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

learn more… | top users | synonyms

2
votes
0answers
27 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 ...
1
vote
1answer
35 views

Integrate the differential form over a cardioid

$\omega=\dfrac{-ydx+(x-1)dy}{(x-1)^2+y^2}$ Calculate $\int_C\omega$ where $C...r=1+\cos\varphi$ (positively oriented) I'm still pretty lost when it comes to differential forms but as far as I ...
0
votes
0answers
23 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 ...
4
votes
0answers
75 views

Generalizing the Hopf invariant to arbitrary manifolds

I recently ran across a qual question about a generalization the Hopf invariant to smooth maps $f: M^{4n-1} \to N^{2n}$ between arbitrary closed connected oriented manifolds of the indicated ...
0
votes
1answer
24 views

Real part of integral over holomorphic 1-form is zero implies the one-form is zero

Suppose we are working on a Riemann Surface $X$, assume of genus g $\geq$ 1. Let $\omega$ be a holomorphic 1-form on $X$, with $$ \textrm{Re} \int_\gamma \omega = 0$$ for every closed contour ...
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
73 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 ...
2
votes
1answer
100 views

Independence of $H(f)=\int_M \alpha \wedge f^* \beta$ on choice of $d\alpha=f^*\beta$?

I came across the following UCLA qual question while studying for my upcoming qual: Let $f: M^{4n-1} \to N^{2n}$ be a smooth map between closed connected oriented manifolds of the indicated ...
3
votes
2answers
55 views

Using Stokes's theorem to calculate a value of integral

Use Stokes's theorem to calculate the integral $$I= \int_\Gamma (x^2+2y)dx+(y+z)dy+(z^2+x^2)dz$$ where $\Gamma$ is the boundary of $$\gamma=\left\{ (x,y,z):3x+y+3z=3,x\ge0,y\ge0,z\ge0\right\} $$ ...
0
votes
1answer
56 views

Question about Alternating forms

So I understand the definition of an alternating form on $\mathbb{R}^m$, but I don't really understand the proof of the lemma. Could someone explain the first observation? Why is it so?
2
votes
1answer
69 views

Construct tensors from differential forms?

Let $(M,g)$ be a Riemannian manifold, differential forms are defined using tensors, could we define a tensor using a differential form? For example, if $\omega$ is a two-form on $M$ which is expressed ...
2
votes
2answers
42 views

An object is travelling in a straight line. Its distance, s meters, from a fixed point at time t seconds is given by the expression

$$s=t^3−t^2−6t$$ a) Find ds/dt when t=3 and interpret this result. b) Find d^2s/dt^2 when t=3 and interpret this result. c) Find the time in seconds when the velocity is 2m/s (d) Using the ...
0
votes
1answer
59 views

Why is the exterior derivative called exterior derivative

I am studying exterior calculus, and I think I have some grasp of what is the exterior derivative. However its name still eludes me - why is it called a derivative? Is it just because the operator $d$ ...
0
votes
0answers
50 views

Antipodal map commutes with antipodal map? [duplicate]

Suppose we have a closed form $d\omega$ on $S^{n}$, and antipodal map $i: S^{n} \to S$ n i.e $i:x \to −x$. How to see that the external differential commutes with antipodal map?
3
votes
1answer
42 views

Well-definedness of a coboundary map between a reduced $L^2$ de Rham cohomology group and a relative cohomology group

I'm working right now with this paper of Carron. And I think I'm stuck at a relatively simple question. On page 11 he is defining a coboundary map $b : H^k_{2, \text{reduced}}(M - K) \to H^{k+1}(K, ...
0
votes
1answer
26 views

Proving the pseudosphere is regular and orientable.

