The exterior-algebra tag has no wiki summary.
4
votes
4answers
86 views
Big Greeks and commutation
Does a sum or product symbol, $\Sigma$ or $\Pi$, imply an ordering?
Clearly if $\mathbf{x}_i$ is a matrix then:
$$\prod_{i=0}^{n} \mathbf{x}_i$$
depends on the order of the multiplication. But, ...
2
votes
1answer
48 views
In Grassmann algebra a la Browne, why are vectors dependent if their wedge product vanishes?
I'm reading John Browne's Grassmann Algebra, Vol 1 : Foundations. Early on, he asserts without proof that if $x$ and $y$ are any two vectors in the underlying (real) vector space such that $x \wedge y ...
0
votes
1answer
25 views
Wedge product of vector fields
Can somebody explain me step by step how can I compute the wedge product $X\wedge Y$ of two vector fields, $X,Y$, in $\mathbb{C}^2$?
We can consider
$$
X=X_1\partial_x+X_2\partial_y
$$
and
$$
...
1
vote
1answer
36 views
Why is $\theta \not \in C^{\infty}(S^1)$?
Why is $\theta \not \in C^{\infty}(S^1)$? I know that since $\int_{S^1} d\theta = 2\pi$ then $d\theta$ is not exact. Thus since $d(\theta)=d\theta$, $\theta$ must not be $C^{\infty}$ but it seems ...
0
votes
1answer
49 views
Complicated calculation on Lie bracket and wedge product
I'm in the following situation: let's assume I am given the following vector fields in $\mathbb{C}^2$, not identically zero:
$$
X=X_1\partial _x+X_2\partial _y
$$
$$
Y=Y_1\partial _x+Y_2\partial _y
$$
...
1
vote
1answer
25 views
Change of base rings for exterior algebra
This may be not a good question. But I really get tough. I am studying basic knowledge about homological algebras and I am dealing with Koszul's Complex and Hilbert's Syzygy Theorem. At the very ...
1
vote
1answer
56 views
Anti-derivation of an exterior algebra
Let $K$ be a commutative ring. $M$ is a free $K$-module of rank $m$. $E(M)$ is the exterior algebra defined by $M$.
$\iota$ is a $K$-homomorphism from $M$ to $E(M)$ such that $\iota(x)=-x$. Since ...
0
votes
0answers
24 views
I get the value of my $2$ form to be $0$.
Let $\omega = (x_1 + \cdots +x_n) \sum_{i<j} dx_i \wedge dx_j$ be a $2$-form on $\mathbb{R}^n, \omega \in \Omega (\mathbb{R}^n)$. Compute the function
$$\omega \left( ...
0
votes
1answer
65 views
wedge product and change of variables
The question is:
Let $\phi：\mathbb{R}^n\rightarrow\mathbb{R}^n$ be a $C^1$ map and let $y=\phi(x)$ be the change of variables. Show that
d$y_1\wedge...\wedge ...
4
votes
2answers
93 views
If $\omega$ is a 2-form on $\mathbb{R}^4$ and $\omega \wedge \omega = 0$, then $\omega$ is decomposable.
I am trying to prove the following from a book I am reading through.
Thm: If $\omega$ is a 2-form on $\mathbb{R}^4$ and $\omega \wedge \omega = 0$, then $\omega$ is decomposable. Note ...
0
votes
1answer
38 views
Proving the Poincare Lemma for $1$ forms on $\mathbb{R}^2$
I am trying to prove the Poincare Lemma for $1$ forms on $\mathbb{R^2}$. So I said that if I doing this, I start of with
$$\omega = f_1(x_1,x_2) dx_1 + f_2(x_1,x_2)dx_2.$$
First thing I want to ...
1
vote
1answer
70 views
Computation of the hom-set of a comodule over a coalgebra: $Ext_{E(x)}(k, E(x)) = P(y)$.
First of all, since every other book somehow mentions that this is trivial, I apologize if it turns out that I am just misunderstanding something in the definitions. So here goes:
The motivation for ...
1
vote
2answers
54 views
Computing wedge products
Compute $\omega = (e_1^* + e_2^* + \cdots+ e_n^*) \wedge (e_1^* + e_2^*) \wedge (e_1^* + e_3^*) \wedge \cdots \wedge(e_1^* + e_n^*)$ in the standard form.
