The differential-forms tag has no wiki summary.
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Existence of a left-invariant $n$-form on a Lie group of dimension $n$
This Do Carmo, Riemannian Geometry, Chapter 1, Exercise 7:
Show that there exists a left invariant differential $n$-form $\omega$ on $G$ ($G$ is a compact connected lie group and $\dim G=n$).
...
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Algorithm/Procedure for finding $\sigma$ such that $\omega=d\sigma$
I know that the Poincare's lemma asserts that under certain conditions a differential form $\omega$ is exact, i.e. it possesses an antiderivative $\sigma$, such that $\omega=d\sigma$.
But as ...
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2answers
53 views
Equality involving exterior product..
suppose you have a differential form $\omega$ writting in local coordinates as $$\omega=\sum_{i=1}^ndx_i\wedge dy_i.$$ Can anyone help me showing the following equality: $$\omega^n=n!(dx_1\wedge ...
4
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1answer
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Smooth homotopic maps and closed forms..
does anyone have any idea for showing the following: Let $f_0, f_1:M\rightarrow N$ smooth homotopic maps between the manifolds $M$ and $N$. Suppose $M$ is compact with no boundary. Show that for every ...
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Maurer-Cartan 1-form
Can anyone help me with the following? Let $\rho$ be the right-invariant Maurer-Cartan 1-form
$$\rho = dg\ g^{-1}$$
I want to show that the MC equation
$$d\rho - \rho \wedge\rho = 0$$
holds.
So ...
3
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1answer
47 views
Curvature tensor of 2-sphere using exterior differential forms (tetrads)
$ds^2= r^2 (d\theta^2 + \sin^2{\theta}d\phi^2)$
The following is the tetrad basis
$e^{\theta}=r d{\theta} \,\,\,\,\,\,\,\,\,\, e^{\phi}=r \sin{\theta} d{\phi}$
Hence, $de^{\theta}=0 ...
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4answers
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Classic example of a non exact form
Let $\dfrac{xdy-ydx}{x^2+y^2}$ be a 1-form defined in $\mathbb{R}^2\backslash\{0\}$.
Where can I find a detailed proof that it is not exact? I would prefer a proof that doesn't use results about ...
2
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2answers
65 views
Symplectic Form Preserved by Orthogonal Transformation
I'm trying to prove that the symplectic form
$$\omega = d(\cos\theta) \wedge d\phi$$
is preserved by the action of $SO(3)$ on $S^2$ where $\phi$ and $\theta$ are spherical polars. Now $SO(3)$ simply ...
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0answers
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Is Cartan's magic formula applicable to time dependent vector fields?
Cartan's magic formula states:
$$\mathcal{L}_v\omega = i_v\mathrm{d}\omega + \mathrm{d}i_v\omega$$
Is this also true for time dependent vector fields? If so: How can I prove it? If not: Is there a ...
4
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2answers
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Orientations on Manifold
This is a very basic definition of orientable and very basic example 20.5 however ı could not understand definition in an good way so ı want you to explain my green writing please :) and my example ...
0
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1answer
48 views
integral of closed differential form
This is my first course in differential forms so it might be a trivial question. If $\mu$ is a $n-1$-form on $n$-dim manifold $X$ the book uses that
$$\int_X{d\mu}=0.$$
Is this expression valid for ...
3
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2answers
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Constructing a 2 form that does not vanish on a space from a one form that does
This may be a silly question...
Given a nonzero one-form $\omega$ that vanishes on a subspace $W$ (with dimension larger than $2$), is it possible to find another one form $\phi$ such that ...
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2answers
47 views
Solve the i.v.p DE
Solve the i.v.p $y^{(4)}-y'''=0 , y(0)=0, y'(0)=0, y"(0)=0, y"'(0)=0$
Would I use the formula $a^{(1/n)}=R^{(1/n)}e^{(e^{i(alpha+2k(\pi))/n})}$
2
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2answers
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Why is $\theta \not \in C^{\infty}(S^1)$?
Why is $\theta \not \in C^{\infty}(S^1)$? I know that since $\int_{S^1} d\theta = 2\pi$ then $d\theta$ is not exact. Thus since $d(\theta)=d\theta$, $\theta$ must not be $C^{\infty}$ but it seems ...
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1answer
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$k$-forms on $\mathbb{R}^n$
Given an expression like
$$
dx_1\wedge dx_2 \wedge dx_4 \left( \begin{bmatrix} 1\\2\\3\ \end{bmatrix} \ , \ \begin{bmatrix} 4\\5\\6 \end{bmatrix} \ , \ \begin{bmatrix} 7\\8\\9 \end{bmatrix} \right) \ ...
