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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2answers
212 views

Applying the Frobenius theorem to a decomposable 2-form

So I have the following problem: Suppose $\omega=\phi \wedge \theta$ is a closed decomposable 2-form on $M$ a manifold (decomposable just means it can be written as a wedge of 1-forms). Suppose $p\in ...
3
votes
1answer
104 views

Wedge product descend to the cohomology

I found this statement in Raoul Bott "Differential Forms in Algebraic Topology": "Because the wedge product is an antiderivation, it descends to cohomology." Apparently this meant to be really obvious ...
3
votes
1answer
236 views

Is this differential form closed / exact?

Could you check if I calculated the exterior derivative of this differential form $\omega$ correctly? $\omega \in \Omega_2 ^{\infty} (\mathbb{R}^3 \setminus \{0\})$ $\omega = (x^2 + y^2 + z^2)^{\...
3
votes
2answers
155 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 ...
3
votes
2answers
55 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 $\omega\...
3
votes
2answers
1k views

Chain rule and differential forms

It is easy to show that the differential forms of order $1$ obeys a form of chain rule. To be precise, $d(f(g(x)) = f^\prime(x) d(g(x))$. This can be for example proved by fixing a co-ordinate basis $...
3
votes
1answer
1k views

The area form of a Riemannian surface

Let $(M,g)$ be an oriented Riemannian surface. Then globally $(M,g)$ has a canonical area-$2$ form $\mathrm{d}M$ defined by $$\mathrm{d}M=\sqrt{|g|} \mathrm{d}u^1 \wedge \mathrm{d}u^2$$ with respect ...
3
votes
1answer
198 views

iterated integration and shuffle product

Recall that a $(r,s)$-shuffle $\sigma$ is a permutation of $r+s$ letters such that $\sigma^{-1}(1) < \cdots< \sigma^{-1}(r)$ and $\sigma^{-1}(r+1) < \cdots < \sigma^{-1}(r+s)$. If $f_1(t),...
3
votes
1answer
317 views

Computing the restriction of a differential form

Define $\omega$ on $\mathbb{R}^3$ by $\omega = x\,dy\wedge dz + y\,dz\wedge dx + z\,dx\wedge dy$. Thus far I have computed $\omega$ in spherical coordinates $(\rho,\phi,\theta)$, as well as computed ...
3
votes
2answers
104 views

Kernel of $\omega^\#$ is $k$-dimensional

Let $M$ be a smooth manifold with coordinates $\{q^i\}_{i=1}^n$ .The variables $(q^i,p_i)$ are coordinates on the cotangent space $\Omega=T^*M$. Any cotangent space carries a natural one-form $\tilde{\...
3
votes
1answer
70 views

De Rahm cohomology of a sphere, help with proof

I am working through Guillemin and Pollack's proof that the de Rahm cohomology of the sphere is $H^p(\mathbf{S}^k) = \mathbf{R}$ for $p = 0$ and $p = k$ and $H^p(\mathbf{S}^k) = 0$ otherwise. Here, $...
3
votes
1answer
84 views

Inclusion, pullback of differential form

Let $\omega=x\,dy\wedge dz +y\,dz\wedge dx+z\,dx\wedge dy$ or in spherical coordinates (unless I had made some mistake) $\omega=r^3\cos \theta\, d\phi\wedge d\theta$. Now I want to find $i^*\omega$ ...
3
votes
1answer
93 views

Integrating the Hopf invariant for $\pi:S^3\to S^2$

I've been working on the last part of problem 9., chapter 9 in Nakahara's Geometry, Topology and Physics all day, with no success, and am in need of some assistance. We are asked to compute the Hopf ...
3
votes
1answer
74 views

Stokes' theorem and symplectic geometry

Let $V = \mathbb{R}^2,$ as a vector space then the Poincaré invariant is an integral $\int_{\gamma} \theta$ where $\theta = p dx $ is the symplectic 1-form and $\gamma$ a closed curve. Now, it is ...
3
votes
1answer
35 views

Exterior derivarive dependent only on point

For any one-form (a linear form on the tangent space of each point) we have its exterior derivative $d\omega$ which is a two-form defined by $d\omega(X,Y)=D_X(\omega(Y))-D_Y(\omega(X))-\omega([X,Y])$ ...
3
votes
1answer
65 views

Integration with 2-forms

Wikipedia says: Let $$ \omega=f_{z}\, \mathrm dx \wedge \mathrm dy + f_{x}\, \mathrm dy \wedge \mathrm dz + f_{y}\, \mathrm dz \wedge \mathrm dx $$ be a 2-form on a surface with parametrization $$\...
3
votes
1answer
54 views

Finding the Lie derivative of a complex valued function

Question: Let vector field $\mathbb{X}$ be given by $\mathbb{X}(x,y)=(-y,x)$ Let $f(x,y)$ be the complex-valued function given by $f(x,y)=(x+iy)^m$ where $m>0$ Show that $L_\mathbb{X}f=imf$ My ...
3
votes
1answer
136 views

How to derive the standard symplectic form on a 2-sphere in cylindrical polar coordinates?