I first thuoght I'd pick a value from ...
1
vote
1answer
23 views
Putting the wedge product in standard/normal form
I have to compute the wedge product of
$$(e_1^* + ze_2^*) \wedge (e_2^* + ze_3^*) \wedge \cdots \wedge (e_{n-1}^* + ze_n^*) \wedge (e_n^* + ze_1^*),$$
and then put it in normal/standard form.
So I ...
3
votes
1answer
66 views
Elements of $\wedge^2V$ expressible in the form $v_1\wedge v_2$
If $V$ is a complex vector space, then an element $w\in \wedge^2V$ is of the form $v_1\wedge v_2$ for some $v_1,v_2\in V$ iff $w\wedge w=0$ in $\wedge^4V$. Could anybody give some intuition/show why ...
1
vote
1answer
28 views
Basis for the space of $p$-forms
According to the book I read, a general $p$-form can be written as:
$$\omega=\omega_{a_1\ldots a_p} dx^{a_1}\wedge\ldots\wedge dx^{a_p},\hspace{0.5cm} a_1>a_2>\ldots>a_p$$
where I have used ...
1
vote
1answer
56 views
decomposable elements of $\Lambda^k(V)$
I have conjecture I have problem to prove or disprove.
Let's $ w \in \Lambda^k(V)$ is k-vector. $W_w=\{v\in V: v\wedge w = 0 \}$ is k-dimensional vector space if and only if $w$ is decomposable.
...
2
votes
1answer
45 views
Without choosing bases, how to show that the determinant is multiplicative in this sense?
I was recently considering this statement:
Let $V$ be a finite-dimensional $k$-vector space, and let $\phi:V\to V$ be an endomorphism. Suppose that $W\subseteq V$ is a subspace that is stable ...
2
votes
0answers
58 views
Computing wedge product of two 1-forms.
Let $L$ be a lattice in $\mathbb{C}$ and let $\pi :\mathbb{C}\to X=\mathbb{C}/L$ be a quotient map. Show that the local formula $dz$ in every chart of $\mathbb{C}/L$ is a well-defined holomorphic ...
2
votes
2answers
49 views
Compute the value of the exterior $2$-form
Compute the value of the exterior $2$-form
$$\omega = (x_1 + x_2)e_1^* \wedge e_2^* + (x_2 + x_3)e_2^* \wedge e_3^* + \cdots + (x_i, x_{i+1})e_i^* \wedge ...
1
vote
1answer
20 views
2-form associated with a skew map
Given a two-form $\omega\in \Lambda^2V$ for some (say finite dimensional) vector space $V$ we may associate with $\omega$ a skew map $f_{\omega}:V\rightarrow V^*$ given by $X\mapsto \iota_X\omega$, ...
0
votes
1answer
37 views
Show that $af∧bg=(ab)f∧g$
Let $v$ be vector space. For $a$ and $b$ are in IR, $f$ is in $A_{k}(V)$ and $g$ is in $A_{l}(V)$
Show that $af∧bf=(ab)f∧g$
Here Will I use the definition of wedge product? Is ti right? How to use? ...
1
vote
1answer
62 views
Use the Fundamental Theorem to deduce the formula for the area of an ellipse.
Use the Fundamental Theorem (Green's Theorem) to deduce the formula for the area of an ellipse. Hint: find a 1-form whose exterior derivative is $ dxdy $.
1
vote
1answer
51 views
If $ \eta $ and $ \varphi $ are closed differential forms, then prove that $ \varphi \wedge \eta $ is a closed differential form.
Let’s assume that $ \eta $ and $ \varphi $ are closed differential forms. Then how can I prove that $ \varphi \wedge \eta $ is a closed differential form as well? Please explain how to solve this ...
1
vote
1answer
91 views
Prove that if $η$ is exact, then $η∧β$ is also exact.
Prove that if $η$ is exact, then $η∧β$ is also exact.
Please give a clear way to solve?
