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

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2answers
21 views

Exercise about differential forms and pull-back: $\omega_p=0$ if and only if $(\omega|_U)_p=0$

Let $M$ be a manifold, $\omega\in\Omega^q(M)$. Let $U\subset M$ be an open subset embedded as manifold in $M$. Let $p\in U$. Show that $\omega_p=0$ if and only if $(\omega|_U)_p=0$. Notation: Let ...
2
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0answers
47 views

Integrating the Hopf invariant for $\pi:S^3\to S^2$

I've been working on the last part of problem 9., chapter 9 in Nakahara's Geometry, Topology and Physics all day, with no success, and am in need of some assistance. We are asked to compute the Hopf ...
2
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2answers
43 views

Under what conditions can a general 2-form be written as a wedge product of two 1-form

Assume we have a 2-form $\omega \in \Lambda^2\mathbb{R}^n$. It is usually stated one can write $$\omega = \alpha \wedge \beta,$$ with $\alpha, \beta \in \Lambda^1\mathbb{R}^n$ only for $n < 4$. How ...
4
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1answer
50 views

Choice of order in the Leibniz rule is arbitrary?

One of the rules which characterizes the exterior derivative is that, for $\varphi$ a real-valued function and $\omega$ a $k$-form, we have $$d(\varphi \cdot \omega) = d\varphi \wedge \omega + ...
2
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2answers
32 views

Line integrals in differential form

I'm a bit confused as to the format of line integrals in differential form (i.e. the form in which Green's theorem is often presented). For example: $$ \oint\limits_\mathcal{C} \left( y^2 \mathrm{d}x ...
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0answers
38 views

Is there a deRham (co)homology for vector-valued differential forms?

Is there a deRham (co)homology for vector-valued differential forms? The deRham (co)homology of differential forms has been well-discussed and well-founded, along with the fact that for the exterior ...
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1answer
64 views

Differential forms- riddle me this.

I stumbled over this answer on math.stackexchange Let $x$ be a point in $\rm M$. Then because $\omega$ is non degenerate at $x$, the antisymmetric matrix $\omega_x$ has full rank on the tangent ...
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3answers
70 views

Exterior derivative

Let $$\omega = \frac{1}{2} \sum_{i,j} \omega_{i,j} dx_i \wedge dx_j$$ be an antisymmetric form, so i.e. $\omega_{i,j} = - \omega_{j,i}.$ Now, in some lecture notes I found that $$d \omega ...
2
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2answers
39 views

Dimension of the $0^{\text{th}}$ de Rham cohomology group of $U$

Let $U\subset\mathbb{R}^n$ be an open set. I proved that $$ H^0_{DR}(U):=\frac{\text{closed forms}}{\text{exact forms}}=\{f\in C^{\infty}(U):\,f\,\,\text{is locally constant} \} $$ I have to show ...
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2answers
35 views

Finite universal covering induces injective maps on cohomology

I am trying to prove the following: Suppose $M$ is a smooth, connected manifold with finite fundamental group and $f : \widetilde{M} \rightarrow M$ is its (smooth) universal cover. Show that $f^* : ...
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1answer
24 views

Hamiltonian vector field and symplectic geometry

I want to show the following theorem: For any Hamilton function $H : M \rightarrow \mathbb{R}$ on some symplectic manifold $M$ and symplectomorphism $f : M \rightarrow M$ we have $X_{H \circ f} = ...
0
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0answers
29 views

Integration by parts in the $x_i$ direction in the integral which uses differential forms

Let $\Omega \subset \mathbb{R}^n$ be open and bounded and let $K \subset \Omega $ be compact. Let $\xi \in C_0^\infty (\Omega )$ with $0\leq \xi \leq 1$ in $\Omega $ and $\xi \equiv 1 $ on $K$. For ...
3
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1answer
55 views

Stokes' theorem and symplectic geometry

Let $V = \mathbb{R}^2,$ as a vector space then the Poincaré invariant is an integral $\int_{\gamma} \theta$ where $\theta = p dx $ is the symplectic 1-form and $\gamma$ a closed curve. Now, it is ...
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0answers
25 views

Is there a treatment/development of the Stokes' Theorem using differential forms and the Henstock-Kurzweil integral i.e. the gauge integral?

I'm working through an analysis text independently to prepare for grad school, and the author has discussed the limitations of both the Riemann and Lebesgue integrals and only hinted at the power of ...
2
votes
2answers
33 views

Pullback of $1$-form in coordinates

Let $$\theta(p) = \sum_{i=1}^n f_i(p) \, dx_i$$ be a $1$-form in local coordinates. then we define $F^*(\omega(p))(X_1,\ldots,X_n) = \omega(F(p))(DF(p)(X_1),\ldots,DF(p)(X_n))$ as the pullback of a ...
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1answer
35 views

Exercise about wedge product and multilinear forms

I'm considering $\omega\in \Lambda^{2q+1}(V^\ast)$, i.e. a multilinear skew-symmetric form. I want to prove that $\omega\wedge\omega=0$. How shall I proceed? Any suggestions? Do I have to write ...
0
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0answers
29 views

Line integral and differential forms

Let $df = \frac{\partial f}{\partial x_1 } dx_1 + \frac{\partial f }{\partial x_2 } dx_2$ be a $1-form.$ I know that that the line integral (along a curve $\gamma:[0,t] \rightarrow \mathbb{R}^2$) is ...
0
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0answers
28 views

Basic exercise differential forms

I have to show that the space of q-differential forms $\Omega^q(U)=\{0\}$ if and only if $q>n$ or $q<0$. Any ideas?
4
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1answer
49 views

What's going on with these identities involving $d$, $\mathcal L_X$, and $\iota_X$?