I have already know that at a point $p\in S^2$ with $u,v$ the tangent vectors at $p$. Then the standard symplectic form is $\omega_p(u,v) :=\langle p,\,u\times v\rangle$ where $u\times v$ is the outer ...
3
votes
1answer
246 views

Differential Forms on submanifolds

Say I take an embedded submanifold of $\Bbb R^n$, like the sphere. Any differential form on $\Bbb R^n$ can be restricted to the sphere. My question is this: is any differential form on the sphere (or ...
3
votes
1answer
78 views

Conceptual Question on different representations of Hyperplanes, Higher Standpoint, Coordinate-free

In a vector space $V$ over some field $F$ a hyperplane is the kernel of some linear transformation $T : V \to F$, i.e. the kernel of an element of the dual space (this could be taken as the definition ...
3
votes
1answer
67 views

Finding the surface element of $S^{3}$

How does one show that the surface element of $S^{3} = \{x=(x_{1}, \dots, x_{4}) \in \mathbb{R}^{4} \mid |x|^2=1\}$ is given by the following $3$-form: $$\omega=x_{1}dx_{2}\wedge dx_{3}\wedge dx_{4}-...
3
votes
1answer
80 views

Recovering a frame field from its connection forms

I have a faced a research problem where I would need to recover a frame field given its connection forms. More precisely, I begin with an orthonormal frame field (given by data) in $\Re^3$ written as $...
3
votes
1answer
116 views

Why generalize vector calculus with $k$-forms instead of $k$-vectors?

The motivation usually given to differential forms is that they generalize vector calculus nicely. That's true, but there are also $k$-vectors, i.e., objects from $\Lambda^k(V)$ instead of $\Lambda^k(...
3
votes
2answers
103 views

What is the value of the 2-form $X(u(Y))-Y(u(X))-u([X,Y])$, is it $2du(X,Y)$?

Let $X,Y$ be two vector fields on a Riemannian manifold $(M,g,\nabla)$ where $\nabla $ is the symmetric metric connection on $M$ and let $u$ be a 1-form. I want to find the value or a simple form or ...
3
votes
1answer
662 views

A $2$-form on $S^2$ is exact if it integrates to zero.

I'm trying to show that a $2$-form on $S^2$ is exact if and only if it integrates to zero, without appealing to de Rham's theorem (basically only using the Poincaré lemma [that every closed form on a ...
3
votes
1answer
200 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
votes
1answer
583 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 \,\,\,\,\,\,...
3
votes
1answer
125 views

differential form is exact on $ U \cup V$

Let $ U ,V \subset \mathbb R ^n$ two simply connected open sets such that $ \displaystyle{ U \cap V}$ is a connected set. If $\omega$ is a closed 1-form wich is exact in $U$ and $V$ prove that: $\...
3
votes
1answer
246 views

Relationship between Rotational Motion and Standard Linear Motion

I am looking to study elementary mechanics (physics) from the point of view of differential forms on manifolds, and moments, and am wondering if there are any texts that generalize the notions of ...
3
votes
1answer
354 views

Intuitive interpretation of these differential forms

Let $\pi: S^2-\{N\}\to \mathbb R^2$ be the stereographic projection map. Let $\sigma:\mathbb R^2\to S^2-\{N\}$ be its inverse. Let $p\in S^2-\{N\}$ and $x_1,x_2\in$ the tangent space of $S^2$ Would ...
3
votes
1answer
241 views

Derivative of an integral of differential form

I have some smooth function $g(x) \colon \mathbb{R}^{n}_{+} \to \mathbb{R}_+$ such that $G_{t} = \{ x \in \mathbb{R}^n_+ \mid g(x) \leqslant t \}$ is compact. I consider a function $$ f(t) = \int\...
3
votes
1answer
261 views

Elementary symmetric polynomials and matrices of 1-forms

Let $A$ be a $n \times n$ matrix of 1-forms (for example, a connection form). Note that $A \wedge A$ is not $0$, but by using the anti-symmetry of the wedge product applied to the entries of $A$ we ...
3
votes
1answer
57 views

Differentiating the pull-back of a one-form

Let $\Omega$ be an open subset of a vector space $V$ and let $\alpha\colon \Omega\to V^*$ be a one-form on $\Omega$. Assuming that $\alpha$ is differentiable, then for any $x\in \Omega$, $D\alpha (x)\...
3
votes
1answer
61 views

Differential forms on a projective curve: two constructions

Let $X$ be a projective curve on a perfect field $k$ ($k$-scheme integral, separated, of finite type) and let $K$ be the function field of $X$. Let's compare the following two constructions: 1. ...
3
votes
1answer
105 views

Intrinsic definition of differential k-form on smooth manifold

Suppose I have a $k$-dimensional manifold embedded in $\mathbb{R}^n$. Munkres defines a $k$-form on $M$ as a a function $\omega$ that assigns an alternating tensor at each point $p \in M$ that acts on ...
3
votes
1answer
66 views

