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

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21 views

singular $1$ cube - Boundary of $2$ chain

This is an exercise from "Calculus on Manifolds" by Michel Spivack (first edition, p.100): If $c$ is a singular $1$-cube in $\mathbb{R}^2-\{0\}$, with $c(0)=c(1)$, show that there is an integer ...
0
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0answers
23 views

Independence of parametrization in defining integral of differential form

This is an exercise from Spivak's Calculus on Manifolds. Questions asks the following : (Independence of parametrization). Let $c$ be a singular $k$ cube and $p:[0,1]^k\rightarrow [0,1]^k$ be ...
1
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0answers
27 views

What is an integral differential form and how do we recognize it as such?

I am reading about embedding theorems of various types of manifolds (Kodaira's embedding theorem being one of them), and one condition that repeats in all of them is that the manifolds should be ...
0
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1answer
17 views

Idea behind the tangential vector space?

I am currently reading a chapter about Pfaff forms, but not really understand, why the author introduces tangential vector spaces - the definition seems rather redundant to me, if I didn't overlook ...
0
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1answer
30 views

Understanding the first fundamental form of a surface, how the parametrization doesn't matter.

The following is an excerpt from Pressley's Elementary Differential Geometry on the definition of the first fundamental form. However, there are some parts of this concept that I'm unclear about. It ...
1
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1answer
21 views

Interior product general rule (differential forms)

How is this general form of interior product on forms $$(i_V\omega^{(p)})=\frac{1}{(p-1)!}V^{\mu}\omega_{\mu\mu_1...\mu_{p-1}}dx^{\mu_1}\wedge dx^{\mu_2}\wedge ...\wedge dx^{\mu_{p-1}}$$?
0
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2answers
30 views

Rewriting a k-form as a wedge product with a 1-form

I am trying to show that a general element of the kth exterior product $\Lambda^kV^*$ (of V an n-dimensional vector space) $$ \alpha = \sum_{i} \alpha_i e_i$$ (where the $\{e_i\}$, for $1\leq i\leq ...
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0answers
24 views

For any closed form $a$ with compact support, there exists a form $b$ w.c.s. in the unit ball such that $a-b$ is exact.

Let $\alpha$ be a closed (differential) $k$-form with compact support in $\mathbb{R}^{n}$. We want to prove that there exists a $l$-form $\beta$ with compact support in the unit ball of ...
2
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2answers
25 views

Exterior derivative of a coordinate function

I'm starting to learn about differential forms. From what I understand the coordinate differential forms $dx^1, \dots, dx^n$ are actually the exterior derivatives of the coordinate functions $x^1, ...
-2
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1answer
33 views

Differential equation

Hello if I have differential equation which is a function of x = differential equation which is function of t Can I say that the differential equation which is function of x = C= the ...
0
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1answer
53 views

Checking that a two-form transforms correctly under Lorentz transformations

This is exercise $7.22$ in Supergravity by Freedman and Van Proeyen, but I did not understand it and would appreciate if you clear it out. Given the below, I still don't get how, if we define the ...
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1answer
26 views

Can we see this integral as the line integral of a 1-form

In Stein and Shakarchi's complex analysis, the following definition is given on pg. 21 Let $z:[a, b]\to \mathbf C$ be a parameterization of smooth curve $\gamma$ in $\mathbf C$ and $f$ be a ...
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0answers
29 views

Is the curl of a vectorfield expressable in terms of a Lie derivative?

I am loving differential forms and the general version of Stokes theorem. However, I am also fond of the intuitive pictures of my older physics days of $grad$, $div$ and $curl$. I recently stumbled ...
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0answers
52 views

Formula for the curvature $2$-form.

I'm currently reading a textbook to do with curvature and $k$-forms. It says that the curvature $2$-form given connection $1$-form, $A$, is $$F =d^A A = dA+A \wedge A$$ It then goes on to say that ...
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0answers
26 views

Application of stoke's theorem - Doubt on calculation of integral

This is an exercise question from Spivak's calculus on manifolds chapter number 4 question 26. Show that $\int_{C_{R,n}}d\theta=2\pi n$, and use stoke's theorem to conclude that $C_{R,n}\neq \partial ...
0
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1answer
30 views

How to find a potential of a differential form?

I need some help in understanding the meaning of this exercise: Determine a potential of the following differential form $$\omega = (3x^2y + z) dx + (x^3 + 2yz) dy + (y^2 + x) dz$$ I don't ...
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0answers
29 views

2 forms and Base

Let$\: V \;$ be a n-dimensional vector space and $\:w\;$ a two form. Proof that there exists a base $\alpha_1,\alpha_2,..\alpha_n, \in V^* \;$ so that $\; \omega =\alpha_1 \wedge \alpha_2 + \alpha_2 ...
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1answer
36 views

What does the notation $g\cdot\omega$ mean in Spivak's Calculus on manifolds?