0
votes
1answer
81 views
how prove $\rho\wedge d\rho=0$ and how to show if $d(f\rho)=0$ for $f$ on $\Bbb R^{n}$ then $\rho\wedge d\rho=0$
$\def\d{\mathrm{d}}\def\R{\mathbb{R}}$Let $ρ$ be a $1$-form on $\R^{n}$ then
Firstly how to prove $\rho\wedge \d\rho=0$
Secondly, how to show that If $\d(f\rho)=0$ for some nowhere vanishing smooth ...
6
votes
0answers
69 views
What is the image of the map $\hom(V,V) \to \hom(\wedge^k V,\wedge^k V)$?
The title says it all. For the uninitiated: Any map $f:V \to W$ induces a map $\wedge^k V \to \wedge^k W$ by $v_1 \wedge \cdots \wedge v_k \mapsto f(v_1)\wedge \cdots \wedge f(v_k)$, so $\wedge^k(-)$ ...
1
vote
1answer
69 views
Associativity property of wedge products
Regarding notation, I have a bit that says: The linear function $e_i^* \in V^*$ determined by $e_i^*(e_j) = \delta_{ij}$ form the basis of $V^*$, which is called the dual
basis for $\epsilon = (e_1, ...
4
votes
1answer
83 views
Are these two definitions of exterior derivative equivalent?
I saw two definition of the exterior derivative of a $k$-form $\omega$.
First definition: $$(d\omega)_p(v_0,\ldots,v_k)= \sum_{i=0}^k(-1)^id(\omega(v_0,\ldots,\hat{v_i},\ldots,v_k))_p(v_i)$$
Second ...
1
vote
1answer
89 views
Tensor and Wedge Product of Vectors
I have a little doubt about tensor product acting on vectors. I was reading Spivak's Calculus on Manifolds, and Spivak introduces the tensor product of multilinear functionals. Latter he introduces ...
3
votes
3answers
81 views
Exterior Algebra as quotient
Given a vector space $W$, I understand what the tensor algebra $T(W)$ is, and I understand that the exterior algebra $\bigwedge W$ is defined as $\bigwedge W := T(W)/N$ where $N$ is the two-sided ...
0
votes
0answers
43 views
a natural linear transformation on the space $V\otimes V$
Let $T\in \wedge^4V$ , and define the tensor $A(T)$, of type (2,2), on the space $V$, ($dimV=8$) by the formula
$(A(T))^{i_1i_2}_{m_1m_2}=\varepsilon ...
0
votes
0answers
68 views
contraction with the metric tensor
What does mean "$\wedge_0^4V $ is the space of 4-vectors whose contraction with the metric tensor of the space $V$ vanishes" how can we formulate this set?
this means $i_gT=0$ for tensor $T$?
4
votes
1answer
88 views
Exterior algebra of a vector bundle
Associated to any vector space $V$ is its exterior algebra $\Lambda(V)$ which has the direct sum decomposition $\Lambda(V) = \bigoplus_{i=0}^n\Lambda^i(V)$ where $n = \dim V$.
My first interaction ...
2
votes
1answer
109 views
Wedge product of forms
Let $\alpha = y^2 dx + dy \in \Omega^1(R^2)$, $\beta = xy dx \wedge dy \in \Omega^2(R^2)$. Is $\alpha \wedge \beta = 0$?
2
votes
1answer
54 views
Is there a smallest sub-Grassmann algebra containing a given vector in Grassmann algebra?
Let $V$ be a $d$-dimensional $\mathbb C$-vector space and the Grassmann algebra
$$\mathcal G (V):=\bigoplus_{n=1}^d V^{\wedge n}$$
where $\wedge$ denotes the antisymmetric tensor product.
I was ...
3
votes
2answers
136 views
$\alpha\wedge\beta = 0$ for all $\beta$ implies $\alpha = 0$ without using the Hodge dual
Let $\alpha$ be a differential $k$-form on an orientable smooth $n$-dimensional manifold. If $\alpha\wedge\beta = 0$ for every differential $(n - k)$-form $\beta$, then $\alpha = 0$ because we can ...
3
votes
0answers
104 views
Multiplication in exterior algebra
Take $V = K^{n}$. Let $\omega$ be a non-zero element of $\bigoplus_{k=1}^n \bigwedge^k V$, where we have excluded the summand $\bigwedge^0 V = K$.
