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

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6
votes
1answer
56 views

$M$ closed $3$-manifold, $\xi$ integrable $2$-dimensional subbundle of $TM$, ensuing properties.

Let $M$ be a closed $3$-manifold, and let $\xi$ be a $2$-dimensional subbundle of $TM$. There is a nowhere zero $1$-form $\alpha$ on $M$ with $\alpha(X) = 0$ for any vector field $X$ which is a ...
0
votes
1answer
57 views

How do I apply Stokes' theorem on this differential form?

I want to calculate the integral $ \int_\gamma \log(1+x^2)dx + \cos(1+y^2)dy + (\sin(x^4) + y + \sin(z^2))dz$ where $\gamma = (\cos(t) + \sin(t), \cos(t)+ 2\sin(t), \cos(t) - \sin(t))$ for $0 \leq ...
5
votes
2answers
86 views

Confusion with the 2-form: $z \, dx\wedge dy$

I'm a bit new to forms and orientations of manifolds, and I'm having a bit of trouble understanding the following simple question. The integral of the two form $w=z \, dx\wedge dy$ over the surface ...
6
votes
3answers
159 views

$2$-dimensional subbundle of tangent bundle of closed $3$-manifold integrable if and only if $\alpha \wedge d\alpha = 0$?

Let $M$ be a closed $3$-manifold, and let $\xi$ be a $2$-dimensional subbundle of $TM$. From here and here, I know that there is a nowhere zero $1$-form $\alpha$ on $M$ with $\alpha(X) = 0$ for any ...
5
votes
1answer
52 views

Any two $1$-forms $\alpha$, $\alpha'$ with property satisfy $\alpha = f\alpha'$ for some smooth nowhere zero function $f$?

This is a followup question to here. Let $M$ be a closed $3$-manifold, and let $\xi$ be a $2$-dimensional subbundle of $TM$. Is there a nowhere zero $1$-form $\alpha$ on $M$ with $\alpha(X) = 0$ ...
4
votes
1answer
60 views

Is there a nowhere $1$-form $\alpha$ on $M$ with $\alpha(X) = 0$ for any vector field $X$ which is a section of $\xi$?

Let $M$ be a closed $3$-manifold, and let $\xi$ be a $2$-dimensional subbundle of $TM$. Is there a nowhere zero $1$-form $\alpha$ on $M$ with $\alpha(X) = 0$ for any vector field $X$ which is a ...
2
votes
1answer
40 views

Implicit formula for the Levi-Civita connection

Let $(M, g)$ be a Riemannian manifold and $X, Y, Z$ smooth vector fields on $M$. Let $\theta_X$ be the $1$-form defined as $\theta_X(Y) = g(X,Y)$ and let $d\theta_X$ be its exterior derivative. Let ...
0
votes
1answer
29 views

How do you find the incidental equation when using the Frobenius method?

If I'm not mistaken, the Frobenius Series is given by $$ y = a_{0}x^{r} + a_{1}x^{r+1} + a_{2}x^{r+2} + \dots + a_{n}x^{r+n} + \dots $$ and so we also have $$ y' = ra_{0}x^{r-1} + (r+1)a_{1}x^{r} + ...
0
votes
2answers
34 views

What is the method used to find the singular points of an ODE?

Suppose you have an ODE, say $$ x^{2} (x+1) y'' + 2y' + xy = 0 $$ How would you find the singular points of this? I've looked online for an explanation of the method used to do this, but have not ...
0
votes
0answers
41 views

Stokes Theorem for non compact subsets of $\mathbb{C}.$

Consider a regular compact subset $K \subset \mathbb{C}$. If $f \in C^1(\mathbb{C})$ one has $$\int_{K} \frac{\partial f}{\partial \overline{z}} = \int_{\partial K} f dz.$$ Now I ask the question for ...
1
vote
1answer
55 views

Prove: The pullback of a volume form on a sphere to a cylinder is a volume form

Prove: The pullback of a volume form on a sphere to a cylinder is a volume form We denote $S = \{ (x,y,z) \mid x^2 + y^2 + z^2 = 1\}$,$ C = \{ (x,y,z) \mid x^2 + y^2 = 1 , |z| < 1 \}$. Given a ...
1
vote
1answer
61 views

Differential forms on a scheme: unclear equation

Disclaimer: In this question I assume that the reader is familiar with the construction of the module of differentials $\Omega^1_{B|A}$ where $B$ is an $A$-algebra. (If you need more details about ...
1
vote
1answer
55 views

Can a closed differential form on a subset of manifold always be extended to the whole manifold?

Here I am using de Rham cohomology. This question occured to me while reading the proof of the exactness of the short exact sequence in the Meyers Vitoris sequence $0 \rightarrow ...
4
votes
1answer
55 views

When is a $(p,q)$-form real?

Let $M$ be a complex manifold, and let $\omega$ be a $(p,q)$-form. Then, in holomorphic coordinates $(z^1,\dots , z^n)$, $\omega$ can be expressed as $\omega = f_{U,V} dz^U \wedge d\bar{z}^V$, where ...
1
vote
1answer
42 views

How can I solve this difference equation, if $p=q$?