Poincare's lemma for 1-form

Let $\omega=f(x,y,z)dx+g(x,y,z)dy+h(x,y,z)dz$ be a differentiable 1-form in $\mathbb{R}^{3}$ such that $d\omega=0$. Define $\hat{f}:\mathbb{R}^{3}\to\mathbb{R}$ by $$\hat{f}(x,y,z)=\int_{0}^{1}{(f(...
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votes
1answer
126 views

Spivak Calculus on Manifolds - Theorem 4-10

Part (4) of Theorem 4-10 in Spivak's Calculus on Manifolds says the following: If $\omega$ is a $k$-form on $\mathbb{R}^m$ and $f: \mathbb{R}^n \rightarrow \mathbb{R}^m$ is differentiable, then $f^...
3
votes
1answer
44 views

$d(\iota_v\rho)=0 \implies d(\phi\iota_v\rho)=d\phi(v)\rho$?

My motivation is physical, but my question is purely mathematical. Everybody knows, that the power of the electric current in a piece of wire is $$P=UI$$ where the wire is regular domain $V$ in a ...
3
votes
1answer
40 views

There is no 2-form on $\mathbb R^3$ whose restriction to every surface gives its volume form

Prove that there isn't such a 2-form $\omega$ on $\mathbb{R}^3$ that $\omega$ restricted to any surface $\Sigma$ gives its volume form. Suppose that there is such a form: $$\omega=f_3(x,y,z)dx \...
3
votes
1answer
123 views

How to visualize differential forms geometrically

I've been attempting to teach myself differential geometry and I have heard that one can visualise them geometrically and that this can sometimes be helpful for an intuitive understanding of them. For ...
3
votes
3answers
115 views

Exterior derivative

Let $$\omega = \frac{1}{2} \sum_{i,j} \omega_{i,j} dx_i \wedge dx_j$$ be an antisymmetric form, so i.e. $\omega_{i,j} = - \omega_{j,i}.$ Now, in some lecture notes I found that $$d \omega (X_F,X_G,...
3
votes
1answer
97 views

Why does a differential form represent a vector field?

I'm trying to learn the Divergence/Stoke's theorem and I can't wrap my head around the meaning of a differential form in this context. What does it mean that a differential form represents a vector ...
3
votes
1answer
93 views

Proving that the pullback map commutes with the exterior derivative

I'm trying to prove that the pullback map $\phi^{\ast}$ induced by a map $\phi:M\rightarrow N$ commutes with the exterior derivative. Here is my attempt so far: Let $\omega\;\in\Omega^{r}(N)$ and let ...
3
votes
1answer
63 views

Lie derivative for a wedge product $\omega_{1}\wedge\omega_{2}$

I have to prove that $L_X\omega_{1}\wedge\omega_{2}=(L_X\omega_{1})\wedge\omega_{2}+\omega_{1}\wedge(L_X\omega_{2})$ using the definition $$L_X\omega=\frac{d}{dt}\vert_{t=0}\varphi_t^*\omega=\lim_{t\...
3
votes
1answer
166 views

Area form a differential form?

I just read that $\omega_x( \eta, \zeta) := \langle x, \eta \times \zeta \rangle $ is the area form on the sphere, where $x \in \mathbb{S}^2$ and $\eta,\zeta \in T_xM.$ All I see is that this is ...
3
votes
1answer
184 views

Volume form for a product manifold.

Given two Riemannian manifolds $(M,g^M)$ and $(N,g^N)$, we can construct a product Riemannian manifold $(M\times N,g^{M \times N})$ as described in Product of Riemannian manifolds? . Is there a ...
3
votes
1answer
65 views

Well-definedness of a coboundary map between a reduced $L^2$ de Rham cohomology group and a relative cohomology group

I'm working right now with this paper of Carron. And I think I'm stuck at a relatively simple question. On page 11 he is defining a coboundary map $b : H^k_{2, \text{reduced}}(M - K) \to H^{k+1}(K, \...
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votes
1answer
141 views

Homework: calculation about differential form

Here is the question: Let $\omega = A dy\wedge dz + B dz \wedge dx + C dx \wedge dy$ in $\mathbf{R}^3$, and $d\omega = 0$. Denote \begin{eqnarray} \alpha = \int_0^1 tA(tx,ty,tz)dt\cdot(ydz-zdy)\\ +\...
3
votes
1answer
61 views

1-form on $S^n$ with non-degenerate differential.

How it can be solved? Whether there is a $1$-differential form $w$ on $S^n$, such that $dw$ is non-degenerate on $T_aS^n$ for each $a \in S^n$?
3
votes
1answer
155 views

Independence of coordinate charts in the definition of the order of a pole of a meromorphic 1-form on a Riemann surface

I am currently reading Forster. Let $X$ be a Riemann suface, and $Y$ an open subset of $X$. Assume we have a meromorphic 1-form $\omega \in \Omega(Y \setminus \{a\})$ with a pole at $a \in Y$. One ...