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

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23 views

closed and bounded form

I have this problem, Let $\omega$ a closed $1$ form in $\mathbb{R^{2}}\setminus {0} $ such that $\omega$ restricted to the set $D$ is bounded with $D=\left \{ x\in\mathbb{R} \text{ such that }\left | ...
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35 views

differentiable curve

I´m a little stuck with this problem, I think is false but I can´t find a counter example, here is the problem Let $\omega$ a 1-form defined in $U\subset \mathbb{R^{2}}$(it can be $\mathbb{R^{n}}$, ...
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43 views

1-forms and zero simple

Let $\varphi$ a differential transformation such that $\varphi (x,y)=(f(x,y),g(x,y))$ and $D\subset U$ such that $\varphi$ restricted to $\partial D=\gamma$ be distinct zero and we define $i(\varphi ...
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21 views

Pullback of area form of manifold by local chart map

Let $M$ be an orientable $2$-manifold in $\mathbb{R}^n$, with area form $dA$. Let $\phi: U \to M$ be a local coordinate system for $M$, with $U \subseteq \mathbb{R}_{uv}^2$. It is stated that ...
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34 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 + ...
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44 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 ...
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86 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 ...
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70 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 ...
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37 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 ...
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39 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 ...
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37 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 ...
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80 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 ...
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28 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 ...
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40 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 ...
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50 views

Commutative differential graded algebra

The situation is the following: Let $V$ be a Lie algebra of finite dimension, say n, and let be the graded commutative algebra $(\bigwedge^{\bullet}V)^*:=(\bigoplus_{k=0}^n\bigwedge^kV)^*$ and define ...
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58 views

How to show the differential form $\nu$ satisfies $\nu(v_1, \ldots, v_n)=\det(a_{ij})$?

In $\mathbb R^n$ consider the differential form $\nu$ satisfying $\nu(e_1, \ldots, e_n)=1$. For every $i=1, \ldots, n$ consider the vector $\displaystyle v_i=\sum_{j=1}^n a_{ij} e_j$. How to show ...
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55 views

Does this Condition Force this Differential 2-form to be 0?

Let $dw$ be a 2-form; $\alpha$ be a 1-form, $g$ a function (i.e., a 0-form), $X$ be a fixed vector field , all living in a 3-manifold, so that $$dw(X,.)=g\alpha .$$ Does this condition force $dw$ to ...
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92 views

differential form and cylindrical coordinate

Problem. If $r, \theta, z$ are the cylindrical coordinate functions on $\mathbb > R^3$ , then $x = r\cos\theta, y = r\sin\theta, z = z$. Compute the volume element dx dy dz of $\mathbb R^3$ ...
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35 views

pullback of differential form over a lie group.

Im given that $\omega \in \Omega^3(SL(2,R))$ satisfies $\omega (I) = -2 dx_1 \wedge dx_2 \wedge dx_3$. Consider the left multiplication $L_A(B) = AB$ as a difeormphism over $SL(2,R)$. I want to ...
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27 views

Litterature: Convex Geometry using differential forms

I've been looking for papers or books that discuss/use differential forms in convex geometry. I have been very unsuccessful in my search and was wondering if anybody here had come across such ...
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67 views

Structure equations on the 3-sphere

On the 3-sphere I have found the vector fields $X_1=(-x_2,x_1,-x_4,x_3)$, $X_2=(-x_3,x_4,x_1,-x_2)$, $X_3=(-x_4,-x_3,x_2,x_1)$, in the basis $\left\{\frac{\partial}{\partial ...
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50 views

invariance of 2-form under $SO(3)$

I'm trying to understand how to derive forms that invariant under action of some group. For example 2-form on $S^2$ and on $\mathbb{R}^3$ is very interesting for me (because I have troubles with it). ...
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53 views

What exactly does it mean to say that “functions cannot be integrated on Riemann surfaces”?

