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

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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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2answers
59 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 $$\gamma=\left\{ (x,y,z):3x+y+3z=3,x\ge0,y\ge0,z\ge0\right\} $$ ...
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1answer
78 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 ...
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1answer
126 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 ...
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1answer
57 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?
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2answers
839 views

Pullback of differential form of degree 1

Good evening, In differential forms (in the proof of the naturality of the exterior derivative), I don't get why if $h\in \Lambda^0(U)$ and $f^*$ is the pullback then, $f^*dh=d(f^*h)$. I wrote ...
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2answers
46 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 ...
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51 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?
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1answer
184 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, ...
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1answer
49 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, ...
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1answer
32 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 ...
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29 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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1answer
45 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?
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1answer
32 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 $$ ...
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1answer
114 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 ...
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0answers
79 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 ...
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1answer
64 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?
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1answer
76 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.
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0answers
45 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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1answer
90 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 ...
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1answer
48 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 ...
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0answers
36 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 ...
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53 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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1answer
71 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 ...
71
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2answers
3k views

Direct proof that the wedge product preserves integral cohomology classes?

Let $H^k(M,\mathbb R)$ be the De Rham cohomology of a manifold $M$. There is a canonical map $H^k(M;\mathbb Z) \to H^k(M;\mathbb R)$ from the integral cohomology to the cohomology with coefficients ...
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34 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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3answers
95 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 ...
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4answers
230 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 ...
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1answer
168 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 ...
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1answer
2k views

How to calculate the pullback of a $k$-form explicitly

I'm having trouble doing actual computations of the pullback of a $k$-form. I know that a given differentiable map $\alpha: \mathbb{R}^{m} \rightarrow \mathbb{R}^{n}$ induces a map $\alpha^{*}: ...
2
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1answer
159 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 ...
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1answer
98 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 $M \to M$ ...
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3answers
160 views

elementary question regarding differential forms

Is it possible to give a high level explanation why changing the order of differentials will give rise to a minus sign ? I.e. why do we have $$ dx\,dt = - dt\,dx $$ (I am going to take a course on ...
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1answer
37 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 ...
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1answer
84 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) = ...
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0answers
79 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 ...
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1answer
89 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 ...
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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 ...
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1answer
439 views

Why are de Rham cohomology and Cech cohomology of the constant sheaf the same

I am comfortable with de Rham cohomology, sheaves, sheaf cohomology and Cech cohomology. I am looking to prove the following theorem: If $M$ is a smooth manifold of dimension $m$, then we have ...
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0answers
70 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 ...
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1answer
72 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 ...
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1answer
25 views

Definition of an integrand

General Question: Say we have integral $$ \int f(z)\ dz $$ Is the integrand in this context (i) $f(z)$ or (ii) $f(z)\ dz$? In any case, is $f(z)\ dz$ a formally defined mathematical object in its ...
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1answer
31 views

Differentiability problem .

Hi can someone help me with the following problem. I am having difficulties evaluating : $$ \frac {d} {dt} f'(u(t)) $$ Is it just $f''(u(t))$ ? Thanks
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1answer
123 views

The kernel of a differential one-form

I'm thinking about the kernel of a differential one-form $\theta\in\Lambda^{1}(M)$: $$ Ker(\theta):=\left\{X\in\mathfrak{X}(M) \;|\; \theta(X)=0\right\} $$ Now suppose $X\in Ker(\theta)$, then is ...
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1answer
64 views

Recovering a frame field from its connection forms

I have a faced a research problem where I would need to recover a frame field given its connection forms. More precisely, I begin with an orthonormal frame field (given by data) in $\Re^3$ written as ...
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62 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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1answer
69 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 ...
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0answers
59 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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1answer
29 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?
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189 views

Explanation for the integral of differential forms

In our course of differential geometry we defined the integral $\int_{U} \omega$ of a differential form $\omega=f dx_1\wedge \ldots \wedge dx_n: T^nU \rightarrow \mathbb R$ with $U\subseteq \mathbb ...