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1answer
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What is a de Rham k-form?
I generally know what a differential k-form is. But what does it mean for a k-form to be a "de Rham" k-form?
Thanks in advance!
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3answers
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Whats the connection between formss and vector fields?
I heard someone talking about how vector fields are the kernels of forms. Can someone give me a detailed explanation about how this works? Thanks.
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2answers
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Inducing orientations on boundary manifolds
Given a $k$-manifold $M$, such that $\partial M$ is a $(k-1)$-manifold, there is a standard way in which $\partial M$ inherits the orientation of $M$. So if $M$ is oriented by the form field $\omega$, ...
6
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1answer
81 views
What is the relation of $\int f dx^1\wedge dx^2\wedge …\wedge dx^n=\int f dx^1…dx^n$
In a book "calculus on manifolds" it is defined that $\int f dx^1\wedge dx^2\wedge ...\wedge dx^n=\int f dx^1...dx^n$ but how it is possible the relate the integrand of a multilinear function ...
2
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1answer
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Computing $n$-th external power of standard simplectic form
I need some help:
Define a 2-form on $R^n$ by $\omega=dx_1\wedge dx_2+dx_3\wedge dx_4+...+dx_{2n-1}\wedge dx_{2n}$. How to compute $\omega^n:=\omega\wedge\omega\wedge\ldots\wedge\omega$?
2
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1answer
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What does $\Omega^\bullet(M)$ mean?
What does $\Omega^\bullet(M)$ mean?
I know that $\Omega^k(M)$ is the set of all differential k-forms.
Thanks in advance!
2
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1answer
64 views
Differential Forms and Area
I think I'm just misunderstanding something here, but in $\mathbb{R}^2,$ there exists a $1$-form (in fact infinitely many) $\omega$ such that for any region $S,$ we have $\int_{\partial S} \omega = ...
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2answers
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Finding a particular form that orients a k-manifold
Suppose one has a $k$-manifold given by $f^{-1}(0)$ for some $C^1$ map $f: U\to \mathbb{R}^{n-k}$ (where $[D f(x)]$ is surjective). How can one construct a form-field $\omega$ that orients the ...
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1answer
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Concept of integration to differential form
How to integrate differential form actually. As far as I know, a differential form is a multilinear function mapping from a vector space to a real number. Let's take $\int_c fdx+gdy$ as an example. It ...
3
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3answers
71 views
Integral depending on a path?
I need to check whether differential form $\omega$ has, in the domain $G$, such property that it's integral doesn't depend on path. In my exercise: $\omega = \frac{ydx -xdy}{x^2+xy+y^2}$ and $G= R^2 ...
4
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2answers
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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 ...
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1answer
72 views
Definition of pull back operation
Let $\varphi:U \rightarrow V$ be a differentiable map between open sets $U \subset \mathbb{R}^n$ and $V \subset \mathbb{R}^m$. Define the pull back operation $\varphi^*: \Omega^{k}(V) \rightarrow ...
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1answer
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Orienting curves with differential forms
Consider the circle given by the equation $x^2+y^2=1$. We can orient this curve by choosing the tangent vector field $(-y,x)^T$, which defines a direction.
Supposedly we can do this with by taking an ...
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1answer
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understanding simple multivariable integrals in terms of differential forms
I am learning a bit about differential forms: defining differential forms in terms of elementary forms, integrating forms over parametrized domains, etc.
I would like to relate this to my previous ...
2
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1answer
46 views
On simply connected domains
During lecture we defined simply connected set in $\mathbb{R}^n$:
$\Omega \subset \mathbb{R}^n$ is simply connected, iff it is connected and for any $C^1$ closed curve $c:[0,1]\rightarrow \Omega$ ...
2
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1answer
38 views
Wedge product of differential and volume form
Let $f(x)$ be a $C^1$ function defined on $\mathbb{R}^n$ and $\nabla f(x) \neq 0$ for any $x \in \mathbb{R}^n$. If $d\sigma$ is the volume form on hypersurface $f(x)=c$ induced from $\mathbb{R}^n$ ...
3
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1answer
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Algebraic Topology Double Complexes
I am going through Bott and Tu and trying to do Exercise 9.13 which says
When a homomorphism $f: K \rightarrow K'$ of double complexes induces $H_d$-isomorphism, it also induces $H_D$-isomorphism.
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A $k$-form is thought of as measuring the flux through an infinitesimal k-parallelepiped
On the wikipedia has written "A $k$-form is thought of as measuring the flux through an infinitesimal k-parallelepiped." How does a $k$-form do this? if this sentence is right, then the flux of which ...
