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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19 views

A natural Poisson bivector on the tangent bundle?

For a smooth manifold $M$, there is a natural $1$-form $\theta$ on $T^*M$ such that $\Bbb d \theta$ is a symplectic form. Somewhat symetrically, on $TM$ there is a natural tangent field $V$. Is it ...
4
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1answer
62 views

Geometric Significance of some features of the Exterior Algebra

I've been tinkering with differential forms for a while now, and I've had a few questions all rolled into one trying to understand them. The exterior derivative is quite natural to me - it looks just ...
2
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2answers
149 views

Interior product and exterior product

I have seen around the internet that this should hold: $$\iota_X(\alpha\wedge\beta)=\iota_X\alpha\wedge\beta+(-1)^k\alpha\wedge\iota_X\beta,$$ where $X$ is a vector field, $\alpha$ a $k$-form, $\...
-2
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2answers
79 views

Why isn't the plücker embedding surjective?

This is basic algebraic geometry, there's a question at the end but please check that my reasoning all the way up to the question also holds: Given a grassmannian $Gr(k,V)$, we can embed it to the ...
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1answer
14 views

Dimension of the projectivization of a wedge?

Please help me understand this: To the best of my knowledge, the projectivization $\mathbb{P}(K^k)$ of a vector space $K^k $ is defined as the set of one-dimensional subspaces of $K^{k+1} $. OK, I ...
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0answers
28 views

How to calculate $\Lambda^r A$? [closed]

How to calculate $\Lambda^r A$ where $ r \in \{1,...,3\} $ and $A \in\operatorname{Mat}_{3\times 3}(\mathbb{R})$, considering exterior algebra?
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1answer
35 views

Exterior algebra of a ring

In the book "Cohen-Macaulay rings" by Bruns and Herzog, the quick introduction of tensor algebra and exterior algebra left me a bit bewildered. After referring to the section on tensor algebra ...
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0answers
40 views

Explain why non-zero bivectors on a plane are called pseudo-vectors of the plane?

I answered it by saying pseudo-vectors changes according to the rotation of the vector on the plane like theta rotating clockwise 90 degrees. Am I right?
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1answer
57 views

Does $A\wedge B=A\wedge C$ imply that $B=C$? [closed]

Does $A\wedge B=A\wedge C$ imply that $B=C$? I need help figuring this out because I am lost on this
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0answers
40 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\...
1
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1answer
51 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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1answer
86 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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2answers
79 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 ...
1
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0answers
54 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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0answers
30 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
23 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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0answers
21 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 ...
2
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1answer
58 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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1answer
55 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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0answers
34 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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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
63 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
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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1answer
54 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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0answers
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. ...
5
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1answer
78 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 ...
2
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1answer
45 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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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 $U$. ...
2
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1answer
41 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 $\...
1
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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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0answers
22 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 ...
0
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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 ...
4
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1answer
42 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 ...
2
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2answers
73 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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0answers
14 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 ...
2
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1answer
32 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 ...
0
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1answer
46 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
44 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
23 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
27 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 ...
86
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7answers
9k views

Exterior Derivative vs. Covariant Derivative vs. Lie Derivative

In differential geometry, there are several notions of differentiation, namely: Exterior Derivative, $d$ Covariant Derivative/Connection, $\nabla$ Lie Derivative, $\mathcal{L}$. I have listed them ...
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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 ...
1
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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 ...
2
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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^...
0
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1answer
30 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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1answer
56 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 ...
1
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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 ...