I've seen statements of this sort used to motivate the introduction of differential forms, and I'm not sure exactly what's meant. Obviously if you start by defining differentiation as an operation ...
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86 views

Is Cartan's magic formula applicable to time dependent vector fields?

Cartan's magic formula states: $$\mathcal{L}_v\omega = i_v\mathrm{d}\omega + \mathrm{d}i_v\omega$$ Is this also true for time dependent vector fields? If so: How can I prove it? If not: Is there a ...
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50 views

A $k$-form is thought of as measuring the flux through an infinitesimal k-parallelepiped

On the wikipedia has written "A $k$-form is thought of as measuring the flux through an infinitesimal k-parallelepiped." How does a $k$-form do this? if this sentence is right, then the flux of which ...
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99 views

Prove $\left.{\partial X\over \partial Y}\right|_Z=-\frac{\left.{\partial Z\over \partial Y}\right|_X}{ \left.{\partial Z\over \partial X}\right|_Y}$

$$\left.{\partial X\over \partial Y}\right|_Z=-\frac{\left.{\partial Z\over \partial Y}\right|_X}{ \left.{\partial Z\over \partial X}\right|_Y}$$ The above is an identity frequently used in ...
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66 views

Integration equivariant form

We consider the action of $S^{1}$ on $S^{2}$ by rotation respect the vertical axes. We want to integrate the $2-$ equivariant form $$\alpha(X)= -X\cos(\phi) +\sin(\phi)\,d\phi\,d\theta.$$ We have ...
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49 views

Finding a function given a differential

I'm still pretty new to this differential business and I have a couple of question concerning this problem I've come across. We're given a differential form $\alpha=p_1 dx_1+p_2 dx_2-H(p_1,p_2)dt$ ...
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74 views

Is there a similar equivalence like the divergence theorem for surface integrals non-linear in the normal vector?

The divergence theorem can be stated as $$\bigcirc \hspace{-1.3em} \int \hspace{-.8em} \int\limits_{\partial\Omega} dA\,n_i = \iiint\limits_\Omega dV\partial_i$$ applied to an arbitrary function ...
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64 views

derivatives of coordinates on a riemann surface

Let $X$ be a compact connected Riemann surface of genus $g>0$. Let $(\omega_1,\ldots,\omega_g)$ be a basis for $H^0(X,\Omega^1)$. Let $x$ be a point in $X$ and let $z:U\longrightarrow B(0,1)$ be a ...
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35 views

Pull-back of a one-form on a sphere.

Let $\iota: S^2 \to \mathbb{R}^3$ be the inclusion map and choose a chart $(U,f)$ on $S^2$, where $U=\{(x,y,z)\in \mathbb{R}^3: z>0\}$ and $$f: U \to \mathbb{R}^2,$$ $$ (x,y,z)\mapsto (x,y). $$ I ...
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36 views

Is there a way to use this interpretation of differential forms on manifolds?

I read Rudin's "Principles of Mathematical Analysis". In the part of Differential Forms, he defined them formally. I particularly enjoyed the formal viewpoint, since everywhere else it seems that the ...
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22 views

differential forms and index

The other question i can´t solve is this, If $\varphi$ a differential transformation such that $\varphi (x,y)=(f(x,y),g(x,y))$ and define $i(\varphi ,D)=\frac{1}{2\pi }\int _{\gamma }\theta _{0}$. ...
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29 views

Integral of differential of $C^1$ $(k-1)-$form in $\mathbb{R}^n$ over orientable $k$-manifold without boundary is zero

I don't get it. Is this really true? I thought compactness of the manifold would be necessary... It is asserted that $\int_M dw =0$, under the hypothesis that $w$ is of class $C^1$ and is a ...
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14 views

Frobenius theorem for singular foliations , what are hypotheses impose over an endomorphism $P:TM\to TM$ that it span a singular foliation in M?