The textbook I'm using define the tractrix by $ T=\{(\sin t, \cos t+\log (\tan (t/2))):0<t\leq\pi\}$ and define the pseudospher being the tractrix roting around the $z$-axis, I have to prove that ...
5
votes
1answer
106 views

Poincare dual of unit circle

I'm trying to self-study Differential Forms in Algebraic Topology by Bott and Tu. I've come across this exercise: Show that the closed Poincare dual of the unit circle in $ R^2-\{0 \} $ is zero, ...
1
vote
0answers
27 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
1answer
89 views

Change of Coordinate Formula for Differential Forms

Let $M$ be a manifold, $x$ local coordinates on an open set $U$, $y$ local coordinates on an open set $V$. In addition, let $(x, \alpha)$ and $(y, \beta)$ be two induced bases for the common part of ...
0
votes
1answer
42 views

Differentiate the following functions

Let $$y(x)= 4 x^3 e^{2x},$$ then $$y'(x) = 4 \times 3 \, x^2 e^{2x} + 4 \, x^3 \times 2 e^{2x} = 12 \, x^2 e^{2x} + 8 \, x^3 e^{2x}$$ Does this look correct?
0
votes
1answer
26 views

the exact form in a manifold

Let $M$ be a compact manifold, $X$ is a vector field on $M$, $\alpha$ is a closed 2-form on $M$, $\phi: M\to M$ is a diffeomorphism such that $\phi^*\alpha=\alpha$, then I want to konw whether $$ ...
4
votes
1answer
80 views

Explanation of differential forms and notation

I'm doing multivariable calculus and I'd love if someone could shed some light on things that confuse me. When we did integrals of real functions with real variables, the $dx$ that was in every ...
4
votes
0answers
62 views

A question on harmonic two-forms

Let $(M^4,g)$ be a closed Riemannian four-manifold with $b_2^+>0$ and $b_2^->0$, is it possible to find two harmonic two-forms $\alpha\in H^2_+(M)$ and $\beta\in H^2_-(M)$, such that ...
0
votes
1answer
43 views

When is an exact 2-form harmonic?

Let $\alpha$ be an exact two-form, $\alpha=d\beta$ for some one-form $\beta$, when is $\alpha$ harmonic? By uniqueness of harmonic forms in cohomology classes, it cannot be harmonic?
2
votes
0answers
44 views

surface element of $S^{3}$

How does one show that the surface element of $S^{3}=\{x=(x_{1},...,x_{4})\in\mathbb{R}^{4}\;|x|^2=1\}$ is given by the following 3-form: $\omega=x_{1}dx_{2}\wedge dx_{3}\wedge ...
2
votes
1answer
56 views

what does it mean for a differential form to be well defined on a manifold?

What does it mean for a differential form to be well defined on some manifold. In particular, why the $2$-form $\omega=d\psi\wedge d\theta$ is well defined on $S^{2}$? Thank you in advance.
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 ...
2
votes
1answer
68 views

De Rham cohomology for $\mathbb{R^2}$

De Rham cohomology groups for $\mathbb{R^2}$. $H^{0}_{dR}(\mathbb{R}^{2})=\mathbb{R}$ since $Z^{0}(\mathbb{R}^{2})$ is the one dimensional space of locally constant functions on $\mathbb{R}^{2}$ and ...
1
vote
1answer
46 views

Is it true that $0$ is the only exact $0$-form

I am totally new to the concepts of forms so sorry if my question is trivial. I came across a statement that ''there are no exact $0$-forms as there is no $-1$ form. So I revisited the definition ...
0
votes
0answers
28 views

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 ...
1
vote
0answers
46 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 ...
0
votes
0answers
55 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: ...
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 ...
0
votes
0answers
15 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 ...
2
votes
4answers
94 views

Show that the form $w$ is closed but not exact

Let $w=\dfrac{-y}{x^2+y^2}dx+\dfrac{x}{x^2+y^2}dy$ Showing that $w$ is closed is easy. Just calculate $dw$ and you'll get 0. But how do I show that $w$ is not exact? In other words, I need to ...
0
votes
1answer
58 views

Defining the integral of differential $1$-forms

Assume $M$ is a smooth manifold, $g:[0,1]\to M$ is a smooth curve on $M$, and $w$ is $1$-form on $M$. Definition: $$\int_gw=\lim_{\Delta\to 0}\sum_{i=1}^nw(x_i)$$ The tangent vectors $x_i$ are ...
2
votes
1answer
104 views

