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

learn more… | top users | synonyms

0
votes
0answers
16 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 ...
5
votes
2answers
406 views

Can calculus of varations be formalised with exterior calculus?

I noticed that a calculus of variations problem is just an integral over a differential form. Therefore, I would think it would be possible to formulate the Euler-Lagrange equations using exterior ...
1
vote
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
votes
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
votes
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 ...
16
votes
3answers
2k views

Geometric interpretation of connection forms, torsion forms, curvature forms, etc

I have just begun learning about the connection 1-forms, torsion 2-forms, and curvature 2-forms in the context of Riemannian manifolds. However, I am finding it hard to relate these notions to any ...
1
vote
1answer
21 views

Interior product general rule (differential forms)

How is this general form of interior product on forms $$(i_V\omega^{(p)})=\frac{1}{(p-1)!}V^{\mu}\omega_{\mu\mu_1...\mu_{p-1}}dx^{\mu_1}\wedge dx^{\mu_2}\wedge ...\wedge dx^{\mu_{p-1}}$$?
6
votes
1answer
191 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 ...
0
votes
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 ...
0
votes
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
votes
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, ...
0
votes
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 ...
3
votes
1answer
567 views

Pullback distributes over wedge product

I'm looking to prove that the pullback of a smooth function distributes over wedge product, i.e. $$\varphi^*(\omega \wedge \eta) = \varphi^* \omega \wedge \varphi^* \eta. $$ Here the question is ...
-2
votes
1answer
32 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 ...
4
votes
1answer
48 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 ...
1
vote
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 ...
1
vote
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 ...
1
vote
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 ...
1
vote
0answers
25 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 ...
2
votes
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 $$ ...
0
votes
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 ...
0
votes
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 ...
0
votes
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 ...
1
vote
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 ...
2
votes
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
votes
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
votes
0answers
57 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
vote
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 $x\in M$ ...
3
votes
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 ...
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 ...
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
votes
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 $d\lambda = \omega$. ...
1
vote
1answer
51 views

Why is the derivative Df(p) defined to $ \in \Lambda^1 (\mathbb{R}^n) $, or how is it a 1-form?

I know that obviously the differential operator $D$ would be a differential form through the word differential. But in Spivak's Calculus on Manifolds he defines a $k$-form $w$ as $w(p) \in \Lambda^k ...
0
votes
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, ...
1
vote
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 ...
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
votes
1answer
143 views

Total confusion about differential one-forms and non-coordinate bases

I asked this question recently (Basis of differential one-form confusion), thought I understood the answer, but now realise I don't. Lee (Introduction to Smooth Manifolds) says that at a point $p$ ...
2
votes
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 ...
2
votes
2answers
122 views

Understanding the definition of a pullback of a differential $k$-form and applying it in $1-d$

I am having trouble understanding the definition of a pullback of a differential k-form in a basic course in differentiable geometry. This is the definition I am given. I believe it is easier to ...
1
vote
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
vote
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 ...
1
vote
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 ...
2
votes
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
votes
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 ...
1
vote
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
votes
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 ...
5
votes
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 ...
1
vote
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, ...
1
vote
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 ...