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

learn more… | top users | synonyms

2
votes
1answer
50 views

understanding of $d(\log f(z))$ in complex analysis

In Gameline's Complex Analysis Chapter 8, the notation $d(\log f(z))$ is used: Here are my questions: In the real case, suppose for any $x\in\mathbb{R}$, one has $f(x)\neq 0$ and $f$ is ...
0
votes
1answer
44 views

Frobenius theorem for differential forms

I have to check the next version of the theorem Frobenius: Let $M$ a smooth manifold and $\{\omega^1,\ldots,\omega^k\}\subset\Omega^{1}(U)$ $l.i.$ on $U\subset M$ and $P(x)=\{ v\in T_x M\vert ...
4
votes
2answers
51 views

Pullback of a complex $ 1$-form

Let $p = \operatorname{exp} : \mathbb{C} \to \mathbb{C}^*$ be a covering and $(U,z)$ a chart of $\mathbb{C}^*$ with $z = x + iy$. Let $\omega = dz/z$ be a one-form on $U$. Problem: Find the pullback ...
3
votes
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 ...
0
votes
0answers
24 views

Prove: $u = \phi_1 ^* \mu_1 \wedge \phi_2 ^* \mu_2$ is a volume form on $M = M_1 \times M_2$.

Given two manifolds, $M_1 \in \mathbb{R}^{n_1}$ and $M_2 \in \mathbb{R}^{n_2}$, with respective volume forms $\mu_1,\mu_2$ prove that $\mu = \phi_1 ^* \mu_1 \wedge \phi_2 ^* \mu_2$ is a volume form on ...
0
votes
0answers
13 views

Find the volume of h(P) in terms of volume of P

Where h:$\mathbb R^n \rightarrow R^n$ is the function h(x) = $\lambda x$ and P is a k-dimensional parallelopiped in $\mathbb R^n$ Attempt at the solution : Let $x_1,....,x_k$ be vectors in $R^n$ ...
1
vote
1answer
40 views

Confusion over the definition of a volume element

I'm fairly new to differential geometry (currently self-teaching) and I'm a bit confused over the definition of a volume form. I've read that, given an $m$-dimensional manifold, a top-form ...
2
votes
1answer
75 views

Verifying Stokes' Theorem on an example

Let $M = \{(x,y,z) \in \mathbb{R}^3 | x^2+y^2+z^2=1,z>0\}$, and let $\omega = xdy$. I would like to verify that $\int_M d\omega = \int_C \omega$, where $C = \partial M$ given by $C = \{(x,y,z) \in ...
1
vote
2answers
60 views

On the definition of the volume form in general vector spaces as given in Spivak, Calculus on Manifolds

For a vector space $V$ denote by $\Lambda^k(V)$ the space of alternating $k$-tensors, or alternating $k$-fold multilinear maps on $V$. I have some difficulty following the intention of the author in ...
3
votes
1answer
77 views

Do we have $\int f dxdy = \int fdydx$ or $\int f dxdy = -\int f dydx$?

If $f : \mathbb R^2 \to \mathbb R$ is an integrable function, then do we have $$ \int f dxdy = \int f dydx $$ or $$ \int f dxdy = -\int f dydx? $$ (I am leaving the domain of integration as it does ...
1
vote
0answers
21 views

Computing $\alpha^*w$ in general

Let $A$ be open in $R^k$, let $\alpha : A \rightarrow R^n$ be of class $C^\infty$. Let $x$ denote the general point of $R^k$, let $y$ denote the general point of $R^n$ If $I = (i_i, ...,i_l)$ is an ...
0
votes
0answers
15 views

Show that $\alpha^*(dy_i) = d(\alpha^*y_i) $

where $y_i:\mathbb R^n \rightarrow R$ is the $i^{th}$ projection function in $\mathbb R^n$ Given: Let A be open in $R^k$; let $\alpha: A \rightarrow R^n$ be a $C^{\infty}$ map. Let x denote the ...
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, ...
1
vote
2answers
62 views