(1) Prove that there exists an $m > 1$ for which ...
3
votes
2answers
142 views
algebraic manipulation of differential form
Suppose $\phi_1, \phi_2, \dots, \phi_k \in (\mathbb{R}^n)^*$, and $\mathbf{v}_1, \dots, \mathbf{v}_k \in \mathbb{R}^n$
$(\mathbb{R}^n)^*$ stands for the space of all linear transformations that goes ...
2
votes
1answer
68 views
Wedge product with a non-degenerate form
Let $\alpha$ be a non-degenerate form in $\Lambda^k(V)$ for some vector space $V$, $\dim V = n$. (Here non-degenerate means that if $x\in V$ is nonzero, then $(y_1 , ... , y_{k-1}) \mapsto \alpha(x , ...
1
vote
1answer
53 views
How can I show that $\det(v_1,v_2,\ldots,v_n)=dx_1\, dx_2\cdots dx_n(v_1,v_2,\ldots,v_n)$?
I wanted to use the definition of a wedge product which says $λ_1λ_2\cdots λ_k(v_1,v_2,\ldots,v_k)=\det(λ_i(v_j))$ with $1<i,j<k$
but I'm not sure if that even can work
5
votes
1answer
114 views
Determinant of the transpose via exterior products
Let $V$ be a finite-dimensional vector space over $F$ and let $\tau:V \to V$ be a linear operator. Here's my definition of the determinant:
If $t:U \to U$ is a linear operator and $\dim(U)=n$ then ...
2
votes
0answers
78 views
Explicit computations of tensor and wedge product
Let $f\colon K^3\to K^3$ be a map in Jordan canonical form having matrix
$$
f=\begin{bmatrix} -1 & 1 & 0 \\ 0 & -1 & 0 \\ 0 & 0& -1 \end{bmatrix}
$$
What is $f\otimes f$? What ...
24
votes
5answers
1k views
Exterior Derivative vs. Covariant Derivative vs. Lie Derivative
In differential geometry, there are several notions of differentiation, namely:
Exterior Derivative, $d$
Covariant Derivative/Connection, $\nabla$
Lie Derivative, $\mathcal{L}$.
I have listed them ...
2
votes
1answer
150 views
Taking the exterior derivative of a 0-form
I'm attempting to show that $dg(\vec{x})=\alpha$, where $\alpha=\Sigma^n_{i=1}f_idx_i$ and $g(\vec x)={1\over{p+1}}\Sigma_{i=1}^nx_if_i(\vec x)$...and $d$ is the exterior derivative. The $f_i$ are ...
3
votes
2answers
76 views
elementary question regarding differential forms
Is it possible to give a high level explanation why changing the order of differentials will give rise to a minus sign ? I.e. why do we have
$$ dx\,dt = - dt\,dx
$$
(I am going to take a course on ...
5
votes
1answer
64 views
general trace relation
Let $V$ be vector space $\dim V=N$, and $A\in End(V)$. Denote
$$
\wedge^k A^m(\mathbf{v}_1\wedge\dots\wedge\mathbf{v}_k)=\sum_{s_1,\dots,s_k=0,1,\sum_j s_j=m} A^{s_1}\mathbf{v}_1\wedge\dots\wedge ...
4
votes
1answer
55 views
Trace of the multiplication operator
Let $V$ be vector space, $\dim V=N$. Define the multiplication operator $L_{\mathbf{b}}$ as $L_{\mathbf{b}}:\omega\to \mathbf{b}\wedge\omega$, where $\omega\in\wedge V$ ($\wedge V$ is the entire ...
3
votes
1answer
179 views
Laplace expansion
This statement is from the book of Winitzki Linear Algebra via Exterior Products. (Section 3.4, page 123) Let $V$ be finite dimensional vector space, $\dim(V)=N$. The determinant of the matrix ...
5
votes
0answers
61 views
Inner product of $p$-forms [duplicate]
Possible Duplicate:
Extension of Riemannian Metric to Higher Forms
I have no problems with understanding the inner product of 1-forms on a Riemannian manifold. We have a metric tensor, it's ...