In chapter $4$ (Integration on chains) of Spivak's Calculus on manifolds he says the following: If $f:\mathbb{R}^n\rightarrow \mathbb{R}^m$ is differentiable and $\omega$ be a $k$ form on ...
0
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1answer
46 views

Integration of one form

$\omega=p(x,y)dx+q(x,y)dy\quad$ a continuously differentiable one form and $d\omega =0$ In addition, for $\alpha(t)=(r\cos t,r\sin t)$, $\int_\alpha \omega =0 $ for some $\; r \in \mathbb R$ I need ...
2
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1answer
36 views

$\omega = x^2dx + xydy + xzdz$ over $S^2 = \{(x,y,z) \in \mathbb R^3 \mid x^2 + y^2 + z^2 = 1\}$

Verify if the following differential forms over $S^2$ are closed and/or exact: $\omega_1 = x^2dx + xydy + xzdz$ $\omega_2 = xdy - ydx$ $\omega_3 = zdxdy - ydxdz + xdydz$. What I have done: since ...
2
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0answers
31 views

$k$-form on $\mathbb{P}^n(\mathbb{R})$

Let $\pi$ be the canonical projection from $\mathbb{R}^{n+1}/\{0\}$ to $\mathbb{P}^n(\mathbb{R})$. Given a $k$-form $\alpha$ on $\mathbb{R}^{n+1}/\{0\}$ find necessary and sufficient conditions such ...
2
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0answers
58 views

When are differential forms related by a base space automorphism?

Let $w$ and $u$ be nowhere-vanishing smooth differential forms fields of degree $n$ on a smooth manifold $M$ (aka smooth sections of $\Omega^n(M)$). When does there exist an automorphism $f: M \to M$ ...
1
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1answer
38 views

Show that $\alpha \wedge \mathrm d \alpha =0$ when $\alpha \in \Omega ^1(M)$ and $d(f\alpha) = 0$ for some nowhere zero function $f$

Let $\alpha \in \Omega ^1(M)=\text{Tens}_1(M)$ be a differential form of degree $1$ on the smooth manifold $M$. Suppose that there is $f\in \mathcal C^\infty (M)$ s.t. $f(x)\neq 0$ for all $x\in M$ ...
6
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1answer
192 views

Find type of a differential form on an almost complex manifold

If $M$ is a nearly Kähler manifold (that is, an almost Hermitian manifold on which $\nabla_X(J)X=0$) we have the three-forms $$ A(X,Y,Z)=\langle\nabla_X(J)Y,Z\rangle \quad\text{and}\quad ...
3
votes
1answer
48 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 ...
3
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1answer
46 views

Finding a one-form $\lambda$ such that $d\lambda = \omega$

Let $\omega = 2xz dy\wedge dz + dz\wedge dx -(z^2 + e^x)dx\wedge dy$. We have just started out with differential forms and need to find a one-form $\lambda$ so that $d\lambda = \omega$. ...
3
votes
1answer
49 views

Integrating a density over a Mobius strip

According to this link one can integrate over a Mobius strip by using "densities". That has me very excited but I can't seem to find a reference on this. Can someone provide a book/ online source ...
0
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1answer
54 views

Why do books refer to $dx_1, \ldots, dx_n$ as differentials and also as covectors?

I'm studying differential forms. In Edwards Advanced Calculus, linear functionals are defined and we learn that each linear functional is a linear combination of the dual basis $ \gamma_1, \ldots, ...
3
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1answer
97 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 ...
2
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1answer
49 views

Possible error in Guillemin and Pollack RE de Rahm cohomology?

The context is Guillemin and Pollack, Chapter 4.6, Cohomology with Forms. Let $U$ be an open subset of $\mathbf{R}^k$ and let $\omega$ be a $p$-form on $\mathbf{R} \times U$, represented as $$ ...
1
vote
1answer
27 views

is the vector space of n- forms of an n-manifold equal to the vector space of compactly supported n-forms?

Let $\Omega^{n}(M)$ be the real vector space of smooth n-forms of an n-manifold $M$. It is a real vector space of dimension 1. $\Omega^{n}_c(M)$ is the real vector space of compactly supported smooth ...
2
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1answer
44 views

Lie derivative and Jacobi bracket for differential k-forms

Prove, by induction on $k$, that the following result holds for $\omega$ a $k$- form on $\mathbb R^n$ $$(L_\mathbb XL_\mathbb Y-L_\mathbb YL_\mathbb X)\omega=L_{[\mathbb X,\mathbb Y]}\omega .$$ Let ...
1
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1answer
70 views

Find a function that makes this differential form exact

We have $\Omega=\mathbb{R^3}\backslash \left\{ (0,0,z):z\in \mathbb{R}\right\}$ and $\omega$ the differential form: $$\omega ...
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1answer
18 views

Exercise in Taylor (PDE, volume 1) - Notation

I struggle to understand the following question. I expect I'm simply being dense about something. Let $F$ be a vector field on $U$, open in $\mathbb R^3,$ $F = \sum_1^3 f_j (x) ...
1
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1answer
38 views

differential forms of 2 sphere

Assume that $w$ is a 1-form on the 2-sphere $S^{2}$ so that $A^{*}w = w$ for all $A \in SO(3)$. Show that $w = 0$ I have tried to apply the definition of pullback and special orthogonal group, but I ...
4
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1answer
49 views

Question about creating a volume form for $SL(2,\mathbb{R})$

This problem comes out of R.W.R. Darling (Differential Forms and Connections) ch.8. In the chapter he shows that if $M$ is an $n$-dimensional differential manifold immersed in $\mathbb{R}^{n+k}$, and ...
2
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1answer
41 views

How does the wedge of dual vectors act on a wedge of vectors?

