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

learn more… | top users | synonyms

1
vote
1answer
109 views

Is the differential form $\omega(x,y)=\frac{2x}{x^2+y^2-4}dx+\frac{2y}{x^2+y^2-4}dy$ exact on his natural domain?

The natural domain $D$ of the differential form $\omega$ is $D=D_1 \cup D_2$ where $D_1=B\bigl((0,0),2\bigr)$ is the open ball of center in the origin and radius $r=2$ and ...
1
vote
1answer
239 views

Recover the differential form by its pullback

I have a smooth manifold $M$ in $\mathbb{R}^n$ given by $M = \{ x \in \mathbb{R}^n \mid g(x) = 0 \}$. Its atlas consists of a single chart $(M,\varphi)$, where $$ \begin{array}{rcl} ...
1
vote
2answers
139 views

Closed differential form.

In the paper, page 5, Line 5, why the form $\eta_j$ is closed? We have to show that $d \eta_j=0$. Here $\eta_j = f(t)dt$ for some function $f$. Is it always true that $d\eta_j$ for any $f$? Thank you ...
1
vote
2answers
30 views

Given an $(n-1)$-form $\varphi$ on a smooth orientable $n$-manifold, there is a vector field $v$ such that $i_v\varphi = 0$.

I am working on the following problem. Let $M$ be a smooth orientable $n$-manifold, $n \geq 2$, and let $\varphi$ be a smooth $(n-1)$-form on $M$. Show that there is a vector field $v$ on $M$ such ...
1
vote
1answer
28 views

How to make an ideal generated by differential forms into a differential ideal?

Let $M=\mathbb{R}^4$ with standard coordinates $x_1,x_2,x_3,x_4$. Let $\alpha=x_2dx_1+x_3dx_3+dx_4$ and $\beta=2dx_2+x_1^2dx_3+x_1dx_4$ How to find a 1-form $\gamma$ such that the ideal generated ...
1
vote
2answers
43 views

Integrating a differential form inside a cylinder

Let S be cylinder given by $x^2+y^2=1$ between $z=1$ and $z=3.$ For $\varphi=e^xdx\wedge dy+ ydz\wedge dx+xdy\wedge dz$ find $\int_S\varphi$. I managed to finish the problem, but I'm getting ...
1
vote
2answers
62 views

The property of an integral manifold of differential form

Let's define the $1$-form $\eta$ on $\mathbb{R}^3$ by formula: $$\eta = A(x,y,z)\;\mathrm{d}x + B(x,y,z)\;\mathrm{d}y + \mathrm{d}z \text{.}$$ And let's assume that $$\mathrm{d}\eta \wedge \eta = ...
1
vote
2answers
55 views

Wedge product and determinants

I am attempting to self-study differential forms this summer, but I ran into this definition for the wedge product in my book, and it doesn't make any sense to me. Note that $\Bbb R^3_p$ is the ...
1
vote
1answer
65 views

A connection to Stoke's Theorem (I think)

This is homework. I just finished a question regarding double integration over the unit sphere involving pullbacks of differential forms to provide context (course is advanced Calculus). The question ...
1
vote
1answer
47 views

Show a 2-form is exact finding a primitive.

I have to show that $\omega=-4xy\:\mathrm{d}x\wedge \mathrm{d}y-2xz\:\mathrm{d}z\wedge \mathrm{d}x +2yz\:\mathrm{d}y\wedge \mathrm{d}z$ is exact finding a primitve of $\omega$ (by Poincare lemma I ...
1
vote
1answer
52 views

Winding number of a linear transformation?

I know that I am computing something incorrectly. I am trying to compute the index of a positive determinant linear bijection. The form I am using is $\omega = \frac{-y dx + x dy}{x^2 + y^2}$. I ...
1
vote
1answer
28 views

If $\omega$ is compactly supported form then so is $d\omega$?

If $\omega$ is a compactly supported differential form then so is $d\omega$. Is it true?
1
vote
1answer
126 views

What is the one form given its value for a vector field?

