to see a relation being established between the Clifford Algebra and the Exterior Algebra of a vector space. Recently reading some texts written by Physicists I've seem applications of ...
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1answer
60 views

Deciding whether a form in the exterior power $\bigwedge^k V$ is decomposable

Let $V$ be a vector space and $\bigwedge^kV$ be the $k$th exterior power. I'm trying to find a condition that characterizes when an element $\omega \in \bigwedge^kV$ is decomposable in the sense that $...
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1answer
49 views

Why is the wedge product of a 1-form and itself $0$? [closed]

Why is the wedge product of a 1-form and itself $0$? Why doesn't this apply to 2-forms?
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23 views

Change of coordinates using the wedge product?

I while back (over a year ago) I was told about wedge products (or something very similar) and how they can be used to change e.g. the curl in Cartesian coordinates to in spherical coordinates. ...
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1answer
44 views

Connecting the regular representation of $\mathfrak{so}(3)$ and the exterior algebra of $\mathbb{R}^3$

It is well known that the regular representation of $\mathfrak{so}(3)$ is the so-called "cross product" matrix $A(x)$ which follows $A(x)y = x\times y$, and $x,y\in\mathbb{R}^3$, while the cross ...
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1answer
26 views

Foiling two tensor products

I have this problem in exterior algebra where I have a function B and B is defined in the following ways. $$ B\left( \left( \begin{array}{c} u \\ v \\ w \end{array} \right), \left( \begin{array}{c} x\\...
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1answer
51 views

Show that $(x\land y)z + (y\land z)x + (z\land x)y=0.$ where $x\land y=(x \times y)\cdot N$.

Let $P\subset \mathbb{R}^3$ be a plane through the origin and $N$ be a unit normal to $P$. For $x,y \in P$, set $x\land y=(x \times y)\cdot N$. Then for any three vectors $x,y,z \in P$, we have $$(...
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0answers
14 views

decomposable p-forms and linear independence

So I am attempting to show that: $$ z_1 \wedge \cdots \wedge z_p = 0 \implies z_1,\ldots,z_p \text{ linearly dependent} $$ my approach is to use the fact that: $$ z_1 \wedge \cdots \wedge z_p = \left[ ...
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0answers
26 views

Equivalent conditions for $\mathfrak{F}$ to be a differential ideal

Heres the question: Let $\mathfrak{F}$ be an ideal of forms on a manifold $M$ locally generated by $r$ independent $1$-forms. Say $\mathfrak{F}$ is generated by $\omega_1,\ldots, \omega_r$ on $U$. ...
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1answer
39 views

Prove the exterior derivative of the following (n-1) form is zero

Let $\omega(x)=\frac{1}{{\parallel x \parallel}^n}\displaystyle\sum_{i=1}^{n}(-1)^{i-1}x_{i} dx_{1} \wedge \dots \wedge \widehat{dx_{i}} \wedge \dots \wedge dx_{n}$ be a differential $(n-1)$ form on $\...
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1answer
20 views

When do exterior and tensor algebras commute with dual spaces?

Suppose $V$ is a vector space, and $V^*$ is its dual space. Furthermore, let $\Lambda(V)$ be the exterior algebra of $V$, and let $T(V)$ be the tensor algebra. When do the following two statements ...
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21 views

What is the motivation for wanting to remove redundant terms that arise from the wedge product of two multilinear functions?

I am currently working through Loring Tu's "An Introduction to Manifolds," specifically the section in which the exterior algebra of multicovectors is introduced, and I am having trouble grasping the ...
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1answer
76 views

Is this an alternate characterization of $\lambda$-rings? Or, what is like a $\lambda$-ring but for symmetric rather than exterior powers?

