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

learn more… | top users | synonyms

2
votes
0answers
39 views

Showing that $\|.\|$ is a norm of the space of 1-forms $\Omega^1(U)$, where $U\subset\mathbb{R}^n$.

Let $U\subset\mathbb{R}^n$ and let $\Omega^p(U)$ denote the vector space of $p$-forms ($p\in\mathbb{N}$). Define the isomorphism $\Phi:\Omega^{1}(U)\to\Omega^{n-1}(U)$ as $$\Phi\left(\omega=\sum_{i=1}^...
2
votes
0answers
32 views

Boundary of oriented $k$-simplex from PMA Rudin

But paragraph which I marked by red line seems to me confusing. Let $k=3$ then $\sigma=[\mathbf{p}_0,\mathbf{p}_1,\mathbf{p}_2,\mathbf{p}_3]$ and $\partial \sigma=[\mathbf{p}_1,\mathbf{p}_2,\mathbf{p}...
2
votes
0answers
43 views

Unnecessary assumption in exercise (from Spivak, Calculus on Manifolds)

I have a question on the following exercise (which is taken from Spivak, Calculus on Manifolds, page 105). If $\omega$ is a $1$-form $f dx$ on $[0,1]$ with $f(0) = f(1)$, show that there is a ...
2
votes
0answers
21 views

Integrating a form over an image

I am trying to integrate the 2-form $$\eta=\frac{1}{\|\mathbb{x}\|^m}(x_1dx_2\wedge dx_3-x_2dx_1\wedge dx_3+x_3dx_1\wedge dx_2)$$ over $Y_\alpha$ where $$\alpha(u, v)=(u, v, (1-u^2-v^2)^{1/2})$$ and $...
2
votes
0answers
31 views

Short question about differential forms / exterior algebra

I am working towards understanding the wedge product of two vectors. Here is what I have so far: Let $V$ be an $n$-dimensional vector space over $\mathbb R$ (or $\mathbb C$, I'm not sure it makes ...
2
votes
0answers
45 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 $$\partial\varphi=\sum_{j=1}^{k+1}(-1)^{j+1}(\varphi\circ \iota^{j,1}-\varphi\circ \iota^{j,0}) $$ ...
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, \...
2
votes
0answers
40 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 ...
2
votes
0answers
52 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 $...
2
votes
0answers
42 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 ...
2
votes
0answers
34 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
64 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 $f: M \to M$ ...
2
votes
0answers
53 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
0answers
35 views

Proving a fact about bilinear forms

The fact I'm trying to prove is that every bilinear form $\omega$ on a set $\omega\subseteq\mathbb{R}^2$ can be written in the form $\omega = f\,dx\wedge dy$, where $f: \Omega\rightarrow\mathbb{R}$ is ...
2
votes
0answers
20 views

Hermitian vector space and relation of associated operators.

Here what i want to do is prove proposition 1.1 in chapter 5, on Wells, Differential analysis on complex manifold, The propositions are follows For Hermitian vector space of complex dimension $n$. ...
2
votes
0answers
27 views

Is there a commutative algebra for which multiderivations are not generated by order 1?

By multiderivations (of order $k$) I mean polylinear skew-commutative operations with values in my algebra which satisfy Leibniz rule in each of the arguments. (That is, the dual module to ...
2
votes
0answers
54 views

Are those two equivalent mathematically speaking?

Starting from the action $$S=-\frac{1}{4}\int{F_{\mu\nu}F^{\mu\nu}d^4x}$$ (until here this is physics but my question is about the math) that is: I got the following equation of motion solving for ...
2
votes
0answers
66 views

Integration is only possible for forms and not for general tensors?

Integration is only possible for forms and not for general tensors? What is the true reason for this? Or can integration of $k$-forms be extended in some natural way to arbitrary $(k,l)$ tensors;if so ...
2
votes
0answers
65 views

A compact $n$-manifold is orientable iff there is an everywhere nonzero $n$-form

Let $M$ be a compact differentiable manifold of dimension $n$ without boundary. Show that $M$ is orientable if and only if there exists a diffential $n$-form $\omega$ defined on $M$ and which is ...
2
votes
0answers
40 views

Does the wedge product of bundle-valued forms induce a universal object?

Given a smooth manifold $M$ and a vector bundle $E$ over $M$, the $C^\infty(M)$-module of $E$-valued $p$-forms on $M$ is defined to be $$\Omega^p(M; E) := \Gamma_M\left( \bigwedge^p T^*M \otimes E \...
2
votes
0answers
127 views

How to define the boundary operator using the exterior derivative?