I read an article on vector fields. the author defined a 1-form on a manifold $M$ as $u(X)=\rho$ when $X$ is a given vector field and $\rho$ is a given real valued function defined on $M$. can we ...
1
vote
3answers
66 views

Differential closed form

Im trying to go alone through Fultons, Introduction to algebraic topology. He asks whether there is a function $g$ on a region such that $dg$ is the form: $$\omega =\dfrac{-ydx+xdy}{x^2+y^2}$$ in some ...
1
vote
1answer
62 views

Visualizing Non-Zero Closed-Loop Line Integrals Via Sheets?

How do I visualize $\dfrac{xdy-ydx}{x^2+y^2}$? In other words, if I visualize a differential forms in terms of sheets: and am aware of the subtleties of this geometric interpretation as regards ...
1
vote
1answer
88 views

A (not so?) simple question about differential forms

Let $M^n$ be a compact orientable manifold and let $\omega$ be a $(n-1)$-form in $M^n$. I want to show that there is $p\in M$ such that $(d\omega)_p=0$. Can somebody help me, please ? Thanks :)
1
vote
1answer
49 views

Is $\omega = dU = sin(x+y)dx+cos(x+y)dy$ an exact form?

In my thermodynamics homework I should prove that $dU = sin(x+y)dx+cos(x+y)dy$ is a function of state. Which means it's integration over any path be constant or in other word $dU$ should be an exact ...
1
vote
1answer
64 views

How to show $[\omega]=0$ implies $[\omega^n]=0$?

I'm trying to prove the following: If $(M, \omega)$ is a symplectic manifold and $[\omega]=0$ then $[\omega^n]=0$, where $[\omega]$ is the De Rham cohomology class of $\omega$. Well what I've done ...
1
vote
1answer
108 views

Relation between de Rham cohomology and integration

This question is a follow-up to When does a null integral implies that a form is exact? . As mentionned in the selected answer, given certain conditions it is possible to find an isomorphism between ...
1
vote
1answer
100 views

symplectic strucutre

Suppose $\omega$ is symplectic structure on $\mathbb R^n$. Let $\omega_0:=\omega|_{x=0}$. Let $\overline{\omega}= \omega_0-\omega$ and for $t\in[0,1]; \omega_t:= \omega+ t\overline{\omega}$. How ...
1
vote
1answer
62 views

$k$-forms on $\mathbb{R}^n$

Given an expression like $$ dx_1\wedge dx_2 \wedge dx_4 \left( \begin{bmatrix} 1\\2\\3\ \end{bmatrix} \ , \ \begin{bmatrix} 4\\5\\6 \end{bmatrix} \ , \ \begin{bmatrix} 7\\8\\9 \end{bmatrix} \right) \ ...
1
vote
1answer
160 views

How general formulation of Stoke's theorem relate to Kelvin-Stokes theorem

I asked a similar question, but I realized the question is too vague and it's better to start a new one: We know that there are two usually used formulations of Stoke's theorem. One is vector ...
1
vote
1answer
204 views

Example of differential form usage of Stoke's theorem

There are many examples that show how Kelvin-Stokes theorem is used. But I would like to see an example that uses differential form usage of Stoke's theorem and is hard or impossible to solve by ...
1
vote
1answer
49 views

Question about Interior product computation

I need to evaluate $\omega=i_X(dx\wedge dy)$ where $X$ is a vector field in $\mathbb{C}^2$ (which means $p=2)$. If I write $X(x,y)=(X_1(x,y),X_2(x,y))$, or simply $X=(X_1, X_2)$, then the interior ...
1
vote
1answer
53 views

Basis for the space of $p$-forms

According to the book I read, a general $p$-form can be written as: $$\omega=\omega_{a_1\ldots a_p} dx^{a_1}\wedge\ldots\wedge dx^{a_p},\hspace{0.5cm} a_1>a_2>\ldots>a_p$$ where I have used ...
1
vote
1answer
79 views

Use Fund Thm to evaluate the integral of $ ze^{x^2} dydz + 3ys dydz + (2-yz^7)dxdy $ over surface of the unit cube, except bottom face.

