For questions about differential forms which commonly arise in differential geometry, and sometimes in multivariable calculus.

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3
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1answer
136 views

De Rham cohomology notation

According to http://en.wikipedia.org/wiki/De_Rham_cohomology, one defines the $k$-th de Rham cohomology group $H^{k}_{\mathrm{dR}}(M)$ to be the set of equivalence classes, that is, the set ...
-3
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1answer
172 views

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 $\dim G=n$). ...
2
votes
1answer
86 views

Question on Differential forms

Let $\Omega\subseteq \mathbb{R}^2$ be an open set and $\omega=\omega^1dx_1+\omega^2dx_2$ an $1$-form on $\Omega$ and $$L=\omega^2\frac{\partial}{\partial x_1}-\omega_1\frac{\partial}{\partial ...
0
votes
1answer
28 views

Question on 2-chain on $\mathbb{R}^3$

Let $\gamma:[0,1]\to\mathbb{R}^3\setminus\{0\}$ be a simplex, with $\gamma(0)=\gamma(1)$. How can I show that exists a $2$-chain $\sigma$ on $\mathbb{R}^3\setminus\{0\}$ such that ...
1
vote
1answer
206 views

Locally exact differential in a disk is exact

I'm reading through Ahlfors' Complex Analysis text for self study, and I found difficulty with a proof. In chapter 4 he defines a locally exact differential as a differential who is exact in some ...
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0answers
51 views

What exactly does it mean to say that “functions cannot be integrated on Riemann surfaces”?

I've seen statements of this sort used to motivate the introduction of differential forms, and I'm not sure exactly what's meant. Obviously if you start by defining differentiation as an operation ...
3
votes
2answers
129 views

Prove that $\frac{1}{2\pi}\frac{xdy-ydx}{x^2+y^2}$ is closed

I would like to prove that $\alpha = \frac{1}{2\pi} \frac{xdy-ydx}{x^2+y^2}$ is a closed differential form on $\mathbb{R}^2-\{0\}$ . However when I apply the external derivative to this expression ...
2
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0answers
167 views

Exact and Closed forms on Manifolds with Boundary

Let $\bar{M}$ be a manifold with boundary and let $M$ be its interior. Is this statement correct? A smooth k-form $\alpha$ on $\bar{M}$ is closed (exact) if and only if its restriction to $M$, i.e. ...
2
votes
2answers
347 views

Hodge dual on orthonormal basis: two inconsistent answers

I'm trying to learn differential geometry using Göckeler & Schücker's book and I have some problems with the hodge star. As an example, say we have two orthonormal bases $e^i$ and ...
0
votes
1answer
235 views

Clarification about differential forms in polar coordinates

In my course about differential forms, we define 1-forms as follows: If $(e_1,..,e_m)$ is the standard basis of $\mathbb{R}^m$ and $\sigma$ a chart around $p\in M$ on the m-dimensional manifold $M$, ...
1
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1answer
108 views

A differential form to compute the k-volume of a k-parallelogram in n dimensions

Computing the k-volume of a k-parallelogram (i.e. a parallelogram spanned by k n-dimensional vectors) in n dimensions is straightforward: Let $P=[\overrightarrow{v_1},...,\overrightarrow{v_k}]$, then ...
1
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1answer
196 views

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 ...
1
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1answer
231 views

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 ...
4
votes
1answer
197 views

Why does $|dz|=-ir\frac{dz}{z}$ when $|z|=r$?

Sometimes I want to compute a line integral over some circle $|z|=r$, where I have $|dz|$ instead of $dz$ given to me. Reparametrizing with $z=re^{it}$, it follows that $dz=rie^{it}dt=izdt$. But I ...
0
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1answer
38 views

index $ n(F;D)$ is odd integer

Let $ F: U \subset \mathbb R ^2 \to \mathbb R^2$ be a map with $ F(x,y) = (f(x,y) , g(x,y))$ ,satisfies $F(-q)=-F(q) \quad \forall q \in D \subset U$ where $D$ is a closed disk with center the origin ...
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1answer
135 views

Tangent Vectors and Differential 1-forms.

I have this 1-form on $\mathbb R^3$ given by $\omega=dz+\frac{x}{2}dy-\frac{y}{2}dx$. If $p_0=(x_0,y_0,z_0)$ and $\vec v=(u_0,v_0,w_0)$, then find the set of tangent vectors $\vec v_{p_0}$ such that ...
1
vote
2answers
72 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 ...
2
votes
2answers
69 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 ...
8
votes
1answer
194 views

How to prove $(0,1)$ form is not $\overline\partial$-exact

On a complex manifold, if we are dealing with the $d$ operator, there's a pretty easy way of showing some form is not $d$-exact, simply by integrating in a closed loop. If you can find a loop that is ...
4
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1answer
68 views

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 ...
3
votes
1answer
296 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 ...
2
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4answers
534 views

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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1answer
73 views

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 ...
6
votes
1answer
114 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 ...
3
votes
2answers
130 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
84 views

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
votes
2answers
183 views

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 ...
5
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1answer
107 views

How to prove that this kind of differential form exists on an algebraic curve?

The following is a problem in Miranda's Algebraic Curves and Riemann Surfaces. Given any algebraic curve $X$ and a point $p \in X$, show that there is a meromorphic $1$-form $\omega$ on $X$ whose ...
0
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1answer
138 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 ...
2
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3answers
73 views

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.
3
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2answers
49 views

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 ...
-1
votes
1answer
188 views

Exterior Derivative question

In http://mathworld.wolfram.com/ExteriorDerivative.html, there is a section that starts from: Define the exterior derivative by $Dt ≡ \frac{\partial}{\partial x} \wedge t$ First of all, what ...
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2answers
50 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 $a^{(1/n)}=R^{(1/n)}e^{(e^{i(alpha+2k(\pi))/n})}$
1
vote
1answer
64 views

$k$-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) \ ...
0
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1answer
81 views

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!
2
votes
1answer
111 views

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 $\omega^n:=\omega\wedge\omega\wedge\ldots\wedge\omega$?
7
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1answer
782 views

Are Clifford algebras and differential forms equivalent frameworks for differential geometry?

I recently discovered Clifford's geometric algebra and its application to differential geometry. Some claim that this conceptual framework subsumes and generalizes the more traditional approach based ...
2
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1answer
109 views

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
votes
1answer
232 views

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 ...
2
votes
1answer
278 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
63 views

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 ...
5
votes
2answers
309 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 ...
3
votes
3answers
79 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 ...
0
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1answer
335 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
50 views

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 ...
1
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1answer
82 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 $c:[0,1]\rightarrow \Omega$ ...
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1answer
184 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
votes
1answer
181 views

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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0answers
49 views

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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0answers
27 views

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 ...