I'm trying to solve the difference equation $$ pE_{k+1} - E_{k} + qE_{k-1} = -1 $$ given the boundary conditions $E_{0} = 0, \; E_{a} = 0$, if $p=q=\frac{1}{2}$ To attempt this, I first found and ...
3
votes
0answers
44 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 ...
1
vote
1answer
38 views

necessary and sufficient condition for the lines of curvature

I'm reading the book "Differential Geometry of Curves and Surfaces" of Manfredo Carmo, and this part confuses me: We have the differential equation of the lines of the curvature: ...
5
votes
1answer
61 views

Where is the error in this proof of the Hodge theorem?

Let $(M,g)$ be a closed smooth Riemannian manifold. The following is the decomposition part of the Hodge theorem: Theorem The canonical map $\mathscr{H}^k(M)\to H^k(M)$ from harmonic $k$ ...
1
vote
0answers
15 views

matrix valued integrating factor of one forms (reference request)

I have $N$ 1-forms $\omega_1(x), \ldots, \omega_N(x)$. I want to know if there exists an invertible linear combination of these forms which yields $N$ closed forms. In other words: does an invertible ...
2
votes
0answers
41 views

Boundary of a $k-$Cell Definition

In the book I am reading, the Boundary of a $(k+1)-$cell $\varphi$ is defined to be the $k-$chain $$\partial\varphi=\sum_{j=1}^{k+1}(-1)^{j+1}(\varphi\circ \iota^{j,1}-\varphi\circ \iota^{j,0}) $$ ...
0
votes
1answer
31 views

Closed Form and Pullback compatibility

given: $U,V \subset \mathbb{R}^N, f\in C^1(V,U)$ a diffeomorphism Let $\omega$ be a k-Form on U and $f^*\omega$ a closed Form. Then with $ 0 = df^*(\omega) = d \omega(df)$ we have ,that $\omega$ ...
0
votes
0answers
19 views

Constructing the Hodge Laplacian from the Laplace-Beltrami one

I know, in principle, how to construct the Hodge Laplacian with the aid of the Hodge star. Is it possible, though, to construct it only from the Laplace-Beltrami operator? I have already found a way, ...
0
votes
0answers
14 views

Total Derivative and Composition

We are given that $C$ is a function of $Y_D$ and $Y_D=Y-Y\tau$. What would be the total differential of $Y=C(Y_D)$? So far I have the following: $$ dY=C_{Y_D}(1-\tau)dY+C_{Y_D}(-Y)d\tau$$ However I ...
2
votes
0answers
27 views

Find $f \in C^1(U,\mathbb{R})$ which satisfiyes the following differential-form

I am quiet new to differential forms so I am not sure if my solution to the following problem is correct. The Problem: given: $U \subset \mathbb{R}^3$ and $U$ is an open set $$ V \in C^1(U, ...
0
votes
3answers
56 views

How to calculate $f(x, y, z)$ given $d f = \frac{\partial f}{\partial x} dx + \frac{\partial f}{\partial y} dy +\frac{\partial f}{\partial z} dz$

On a manifold with local coordinates $(x_1, \ldots, x_n)$ I have a closed 1-form $\omega$ for which $d \omega = 0$ holds. This means There must be a function $f(x_1, \ldots x_n)$ for which $d f = ...
2
votes
1answer
32 views

Basis representation of differential-form $f(x_1,x_2):=(x_1+x_2, -x_1, -x_2)^T$

I am trying to learn differential forms. I have read some scripts about differential forms and now I am trying to solve some problems. So the problem is: given $f: \mathbb{R}^2 \to \mathbb{R}^3, ...
2
votes
0answers
24 views

Compute the degree of $\phi:(x_1,x_2,x_3,x_4) \mapsto (x_1,-x_3,-x_2,x_4)$. Does $\phi$ preserve orientation?

Consider the map $\phi:S^3 \rightarrow S^3$ given by $\phi(x_1,x_2,x_3,x_4)=(x_1,-x_3,-x_2,x_4)$. Compute the degree of $\phi$. Does $\phi$ preserve orientation? First I want to point ...
0
votes
1answer
35 views

Where is my mistake in this exterior derivative of a $2$-form not being a tensor?

Using the invariant formula for the exterior derivative, one gets that for a $3$-form $\omega$ its derivative is evaluated on vector fields according to $$3 \ {\rm d} \omega (X, Y, Z) = X \ \omega ...
2
votes
0answers
40 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
votes
0answers
25 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 ...
2
votes
0answers
40 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
50 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
vote
1answer
27 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
votes
2answers
37 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
31 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
31 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
votes
1answer
43 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
votes
1answer
55 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 ...
1
vote
1answer
32 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
37 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
57 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
0answers
27 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
votes
1answer
58 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 ...
1
vote
1answer
42 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
votes
1answer
57 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
votes
1answer
38 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
33 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
61 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 $f: M \to M$ ...