I am looking for a way to define the boundary operator $\partial : M^n \to N^{n-1}$ from an $n$-dimensional manifold $M$ to its boundary $N$ using the the expression \begin{equation*} \int_M d \alpha ...
2
votes
0answers
81 views

Clarification of vector valued forms (sections and tensor products)

I am seeking a bit of clarification on vector valued forms. Intuitively, a vector valued differential form is a differential form on $M$ whose values lie in some vector space. More accurately, Let $...
2
votes
0answers
39 views

Why do entries of Maurer-Cartan 1-form span space of left-invariant 1-forms

Suppose $G$ is a $k$-dimensional Lie group of $n\times n$ matrices of the form $[g_{ij}(x_1,...,x_k)]$ where $g_{ij}$ are smooth functions. As a follow-up to the question I posted recently, I now ...
2
votes
0answers
33 views

Differentiability of a map into $\wedge T^*M$

Let $M$ a differentiable manifold and $p \in M$. Consider $(U;x_1, \dots, x_n)$ a coordinated system around $p$. $\{dx_\phi\}_\phi$ with $\phi \in \cal{P}(\{1, \dots n\})$, $dx_\phi=dx_{i1} \wedge \...
2
votes
0answers
108 views

Differential one-forms and change of coordinates

Consider two differential one forms: $$\omega=\sum_{i=1}^N \omega_i dx^i$$ $$\omega'=\sum_{i=1}^N \omega'_i dx'^i$$ As I recall from my analysis courses, the symbols $dx$ are a particular notation ...
2
votes
0answers
306 views

Compute the wedge product

This is my first time computing the wedge product, I am not sure if I have done it correctly as I do not have solutions, if I have gotten the answer right or am doing the right method please say. ...
2
votes
0answers
145 views

Pullback of a 1 form on the circle

Q: Let $M$ be a smooth compact manifold, and suppose there is a smooth map $F:M \rightarrow S^{1}$ whose derivative is non-zero at every point. Prove that the de Rham cohomology space $H^{1}(M)$ is ...
2
votes
0answers
82 views

Homotopy, Stokes Theorem and Orientation

I have a problem in which the theory and the computation disagree about a minus sign. My question requires a little setting up. I have a complex valued 2-form $$ \omega = \alpha(\xi_1,\xi_2)\, \...
2
votes
0answers
89 views

An error applying the Stokes theorem?

M is the surface $z=x^2+y^2$ with standard orientation for $x^2+y^2\leq 1$ and $\varphi = 4x^2ydy+z^2dz$ I'd like to verify that $\int_Md\varphi=\int_{\partial M}\varphi$, which I did, but according ...
2
votes
0answers
131 views

Prove Green's theorem for circles

So the problem is in the title. The rules are that I can't split the circle into "rectangles" and I can't use pull-back. I tried to do something similar to the proof on unit squares. The problem is ...
2
votes
0answers
82 views

A few identities on differential forms

I just started studying differentiable manifolds, and I've encountered a few allegedly simple questions that aren't really simple to me. If possible, I'd be very happy to see full proofs (without ...
2
votes
0answers
116 views

The first proof for Poincare lemma in history

How can I get a reference about the first proof of Poincare lemma in history? I already know some methods of proof, but I do want to know the original approach. Thanks for your help!
2
votes
0answers
238 views

Volume form on $\mathbb{S}^2$

Let $\omega = x_1 \,dx_2 \wedge dx_3 + x_2 \,dx_3 \wedge dx_1 + x_3\, dx_1 \wedge dx_2$, with $(x_1,x_2,x_3) \in \mathbb{S}^2$, be a volume form on $\mathbb{S}^2$, and let $f : \mathbb{S}^1 \times ]-\...
2
votes
0answers
64 views

Help with local coordinates computations-differential forms

I am reading a lecture notes on differential forms and tangent bundles on complex manifolds and I got stuck on one line where the author does not explain how he did the computation: Let $\...
2
votes
0answers
57 views

Laplacian on a warped product.