Let $M$ a smooth manifold. Given a morphism $P:TM\to TM$, i.e, $\pi\circ P\equiv Id$ and is linear over the fibers. If we suppose that $P^2=P$, then for each $x\in M$, $P_{x}:T_xM\to T_xM$ is a ...
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27 views

Integrating differential forms over a box

I've only ever seen examples of integrating a differential form over a curve C involving defining a parameterization. I have seen people integrate 1 forms over a box without defining a ...
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32 views

On the existence of integrating factor for 1-form

Question: Is there any holistic approach to determining the existence/finding the integrating factor for 1-forms? I never took any course in differential equations...I guess this is something ...
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60 views

How can we compute the order of 1-form on Riemann surfaces

Let X be a hyperellictic curve defined by $y^2=h(x)$. Let $\pi:X\rightarrow\mathbb{P}^1$ be the double covering map seding $(x,y)$ to $x$. Let $\omega=\pi^*(dx/h(x))$. How can we compute the orders of ...
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25 views

Integrating a differential form over two oriented line segments

The problem is the following: Integrate the differential form $(\cos x \arctan e^x-y)dx+(2xy-y^2)dy$ over two oriented line segments $AB$ and $BC$ where $A=(0, -1), B=(1, 0)$ and $C=(0,1)$ I'm ...
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coordinates for $T^*M$, $T^*T^*M$, $T^*T^*… T^*M$

This relates to my previous question on the tautological one form. I'm reading John Lee's Smooth Manifold book. Anyway, I wanted to clear up on some notation, which is confusing at first, but I ...
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66 views

Given a closed form, show it is exact

I came across an old exam problem and I wonder how to approach it: 1) Show that if $f$ is a $k$-form on $\mathbb{R}^{n}$ and $df=0$, $k>0$ then $f=dg$ for some $k-1$-form $g$. My first thoughts: ...
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25 views

Differential forms and the Lebesgue measure. Parametrization.

Assume $f=\sum_I a_Idx_I$ is a continuous differential $k$-form on $\mathbb R^n$. Here $I$ ranges the increasing $k$-indexes. Fix a point $x_0=(x^0_1,\dots,x^0_n)$ and a multi-index ...
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37 views

differential form and preserving volume

I was reading a beamer about differential foms and I found the following problem. if $M$ is an orientable manifold and $\omega \in \Omega^n(M)$ is the volume form and Let $X$ any vector field. Then ...
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27 views

Show that $d\log f$ is a 1-current on a 1-dimensional complex manifold

I am having trouble with this problem (and it might be because I have the formulation slightly off). I need to show that $d\log f$ is a 1-current on a 1-dimensional complex manifold $M$. This means ...
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54 views

Proving $d(f^*e_i')=0$

Let $f:M\to N$ be a differentiable map between two manifolds. $e_i$ is a basis vector of $N$ with respect to some chart and $e_i'$ its dual (i.e. $e_i'(e_j)=\delta_{ij}$). How do I prove the ...
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68 views

Is a Differential 1-form ( on M^3)Dual to a Contact Vector Field a Symplectic Form?

say $w$ is a global contact 1-form on a 3-manifold $M^3$ , meaning $w \wedge dw \neq 0$ at any point in the manifold , and let $X$ be the vector field dual ( under, say, a choice of Riemannian metric ...
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67 views

differentiable equal r-forms

Let $\alpha$, $\beta$ be two $r$-forms continuous in $U\subseteq \mathbb{R}^n$ open. If $\int_M \alpha =\int_M \beta $ for all surface $M\subseteq U$   dimension $r$, compact, with boundary, then ...
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227 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 ...
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64 views

Show that $\int_Md\omega=0$.

Let $\omega$ be a continiously differentiable $(k-1)$-form in the open set $U\subset\mathbb{R}^n$ and $M\subseteq U$ an orientated compact k-dimensional manifold. Show that $$ ...
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96 views

Existence of Spin Group

"In mathematics the spin group Spin(n) 1[2] is the double cover of the special orthogonal group SO(n), such that there exists a short exact sequence of Lie groups As a Lie group Spin(n) therefore ...