This is a question about $\lambda$-rings. A $\lambda$-ring is a commutative ring together with operations $\lambda^n$ for each whole number $n$ which are analogous to the $n$th exterior power and ...
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0answers
9 views

Exterior algebra subspace of all grade-n wedge products of a vector

Let $V$ be a finite-dimensional vector space, and let $\Lambda(V)$ be its exterior algebra. Then if $S_k = \text{span}(k_1,k_2,...,k_n)$ and $\hat k = k_1 \wedge k_2 \wedge ... \wedge k_n$, there is ...
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1answer
40 views

Noncanonical isomorphism of spaces of differential forms

Let $\pi: V \to M$ be a smooth $n$-dimensional vector bundle over $M$. Are the spaces of differential forms $\Omega^i(V)$, $\Omega^i(V^*)$ noncanonically isomorphic? If so, how do I see this? Is there ...
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2answers
71 views

Frank Warner's definition of the Hodge star

Frank Warner's book, chapter 2, excercise 13 states the following: If $V$ is an oriented inner product space ($n$ dimensional) there is a linear map $\ast \colon \Lambda (V) \to \Lambda (V)$, ...
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10 views

Parameterizing the set of subquotients of a Hilbert space

If you want to parameterize the set of subspaces of a finite-dimensional Hilbert space $V$, and naturally induce a norm derived from $V$ on the resulting moduli space, the classical approach is to ...
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1answer
31 views

Prove that $p-$form $\omega$ on $M\times N$ is $\delta \pi(\alpha)$ iff $i(X)\omega=0$ and $L_X\omega=0$

Let $M$ be connected and let $\pi:M\times N \rightarrow N$ be the natural projection. Prove that $p-$form $\omega$ on $M\times N$ is $\delta \pi(\alpha)$ for some $p-$form $\alpha$ on $N$ if and only ...
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1answer
43 views

Prove the pullback of the wedge product is the wedge product of the pullbacks.

Let $F:V \rightarrow W$ be a linear map. Show that $F^{\ast}(\omega \wedge \eta)=(F^{\ast}\omega) \wedge (F^{\ast}\eta)$ for all $\omega \in \Lambda^{p}(W) , \eta \in \Lambda^{q}(W)$. Where $F^{\ast}...
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1answer
42 views

Exterior derivative of a two-form with conditions

With the use of this formula $$d\omega(X_1, \dots, X_{r+1}) = \sum_{i=1}^{r}(-1)^{i+1}X_i\omega(x_1,\dots,\hat{X}_i,\dots,X_{r+1})+\sum_{i<j}(-1)^{i+j}\omega([X_i,X_j],X_1,\dots,\hat{X}_i,\dots,\...
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0answers
22 views

Exterior power of a free module [duplicate]

Suppose $M$ is a free R module where $R$ is commutative and unital. Is $\Lambda^n M$, the nth exterior power of $M$ free?
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1answer
26 views

Wedge product on free module

Let $R$ be a unital commutative ring, let $M$ be a free module of rank $n$ over $R$ with a basis $e_1,\dots,e_n$. Suppose that $m_1,\cdots,m_n$ are elements of $M$ such that $$m_1\wedge\cdots\wedge ...
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1answer
33 views

Powers of two in basis vectors indexes for wedge products

I am reading Fundamentals of Grassmann Algebra for game developers and it dawned on me that a slight tweak in the notation would make things easier. Remember that for n dimensions that number of basis ...
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2answers
36 views

Prove $\det(T^*)=\det(T)$ using exterior algebta

Let $T:V\to V$ be a linear operator on a finite-dimensional vector space $V$. $T^*:V^*\to V^*$ is the transpose of $T$ (i.e., $T^*(\lambda)=\lambda\circ T$). I want to find a coordinate-free proof ...
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0answers
63 views

exterior derivative of 1-form on surface for non-regular mapping?

I am studying alone the elementary differential geometry written by Barrett O'Neill. This time I'm totally lost. I can't even get any idea from the hint...;( How is the problem related to Lemma 4.5 ...
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1answer
34 views

Derivative is an alternating 1-tensor?

I am reading through spivak, and he states, if $f: \mathbb{R^n} \to \mathbb{R}$ is differentiable, then $Df(p): \mathbb{R^n} \to \mathbb{R}$, and since this is linear, we have that $Df(p) \in \Lambda^...
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1answer
27 views

Alternating k-tensors and determinant (theorem 4-6, Spivak)

I am having troubles understanding this proof. I don't understand how, if $\eta \in \Lambda^n(\mathbb{R^n})$ implies that $\eta = \lambda \cdot \det$ - really I don't see how determinants come from ...
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0answers
30 views

