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

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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 ...
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1answer
53 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 ...
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1answer
130 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
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1answer
228 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 ...
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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 ...
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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 ...
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1answer
79 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 ...
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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 ...
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2answers
98 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 ...
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583 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 ...
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1answer
195 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 ...
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1answer
566 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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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: ...
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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 ...
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1answer
351 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 ...
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1answer
240 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) = ...
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1answer
257 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 ...
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1answer
53 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 ...
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1answer
58 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. ...
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1answer
89 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 ...
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1answer
62 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 ...
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1answer
118 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 ...
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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
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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 ...
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1answer
115 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 ...
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1answer
34 views

Restriction of differential $1$-forms to open subsets?

A vector field on a manifold $M$ is a linear map $X:C^\infty(M)\longrightarrow C^\infty(M)$ with an additional property. The set $\mathfrak{X}(M)$ of all vector fields on $M$ is a ...
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3answers
111 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 ...
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1answer
84 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 ...
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1answer
85 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 ...
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1answer
61 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 ...
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1answer
144 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 ...
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1answer
166 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 ...
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1answer
64 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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1answer
138 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)\\ ...
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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
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1answer
154 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 ...
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0answers
46 views

Does integration wrt to a differential form always come from a measure?

More precisely, is there an $n$-manifold $M$ with an $n$-form $\omega$ such that there is no measure $\nu$ on $M$ satisfying $$\int f \omega = \int f d\mu $$ for all compactly supported smooth ...
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0answers
22 views

On the meaning of formal sums of $k$-cubes, i.e. $k$-chains (in integration on manifolds)

A singular $k$-cube in $A \subseteq \mathbb R^n$ is a continuous function $c : [0,1]^k \to A$. A singular $0$-cube in $A$ is then a function $f : \{0\}\to A$, what amounts to the same thing, a point ...
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0answers
25 views

Kodaira decomposition of 1-form on a Real manifold

I recently stumbled across the Hodge decomposition theorem, which states that on any compact orientable manifold, for any form the following holds $$ \omega = \text{d}\alpha + \delta \beta + \gamma $$ ...
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45 views

Complex Analysis with differential forms

I'm studying a little of Complex Anlysis and I have seen that I can thing the integrals of complex functions as integrals of differential forms in $\mathbb{R}^n$. For example I know that Cauchy ...
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1answer
169 views

Why is Schouten-Nijnhuis bracket trivial on Poisson cohomology?

For a commutative algebra $A$, let a biderivation $P$ be called a Poisson structure if $[[P,P]]=0$ (the bracket is Schouten-Nijenhuis). Then one obtains a complex of multiderivations with $[[P,{}]]$ ...
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1answer
70 views

If $\alpha \wedge \omega = 0$ then $\alpha = f \omega$ for some $f$

Question: Let $\alpha$,$\omega$ be $1$-forms of class $C^1$ in $\mathbb R^3$. If $w(x) \neq 0$, for every $x \in \mathbb R^3$ and $\alpha \wedge \omega = 0$. Then $\alpha = f\omega$, where $f : ...
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0answers
57 views

Suggestion for reference book for differential forms, differentiable manifolds and other topics

I am currently taking a course on multivariable calculus and our professor is following the book by Do Carmo: Differential forms and applications. I feel the text is too rigorous, which I really ...
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91 views

Find the interior product of a basic p-form $\alpha = dx_1 \wedge dx_2 \wedge \ldots \wedge dx_p$ a and a vector field $X$

I'm reading Differentiable Manifolds by Nigel Hitchin, that is, his class notes for an Oxford course freely available here. In particular, I'm trying to understand the interior product on manifolds, ...
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57 views

Integration of forms on product manifolds

Let $M$ be a compact, connected and oriented smooth manifold of dimension $m$ and let $\pi_1,\pi_2:M\times M\rightarrow M$ be the projections to each factor. Given $\alpha\in H^m(M;\mathbb{R})$, I ...
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340 views

push forward of differential form/ integration over fiber

It is elementary that differential forms can be pulled back via a smooth map between manifolds. However, I was reading a paper and came across a construction about push forward of a differential form ...
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1answer
153 views

Vectors, Forms, Multivectors, and Tensors

In researching some of the ways that vectors (and vector fields) generalize I find that there are apparently many different objects that generalize them -- matrices, differential forms/ covectors, ...
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1answer
70 views

An exercise on differential forms

Let $\omega$ be a $1$-form in $U \subset \mathbb{R}^2$. A local integrating factor in $p$ for $\omega$ is a function $g: V \rightarrow \mathbb{R}$ defined in a neighbourhood $V$ of $p$ such that ...
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0answers
36 views

The lie bracket as a limit

Let $M$ be a manifold and $f_t$ be the monoparametric group of local transformations generated by a vector field $X$. Suppose $w$ is a $k$-form and $x_1,\dots,x_k$ are vector fields. How can we see ...
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131 views

Differential Form Pullback Definition

I'm having some difficulty following how Spivak (Calculus on Manifolds) has induced the pullback on page 89. From reading elsewhere online it seems convention is to define the induced map of the ...