Use the Fundamental Theorem to evaluate the integral of $ ze^{x^2} dydz + 3ys dydz + (2-yz^7)dxdy $ over the surface of the unit cube, except the bottom face.
1
vote
1answer
85 views

Find a $1$-form whose exterior derivative is $2x^2y^2dydz +3xz^2dzdx - 4xy^2dxdy$

Find a $1$-form whose exterior derivative is $$2x^2y^2dydz +3xz^2dzdx - 4xy^2dxdy$$ An exterior calculus question. I am trying to learn some algebraic topology, and have hit a bump with some (I ...
1
vote
1answer
94 views

Why is this integral zero? (Inner product between two 1-forms on a Riemann surface)

I have a quick question regarding the proof of Proposition II.3.2 in Farkas & Kra (pg. 40). The proposition is that if $\alpha$ is a square-integrable, $C^1$ 1-form, then $\alpha$ lives in a ...
1
vote
1answer
161 views

elementary questions about differential forms

QUESTION 1: So I know that if $\omega$ is an alternating $p$-form for odd $p$ on some vector space $V$, then $\omega\wedge\omega = 0$. But...isn't the same true for any $p$? Ie, take for example $p ...
1
vote
1answer
327 views

Existence of orthonormal frame

Let $M$ be a surface with Riemannian metric $g$. Recall that an orthonormal framing of $M$ is an ordered pair of vector fields $(E_1,E_2)$ such that $g(E_i,E_j)=\delta_{ij}$. Prove that an orthonormal ...
1
vote
1answer
198 views

Proving Fréchet differentiability

Am learning about Fréchet differentials and was wondering if for a real matrix $X$ and positive semidefinite real matrices $A,B$ the function $f(X)=TrX^TAX-X^TBX$ is twice Fréchet differentiable or ...
1
vote
1answer
144 views

What exactly are the definitions of $\varphi^*dz$ and $\varphi^*d\bar{z}$?

Suppose $\varphi$ is a smooth map on $\mathbb{C}$. For a function $f$, we define a $0$-form $\varphi^*f$ as $\varphi^*f=f\circ\varphi$. Also, $$ \varphi^*\,dx=\frac{\partial\varphi_1}{\partial ...
1
vote
1answer
117 views

Prove that $f^*\omega=0$

I'd like some help to prove the following: $f\colon U \subset \mathbb{R}^m \to \mathbb{R}^n$, differentiable; $m<n$ ; $\omega$ is a $k$-form in $\mathbb{R}^n$, $k>m$. Show that $f^*\omega = 0$ ...
1
vote
1answer
212 views

Elementary symmetric polynomials and matrices of 1-forms

Let $A$ be a $n \times n$ matrix of 1-forms (for example, a connection form). Note that $A \wedge A$ is not $0$, but by using the anti-symmetry of the wedge product applied to the entries of $A$ we ...
1
vote
1answer
230 views

The chain rule for a function to $\mathbf{C}$

Let $f:U\longrightarrow \mathbf{C}$ be a holomorphic function, where $U$ is a Riemann surface, e.g., $U=\mathbf{C}$, $U=B(0,1)$ or $U$ is the complex upper half plane, etc. For $a$ in $\mathbf{C}$, ...
1
vote
1answer
204 views

If $S^{2n+1}$ is covering space of $X$, then $X$ is orientable.

Is there any direct way to prove that $n$-manifold is orientable? In AT we can just calculate $n$'th homology group and check whether it's $\mathbb Z$ or $0$. But I want a geometric method, using ...
1
vote
0answers
24 views

Reference Request for differential ideals of Pfaffian forms on jet bundles

My setting is the following: Given two families of differential forms $\omega^i$ (with $i=1,...,m$) and $\mathrm d F^j$ (with $j=1...n-m$), define $$\eta^i := \sum_{j=1}^m a^i_j\omega^j + ...
1
vote
0answers
35 views

differential form: a question in Novikov's book

I met a problem when reading the book of Novikov et al. "Modern geometry, vol 1". In page 200 of the book (GTM 93, English version), they say for 2-form $F$, $$ (F\wedge F)_{0123}=-{1\over ...
1
vote
0answers
55 views