Unifying Transformations in Complex 3-Space

I am currently researching the vector space $\mathbb{C}^{3}$ and I was wondering if it is possible to generate a scheme of unifying the rigid transformations in $\mathbb{C}^{3}$. I know that in the ...
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1answer
54 views

Anisotropic scaling in geometric/Clifford algebra

Take the geometric algebra over $\Bbb R^n$. Suppose we have a blade multivector in this algebra. Now we want to anisotropically scale this multivector. Is there a general closed-form expression for ...
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0answers
21 views

Computing $\alpha^*w$ in general

Let $A$ be open in $R^k$, let $\alpha : A \rightarrow R^n$ be of class $C^\infty$. Let $x$ denote the general point of $R^k$, let $y$ denote the general point of $R^n$ If $I = (i_i, ...,i_l)$ is an ...
3
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1answer
43 views

Bases and wedge product

Let $\{v_1,\dots, v_r\}$ and $\{w_1,\dots, w_r\}$ be linearly independent sets of a vector space $V$. If $v_1 \wedge \dots \wedge v_r =c w_1 \wedge \dots \wedge w_r $ for some nonzero number $c$, ...
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0answers
22 views

The determinant of $X_I$ = wedge product of elementary tensors

Let $(x_1,\dots,x_k)$ be vectors in $\mathbb{R}^n$; let $X$ be the matrix $X = [x_1 \cdots x_k]$. If $I = (i_1, \dots, i_k)$ is an arbitrary $k$-tuple from the set $\{1,\dots, n\}$, show that $\phi_{...
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1answer
33 views

Exterior power of representation and invariance

Let $G$ be a group, $(\rho,V)$ a finite dimensional real representation of $G$, and $W$ a subspace of $V$ of dimension $k$. Assume that $\Lambda^k W$ is a $G$-invariant (and one dimensional) subspace ...
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1answer
39 views

Computing a scalar $c$ such that $cx= c(e_1 \wedge \cdots \wedge e_{2n})=\omega \wedge \cdots \wedge \omega$.

Let $n \geqslant 2$ be an integer and consider the vector space $V=F^{2n}$ with the standard basis $e_1,\ldots,e_{2n}$. Now the second exterior power $\Lambda^2(V)$ contains the element $$\omega=e_1 \...
2
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1answer
31 views

Linear relations among wedge products.

Let $v_1,v_2,v_3,v_4$ be $4$ vectors in a two dimensional space $V$. Then one can work out by hand that: $$(v_1\wedge v_2)(v_3\wedge v_4) + (v_1\wedge v_3)(v_4\wedge v_2) + (v_1\wedge v_4)(v_2\wedge ...
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2answers
35 views

If $u \in \Lambda^{p}(V)$ and $v \in \Lambda^{q}(V)$, does $u \wedge v = (-1)^{pq} v \wedge u$?

Consider the exterior algebra $$\Lambda^{\ast}(V)= \bigoplus_{m=0}^\infty \Lambda^m(V),$$ for a finite dimensional $F$-vector space. If $u \in \Lambda^{p}(V)$ and $v \in \Lambda^{q}(V)$, does $u \...
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1answer
32 views

Function of vector space product to exterior power of that vector space

Let $W$ be a $n$-dimensional vector space with basis $w_1,...,w_n$ and $f:W\times W\rightarrow\bigwedge^2W$ with $f(x,y)=x\wedge y$. 1 For $n=3$, how do I express $f(aw_ 1+bw_2+cw_3,dw_ 1+ew_2+...
2
votes
1answer
39 views

Does every element in exterior algebra have the form $v_1 \wedge v_2 \wedge … \wedge v_k$? [duplicate]

Does every element in $\wedge ^k V$ can be expressed as the form $v_1 \wedge v_2 \wedge ... \wedge v_k$ ? Here $V$ is a n-dim vector space, and $v_i$ are vectors in $V$. Intuitively it is right, but ...
4
votes
1answer
62 views

Is there a nowhere $1$-form $\alpha$ on $M$ with $\alpha(X) = 0$ for any vector field $X$ which is a section of $\xi$?

Let $M$ be a closed $3$-manifold, and let $\xi$ be a $2$-dimensional subbundle of $TM$. Is there a nowhere zero $1$-form $\alpha$ on $M$ with $\alpha(X) = 0$ for any vector field $X$ which is a ...