Relation between exterior derivative and Lie bracket

There is a formula connecting the exterior derivative and the Lie bracket $$d\omega (X,Y) = X \omega(Y) - Y \omega(X) - \omega([X,Y]).$$ What is a good way to remember this? By which I mean, what ...
0
votes
0answers
20 views

clarification on what order of differential form means

I'm confused about what the order of a differential form is, from my understanding it is the k-tuple of vectors that the function takes in, but this doesn't seem right. For instance, according to my ...
1
vote
1answer
48 views

Proof that $x \ dy \wedge dz$ is NOT exact on any open subset of $\mathbb{R}^{3}$

Consider the smooth differential 2-form $\omega = x \ dy \wedge dz$ defined on $\mathbb{R}^{3}$ (using the coordinates $(x,y,z)$ of course). I'm trying to show that $\omega$ is NOT exact on any open ...
0
votes
0answers
30 views

If a k-form vanishes in the neighborhood of p then it vanishes at p

Let w be a k-form defined in an open set A of $\mathbb R^n$. We say that w vanishes on x if w(x) is the zero tensor. Show that if w vanishes at each x in a neighborhood of $x_0$ then dw vanishes at ...
0
votes
1answer
42 views

Finding Integrating factor

I have been solving this ODE $ydx+(y+\tan(x+y))dy=0$ My approach was like this; As this equation is not exact; $\frac{\partial}{\partial y}y = 1$ $\frac{\partial}{\partial ...
0
votes
0answers
24 views

Writing a linear transformation in terms of elementary alternating k-tensors

Let T: $\mathbb R^m \rightarrow R^n $ be the linear transformation T(x) = Bx If $\alpha_I$ is an elementary alternating k-tensor on $\mathbb R^n$, the $T^*\alpha_I$ has the form $T^*\alpha_I = ...
1
vote
1answer
46 views

Lie derivative of a two-form

Let $M$ be a manifold, $\alpha$ a two form and $x,y,z$ vector fields. How can we show that $$L_x(\alpha(y,z))=(L_x \alpha)(y,z)+\alpha(L_x y,z)+\alpha(y,L_x z)$$ (where $L_x$ is the Lie derivative ...
1
vote
0answers
21 views

The determinant of $X_I$ = wedge product of elementary tensors

Let $(x_1,\dots,x_k)$ be vectors in $\mathbb{R}^n$; let $X$ be the matrix $X = [x_1 \cdots x_k]$. If $I = (i_1, \dots, i_k)$ is an arbitrary $k$-tuple from the set $\{1,\dots, n\}$, show that ...
0
votes
1answer
33 views

Checking alternating tensors on $\mathbb R^4$

I'm trying to solve the following question : Which of the following are alternating tensors in $\mathbb R^4$ and express those that are in terms of the elementary tensors on $\mathbb R^4$: ...
-7
votes
1answer
127 views

Does there exist a smooth curve satisfying certain conditions?

Let $\gamma := dc - a\,db$ be a $1$-form on $\mathbb{R}^3$ with coordinates $a$, $b$, $c$. Have $\omega$ be the $2$-dimensional subbundle of $T\mathbb{R}^3$ consisting of vectors in the kernel of ...
0
votes
0answers
46 views

Why don't I get the same answer when I calculate the pullback vs integral over a manifold?

Let's take the differential form $\omega = xy \, dx \wedge dy$. We say that $M$ is the surface $z = x^2 + y^2 \leq 1$ with the standard orientation. I can calculate $\int_M \omega$ via pullback and ...
2
votes
1answer
35 views

What's the exterior derivative of the two form: $x \,dy \wedge dz + y \,dx \wedge dz + z\,dx \wedge dy$?

What's the exterior derivative of the two form: $x \,dy \wedge dz + y\, dx \wedge dz + z\,dx \wedge dy$? I saw some calculations on this site and I'm pretty sure that the answer is $3\, dx\wedge ...
1
vote
1answer
41 views

Prove that the form $f(x^2+y^2)x\,dx +f(x^2+y^2)y\,dy $ is closed for a continuous $f$.