Let $(M, g)$, $(N, h)$ be complete Riemannian manifolds (not necessarily compact). Let $f : M \rightarrow (0, \infty)$ be a smooth function, and finally let $$\overline{M} = M \times_f N$$ be the ...
2
votes
0answers
326 views

$\alpha \wedge \beta = 0$ iff $\beta = \alpha \wedge \gamma$

I have been given the following problem: Let $\alpha$ be a nowhere-zero 1-form. Prove that for a (p+1)-form $\beta$ $(p\geq0)$, one has $\alpha \wedge \beta = 0$ if and only if $\beta = \alpha \...
2
votes
0answers
81 views

What is the differential of a function?

I'm reading Do Carmo's Differential Forms and Applications (1st ed) and on page 6 he takes a differentiable map $f: \mathbb{R}^{n} \rightarrow \mathbb{R}^{m}$, a point $p \in \mathbb{R}^{n}$ and a ...
2
votes
0answers
230 views

Exact and Closed forms on Manifolds with Boundary

Let $\bar{M}$ be a manifold with boundary and let $M$ be its interior. Is this statement correct? A smooth k-form $\alpha$ on $\bar{M}$ is closed (exact) if and only if its restriction to $M$, i.e. $\...
2
votes
0answers
874 views

Top deRham cohomology group of a compact orientable manifold is 1-dimensional

Let $M$ be a compact smooth orientable manifold of dimension $n$. I am looking for a simple proof that $H_{dR}^n(M) \cong \mathbb R$. Equivalently, an $n$-form which integrates to 0 is exact. I can ...
2
votes
0answers
68 views

How to apply Gauss's theorem when the metric is unknown

Let $f:U \to \mathbb{R}^3$ be a surface, where $U=\{(u^1,u^2)\in \mathbb{R}^2:|u^1|<3, |u^2|<3\}.$ Consider the two closed square regions $F_1=\{(u^1,u^2)\in \mathbb{R}^2:|u^1|\leq1, |u^2|\leq1\}...
2
votes
0answers
181 views

Calulation of pullback of form

If $M$ is $2n+1$ dimensional manifold, and $M'= M\times \mathbb R$ Let $x_1,y_1,... x_n, y_n,t', t$ be coordiante of $M'$. With $t$ for coordinate for $\mathbb R$. Let $$ \omega= \sum_{i=1}^n dx_i\...
1
vote
0answers
18 views

Explicit isomorphism $H (E, E^0) \cong H_{cv} (E)$

Yo! I've been trying to understand better why $$H (D, D-\{x\}) \cong H_c (D)$$ for a disk $D$ and, more generally, why $$H (E, E^0) \cong H_{cv} (E)$$. In general, the first isomorphism can be seen as ...
1
vote
0answers
41 views

Challenging calculation of a Jacobian for an unusual matrix coordinate transformation

I am studying a random matrix ensemble and I am having trouble performing a coordinate transformation. My question is very straightforward, but perhaps a bit technical. I have the following integral--...
1
vote
0answers
39 views

Computing $\alpha\wedge d\alpha$ where $\alpha = dz + x\,dy$

Let $\alpha = dz + xdy$ be a one-form on $\mathbb R^3$. I would like to compute the wedge product $\alpha\wedge d\alpha$ as explicitly as possible but I am not sure if I am doing this correctly. $$d\...
1
vote
0answers
33 views

Do I understand the divergence theorem correctly?

Suppose the area, volume or hyper volume covered by a vector is $$ \mathrm{V}\left(\vec{u}\right) = u_x \times u_y \times \ldots $$ And the area, volume or hyper volume covered by a matrix is $$ \...
1
vote
0answers
33 views

Moduli space of differential forms

In this paper, Debarre-Voisin refer to the "moduli space" of differential 3-forms $\sigma \in \bigwedge^3(V_{10}^*)$ on a fixed vector space $V_{10}$ of dimension 10, and state that this space is $\...
1
vote
0answers
49 views

Meaning of alternation in definition of wedge product

Spivak defines the wedge product as $\omega \wedge \nu = \frac{(k+l)!}{k!l!} \text{Alt}(\omega \otimes \nu)$ and I have been running into some conceptual issues here. The alternation is defined as $...
1
vote
0answers
38 views

Stokes' Theorem from PMA Rudin. Confusing moment with simplex

I am reading the proof of Stokes' theorem from PMA Rudin but one moment seems to very weird. Why Rudin considers the case when $\sigma=[0,\mathbf{e}_1,\dots, \mathbf{e}_k]$? After all $\sigma$ may ...
1
vote
0answers
22 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 ...
1
vote
0answers
22 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 $\phi_{...