Integrability of 1-forms and Stokes' Theorem

Let $\alpha$ be a $1$-form defined on a manifold $M$ and $\Delta = ker (\alpha)$. The classical theorem of Frobenius says that $\Delta$ is integrable if $\alpha \wedge d\alpha =0$ i.e if $d\alpha$ is ...
1
vote
0answers
39 views

Residue sequence

I'm reading book Compact complex surfaces. In the first section of the second chapter they consider a curve $C$ on a surface $X$ (for simplicity I assume that $X$ and $C$ are smooth), then tensoring ...
1
vote
2answers
134 views

Is a 1-form locally expressible as $dx$?

Given a 1-form $\alpha$ which is non-zero at every point of a manifold $M^n$, is it true that locally I can express the form as $dx$? (that is around each point there is a coordinate neighborhood such ...
1
vote
0answers
57 views

A 1-form on a smooth manifold is exact if and only if it integrates to zero on every closed curve

I am stuck on the following problem, which comes from a old qualifying exam. Prove that a 1-form $\phi$ on $M$ is exact if and only if for every closed curve c, $\int_{c} \phi =0$. One way is an ...
1
vote
0answers
33 views

Confused about why $dd=0$ in de rham cohomology?

In Bredon's proof that $dd=0$, he lets $\omega=fdx_1\wedge\cdots\wedge dx_p$, and calculates $$ dd\omega=\sum_{j=1}^n\sum_{i=1}^n\frac{\partial^2 f}{\partial x_i\partial x_j}dx_j\wedge dx_i\wedge ...
1
vote
0answers
34 views

Two isomorphisms of the space of vector fields/forms

Suppose you have a diffeomorphism $$ \phi: M \to N. $$ As far as I understand, this gives rise to two distinct isomorphisms $$ a : \mathcal A^1(M) \cong \mathcal A^1(N) :b $$ between the space of ...
1
vote
1answer
18 views

linear form $\phi : \mathbb{R^3} \times \mathbb{R^3} \to \mathbb{R}$ is alternate if and only if $\phi(v,v)=0$,for all $v \in \mathbb{R^3}$

Prove that a bilinear form $\phi : \mathbb{R^3} \times \mathbb{R^3} \to \mathbb{R}$ is alternate if and only if $\phi(v,v)=0$,for all $v \in \mathbb{R^3}$. My thought:- If $\phi : \mathbb{R^3} ...
1
vote
0answers
32 views

Stoke's theorem explanation

Firstly, Wikipedia informs me that Stoke's theorem might mean two different things so I'm pretty sure I'm talking about the general one. I'll just state what I understand from it and I'd like someone ...
1
vote
0answers
30 views

Wedge product of Lie algebra valued differential forms [duplicate]

Let $\mathfrak{g}$ be the Lie algebra of a matrix Lie group. Furthermore, let us consider the following $\mathfrak{g}$-valued $p$-form and $\mathfrak{g}$-valued $q$-form: \begin{equation} ...
1
vote
0answers
72 views

differential forms and orientation in Bott and Tu

I am reading Bott and Tu book on my own. If you have the text you can answer my question perhaps, if not it would be too long to explain it. I have gotten to page 29. I am confused about proposition ...
1
vote
0answers
26 views

prove that ''pullback'' maps forms to forms

Suppose we have $A: V_{1}\to V_{2}$ where $V_{1},V_{2}$ are real vector spaces. Then $A^{\star}:\mathcal{J}^{k}(V_{2})\to \mathcal{J}^{k}(V_{1})$ where $\mathcal{J}(V):=\{\text{space of all ...
1
vote
0answers
37 views

integral of a 2-form over an oriented manifold

An old exam question: Let $M$ be oriented submanifold of $\mathbb{R}^{n}$ of dimension $k$, let $\omega$ be a $k$-form on $M$. 1) Define $\displaystyle\int_{M}\omega$ 2) let ...