Prove that the form $f(x^2+y^2)x\,dx +f(x^2+y^2)y\,dy $ is closed for a continuous $f$. I'm not even sure if the exterior derivative exists in this case, but I need to prove that the integral of the ...
1
vote
1answer
24 views

Question on the image of the parameter space under two non-commuting flows.

Let $\phi^t_1$ and $\phi^s_2$ be two smooth flows on $\mathbb{R}^2$ defined for $t\in[-1,1]$ and $s\in [-1,1]$. Assume (1) $\phi^0_1\big((0,0)\big)=\phi_2^0\big((0,0)\big)=0$ (2) $(\phi^t_1)'|_0$ ...
1
vote
1answer
23 views

Series representation of a differential form

I have a problem understanding the general series representation of a p-form. For 1-form things are pretty clear to me: For $h = (h_1, \dots, h_n)^T \in \mathbb{R}^n $ and $ h = \sum\limits_{i = ...
0
votes
0answers
23 views

Extension of divergence free vector field as a divergence free vector field.

Let $M$ be a compact smooth Riemannian manifold of dimension $n$. Assume that $M$ is isometrically embedded in $\mathbb{R}^m$ for some sufficiently large $m$ via the map $\iota$. Let $X:M\to TM$ be ...
3
votes
1answer
78 views

Does “every closed form is exact” imply simply connected for a connected open set in $\mathbb{R}^n$?

I know that if $U\subseteq\mathbb{R}^n$ is an open simply connected set, every closed 1-form $\omega\in\Omega^1(U)$ is also exact. I was wondering: does the converse hold? So if every closed ...
2
votes
1answer
61 views

What is the difference between vector and point in differential geometry

I am reading the thread but I can't comment. So I would like to open a new one and ask for clarification. Could you explain the difference between "vector" and "point" here? His is the answer of John ...
4
votes
1answer
49 views

Smooth representative $f: S^{2n - 1} \to S^n$, do we have $f^*\omega = d\alpha$?

Let $[f] \in \pi_{2n - 1}(S^n)$. Choose a smooth representative $f: S^{2n - 1} \to S^n$. Let $\omega$ be a smooth $n$-form on $S^n$ with$$\int_{S^n} \omega = 1.$$Do we have that$$f^*\omega = ...
3
votes
1answer
56 views

Diffeomorphism group $\text{Diff}_\omega(D^2, \partial D^2)$, exact differential form.

Let $D^2$ denote the closed unit disk in $\mathbb{R}^2$. Let $\omega = dx \wedge dy$ denote the standard area form on $\mathbb{R}^2$ (and on $D^2$ by restriction). Let $\phi$ be a diffeomorphism of ...
3
votes
2answers
107 views

Does there exist a $1$-form $\alpha$ with $d\alpha = \omega$?

Let $\omega := dx \wedge dy$ denote the standard area form on $\mathbb{R}^2$. As the question title suggests, does there exist a $1$-form $\alpha$ with $d\alpha = \omega$?
0
votes
0answers
26 views

From a $p$-form to it series representation with a determinant

I want to understand the concept of $p$-forms, but I have yet trouble to relate this concept to things I already know/understand. Def.: $U \subset \mathbb{R}^n$ be open. $ w : U \times \mathbb{R}^n ...
5
votes
2answers
91 views

How to compute the pullback of $(2xy+x^{2}+1)dx+(x^{2}-y)dy$ along $f(u,v,w)=(u-v,v^{2}-w)$?

I'm trying to do my first pull-back of a differential form. I know that $\omega=(2xy+x^{2}+1)dx+(x^{2}-y)dy$ is a differential form on $\mathbb{R}^{2}$. I have $f : \mathbb{R}^{3} \to \mathbb{R}^{2}$ ...
0
votes
0answers
32 views

Field and dimension of the set of all differential forms (as a vectorspace)

Wikipedia says The set of all differential $k$-forms on a manifold $M$ is a vector space, often denoted $\Omega^k(M)$. Over which field is this vector space defined? Let $\dim(M)=:n$. What is ...
6
votes
1answer
57 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
88 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
166 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
53 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
61 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
42 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
35 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
35 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
62 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
62 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 ...
2
votes
1answer
59 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 ...