Suppose $dx_i$ is the dual basis in $R^n$ so that $$dx_i (e_j) = \delta_{ij}.$$ It makes sense to me how 1-forms work: a 1-form evaluated at a point gives some linear functional, which takes ...
0
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2answers
50 views

Vector-valued differential forms

Given a smooth real manifold $M$, and a real vector space $V$, when we talk about a $V$-valued $k$-form on $M$, do we mean an element of $\Gamma(\wedge^{*k}M)\otimes V$, where $\Gamma$ denotes ...
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0answers
37 views

Show that if $\omega$ is a 1-form differential, then $\left\vert\int_{C}\omega\right\vert\leq ML$

Show that if $\omega$ is a 1-form differential define on $U\subset\mathbb{R}^{n}$, $c:[a,b]\to U$ is a differentiable curve and $\vert\omega(c(t))\vert\leq M$, for all $t\in [a,b]$, then ...
4
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0answers
86 views

Exterior Differential (and its Equivalent Differential Operator) of an Integral 0-Form

I am reading Witten's 1982 paper "Supersymmetry and Morse Theory," and while I am slowly learning the material as I read through the paper, I have come across an equivalence that, while it should be ...
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1answer
55 views

The exterior derivate and pullback commute

The above question is from a past exam. I am having trouble with the fine details, ie what $F*dw$ and $dF*w$ actually look like. Can anybody show me how this question is solved? I have solved it ...
5
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0answers
42 views

Definitions/ intuition for differential forms

I've read about differential forms in the Princeton Companion to Mathematics by Gower and in baby Rudin and I'm having trouble reconciling the two expositions. Rudin says a k-form in $E$ ($\subset ...
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3answers
33 views

Help with the definition of a bilinear form $\omega$

According to this for $V$ a $2n$ (real) dimensional space any bilinear form $\omega: V \times V \to \mathbb{R}$ induces a linear map $\tilde{\omega}: V \to V^*$ via $$ \tilde{\omega}(v) := \omega(v, ...
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2answers
52 views

Stoke theorem and exterior derivative

$w=x \, dy\wedge dz - 2z f(y) \, dx \wedge dy + y f(y) \, dz \wedge dx$ where $f: \mathbb{R} \to \mathbb{R}$ belong $C^1$ (differentiable and derivative is continuous) with $f(1)=1$. Find $f$ so that ...
2
votes
1answer
78 views

If $d(f\omega)=0$, then $\omega \wedge d(\omega)=0$

Here's the question: Suppose that $\omega$ is a $k$-form on an open set $U$ of $\mathbb{R}^n$ and $f:U \to \mathbb{R}$ is a $C^\infty$ function such that $f(x) \neq 0$, for all $x \in U$, and ...
0
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0answers
27 views

Show that a derivation mapping k-forms to (k+1)-forms is zero.

This is question 7.4-6 in "Manifolds, Tensor Analysis and Applications", by Marsden, Ratiu and Abraham. It says: show that a derivation mapping $\Omega^k(M)$ to $\Omega^{k+1}(M)$ for all $k$, is ...
1
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1answer
48 views

Give a volume form on $\mathbb{RP}^3$ [closed]

I was asked to determine a volume form on $\mathbb{RP}^3$. I would really appreciate any help. Thanks in advance.
1
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1answer
33 views

Conservative Covector Fields are Exact

$\newcommand{\R}{\mathbf R}$ I am trying to understand the proof of the following: Theorem. Let $M$ be a smooth manifold and $\omega$ be a smooth covector field on $M$. Then $\omega$ is ...
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1answer
72 views

the $\partial\bar{\partial}$-lemma dilemma

In the question here Simplifying the Kahler form, user290605 asked a question about how is that when we take the differential of Kahler form:$$\mathcal{K}=\frac{\sqrt{-1}}{2\pi}g_{i\bar{j}}dz^i\wedge ...
4
votes
2answers
85 views

How to prove $d\omega=(\nabla_\mu\omega)_\nu dx^\mu\wedge dx^\nu$ without using coordinates

This is exercise 7.8 b) of Nakahara's GTaP: Let $\omega\in\Omega^1(M)$ be a 1-form on a Riemannian manifold with Levi-Civita connection $\nabla$. Prove that $$ ...