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An explicit $\Lambda_R^\ell(M)$ when $M$ is not free
Let $\Lambda_R^n(M)$ be the nth exterior power of an $R$-module $M$. Let us assume $M$ is finitely generated.
When $M$ if free, say, $M=R^{\oplus d}$, we know \begin{equation}
\Lambda_R^n(M)\cong ...
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3answers
69 views
Differential Forms, Exterior Derivative
I have a question regarding differential forms.
Let $\omega = dx_1\wedge dx_2$. What would $d\omega$ equal? Would it be 0?
2
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1answer
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Formal finite sum for integration on k-chains
This question is related to the integration of differential forms on chains, as exposed in the document at: http://www.math.upenn.edu/~ryblair/Math%20600/papers/Lec17.pdf . The author gives the ...
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How general formulation of Stoke's theorem relate to Kelvin-Stokes theorem
I asked a similar question, but I realized the question is too vague and it's better to start a new one:
We know that there are two usually used formulations of Stoke's theorem. One is vector ...
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Example of differential form usage of Stoke's theorem
There are many examples that show how Kelvin-Stokes theorem is used.
But I would like to see an example that uses differential form usage of Stoke's theorem and is hard or impossible to solve by ...
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1answer
28 views
Question about Interior product computation
I need to evaluate $\omega=i_X(dx\wedge dy)$ where $X$ is a vector field in $\mathbb{C}^2$ (which means $p=2)$. If I write $X(x,y)=(X_1(x,y),X_2(x,y))$, or simply $X=(X_1, X_2)$, then the interior ...
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1answer
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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 ...
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Baby Rudin, Chapter 10, Problem 23 (d) - Differential forms.
In problems 21 and 22, Rudin defines the differential forms $\eta=\dfrac{xdy-ydx}{x^2+y^2}$ and $\zeta=\dfrac{x dy \wedge dz+ydz \wedge dx+z dx \wedge dy}{r^3}$ and the reader is asked to prove ...
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Is this a trivial Stokes exercise?
I'm working on this exercise from an old Spivak Differential Geometry book which I must be misunderstanding. The question reads:
Let $M$ and $N$ be compact $n$-dimensional manifolds with boundary and ...
2
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2answers
83 views
Is $\omega_n$ exact in $\mathbb R^n -\{0 \}$?
For $n \ge 2$ consider the differential form $\omega_n=r^{-n} \sum_{i=1}^n(-1)^{i-1}x_idx_1 \wedge \ldots \wedge dx_{i-1} \wedge dx_{i+1} \wedge \ldots \wedge dx_n$, defined on $\mathbb R^n \setminus ...
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Dimension of de Rham Cohomology groups?
Is there a simple way to prove that the de Rham cohomology groups of a compact manifold $M$ have finite dimension as $\mathbb{R}$-vector spaces?
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1answer
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Use Fund Thm to evaluate the integral of $ ze^{x^2} dydz + 3ys dydz + (2-yz^7)dxdy $ over surface of the unit cube, except bottom face.
Use the Fundamental Theorem to evaluate the integral of $ ze^{x^2} dydz + 3ys dydz + (2-yz^7)dxdy $ over the surface of the unit cube, except the bottom face.
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1answer
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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 $.
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Let $ \omega = (x^2 - y)dx + (x - y^2)dy $. Verify $ \Lambda $* is a contravariant functor from finite dimensional vector spaces to graded algebras.
Let $ \omega = (x^2 - y)dx + (x - y^2)dy $. Verify that $ \Lambda $* is a contravariant functor from finite dimensional vector spaces to graded algebras.
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1answer
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Is $\zeta=\frac{x dy \wedge dz+y dz \wedge dx+z dx \wedge dy}{r^3}$ exact in the complement of every line through the origin?
$r=\sqrt{x^2+y^2+z^2}$ of course.
If the line is the $z$ axis, it is given in the book (Rudin) that $\zeta=d \left( -\dfrac{z}{r} \dfrac{xdy-ydx}{x^2+y^2} \right)$
I've managed to figure out 2 ...
4
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1answer
100 views
Relating volume elements and metrics. Does a volume element + uniform structure induce a metric?
AFAIK a metric uniquely determines the volume element up to to sign since the volume element since a metric will determine the length of supplied vectors and angle between them, but I do not see a way ...
0
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1answer
73 views
Finding the winding number of a curve
Let $\gamma(t)=(r \cos t,r \sin t)$, for some $r>0$, and let $\Gamma$ be a $C^2$-curve in $\mathbb R^2-\{\bf 0\} $, with parameter interval $[0,2 \pi]$, with $\Gamma(0)=\Gamma(2 \pi)$, such that ...