De Rham cohomology, and forms on manifolds

In String Theory and M-Theory by Becker, Becker and Schwarz, they introduce a group, $$C^{p}(M)$$ which they denote the group of all closed $p$-forms on the manifold $M$. Furthermore, they state ...
2
votes
1answer
100 views

Pullback of Differential-Form

Given a differential form $$x\,dy\wedge dz-y\,dx\wedge dz+z\,dx\wedge dy$$ I am supposed to prove that the it's pullback by a linear map of determinant one leaves it invariant. For example, if ...
1
vote
1answer
77 views

diffeomorphism preserve a volume form

Let $\omega_1$, $\omega_2$ two volume form on a compact manifold $M$, we know that there exists a never-vanishing function $f$, s.t. $\omega_1=f\omega_2$. If $h$ is a diffeomorphism $M \to M$ ...
0
votes
1answer
39 views

Differential Forms / Stokes' Theorem Problem

Problem: Let $w = (x + y)dz + (y + z)dx + (x + z)dy$ and let $S$ be the upper part of the unit sphere; that is, $S$ is the set of $(x,y,z)$ with $x^2+y^2+z^2 =1$ and $z\ge0$. $\delta$$S$ is the unit ...
0
votes
1answer
34 views

How to show the identity $d(*df)=(\Delta f)\nu$ where $\nu=dx_1\wedge\ldots \wedge dx_n$?

My textbook defines the Hodge star operator as follows: Given $\omega\in \Omega^k(\mathbb R^n)$ we define $*\omega\in \Omega^{n-1}(\mathbb R^n)$ setting $$*(dx_{i_1}\wedge\ldots\wedge ...
1
vote
3answers
84 views

A question on differential topology

Let $\mathbb{C}P(1)$ denote the complex projective line. I am attempting to show that there does not exist a nonzero holomorphic differential $1$-form on $\mathbb{C}P(1)$. My intuition is as ...
1
vote
1answer
47 views

If $\omega\wedge\beta$ is exact for every closed form $\beta$, then $\omega$ is exact.

Let $\omega$ be a closed $k$-form. Then: If $\omega$ is exact, for every closed form $\beta$, the form $\omega\wedge\beta$ is exact. Proof: Let $\omega=d\alpha$. Now $d(\alpha\wedge\beta) = ...
4
votes
0answers
61 views

show that $\omega$ is exact if and only if the integral of $\omega$ over every p-cycle is 0

Let $M$ be an oriented smooth manifold and $\omega$ a closed $p$-form on $M$. Show that $\omega$ is exact if and only if the integral of $\omega$ over every $p$-cycle is $0$. In particular, how to ...
2
votes
1answer
56 views

Product rule for differential forms?

How would I work out $d(\omega F)$ where $\omega$ is a 1-form and $F$ is a vector valued function? I know that $d(f \omega) = df \wedge \omega + f d\omega$, for a smooth function $f$. I suppose this ...
1
vote
1answer
29 views

How to show $\nu=dx_1\wedge\ldots \wedge dx_n$?

Let $\nu$ be the $n$-form in $\mathbb R^n$ satisfying $\nu(e_1, \ldots, e_n)=1$ where $\{e_1, \ldots, e_n\}$ is the canonical base of $\mathbb R^n$. Let $\displaystyle v_i=\sum_{j=1}^n a_{ij}e_i$. How ...
1
vote
0answers
103 views

Line integral - should I parametrize the square?

I have the following $1-\text{form}$ defined: $$\omega = \displaystyle\frac{2xy}{(1-x^2)^2+y^2}\mathrm{dx}+\displaystyle\frac{1-x^2}{(1-x^2)^2+y^2}\mathrm{dy}$$ I'd like to find ...
1
vote
1answer
53 views

Flat Surfaces in $\mathbb{R}^3$ Can Be Bent Only Along Straight Lines

This is a problem out of Elementary Differential Geometry by Barrett O'Neill (Chapter 6 Section 3 Number 2). Let $M$ be a flat surface in $\mathbb{R}^3$ with principal curvatures $k_1$ and $k_2$, ...
2
votes
0answers
58 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 ...
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 ...