Let $\Omega^k$ denote the smooth $k$-forms on a given smooth manifold. Then we have the following operators: Exterior derivative: $d:\Omega^k\to\Omega^{k+1}$ (takes you to the right in the de Rham ...
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1answer
33 views

Stokes' theorem generalized the FTC part 2. Is there a known generalization for part 1?

Stokes' theorem generalizes the fundamental theorem of calculus (part 2) using differential forms. Is there a known generalization of part 1? edit In case anyone is unaware, The fundamental theorem ...
3
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2answers
56 views

would like proof of Theorem 4-10, from Spivak's Calculus on Manifolds

I'd like to understand a way to prove part 2 of this theorem. 2) If $\omega$ is a $k$-form and $\eta$ is an $l$-form, then $d(\omega\wedge\eta)=d\omega\wedge\eta+(-1)^k\omega\wedge d\eta$. I see ...
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0answers
21 views

Show that $d^2(w)=0$ for every $k$-form in $\mathbb{R}^n$

The question says to start showing this for 0-forms, and I did it. But how can I extend it for any k-form?
2
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3answers
35 views

Finding a $1$-form on a hyperboloid of one sheet which is closed but not exact

I want to find a closed 1-form $w$ on $$M=\{(x,y,z):x^2-y^2-z^2=-1\}\subset \mathbb R^3$$ which is not exact. I think that $$\frac{x\,\mathrm{d}x+y\,\mathrm{d}y+z\,\mathrm{d}z}{(x^2+y^2+z^2)^{3/2}}$$ ...
0
votes
1answer
55 views

Show that $f^*\omega = \det(df) \, dx_1\wedge\cdots\wedge dx_n$

Let $f:\Bbb{R}^n\to \Bbb{R}^n$ be a differentiable map given my $f(x_1,\ldots,x_n)=(y_1,\ldots,y_n)$, and let $$\omega=dy_1\wedge\cdots\wedge dy_n.$$ Show that $$f^*\omega = \det(df) \, ...
2
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1answer
38 views

Linear Maps from $V$ to $V^*$ defined by a 2-form

I came across this idea at the very beginning of a book and I don't quite seem to grasp it. It states given $\omega \in \bigwedge ^2 (V) $ you can define a linear map $ \omega^\#: V \to V^* $ by ...
3
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1answer
54 views

An exercise on differential forms

Let $\omega$ be a $1$-form in $U \subset \mathbb{R}^2$. A local integrating factor in $p$ for $\omega$ is a function $g: V \rightarrow \mathbb{R}$ defined in a neighbourhood $V$ of $p$ such that ...
1
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1answer
38 views

Closed form defines locally a function?

In my particular example, I have a volume form on a one-dimensional manifold, so i.e. this volume form $dx$ is closed. Now, I was wondering, does the integral function $f(y):=\int_{x_0}^{y} dx$ define ...
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3answers
50 views

The Euclidean Metric on $\mathbf R^3$ Induces an Index-Lowering Isomorphism $b:\mathfrak X(\mathbf R^3)\to \Omega^1(\mathbf R^3)$.

In Lee's Introduction to Smooth Manifolds, Second Edition, the line just before Equation 14.25 reads The Euclidean metric on $\mathbf R^3$ induces an index-lowering isomorphism $b:\mathfrak ...
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2answers
28 views

Closedness of a complex $1$-form defined by homogeneous functions

How can I prove this 1-form is a closed one within the specific subset $A\subset \mathbb C$ ? $$\omega=\frac{f(x,y)}{xf(x,y)+yg(x,y)}dx+\frac{g(x,y)}{xf(x,y)+yg(x,y)}dy$$ Where $f, g\in C^1$ are ...
0
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1answer
21 views

Notation for a projection of a differential form

Let $\omega = a_1 dx_1 + a_2 dx_2 + b_1 dy_1 + b_2 dy_2$. Is there any established notation to denote a mapping that "filters out" the $dy_i$-Terms? To be more precise, I invent my own one. Assume ...
4
votes
1answer
51 views

Can a volume form on a submanifold be extended to a parallel form in a neighbourhood?

Let $(M^{n+1},g)$ be a Riemannian manifold and let $\Sigma^n \hookrightarrow M$ be a smooth, closed, embedded submanifold. Let $\Omega$ be the volume form of $\Sigma$. It is well-known that a volume ...
3
votes
0answers
24 views

The lie bracket as a limit

Let $M$ be a manifold and $f_t$ be the monoparametric group of local transformations generated by a vector field $X$. Suppose $w$ is a $k$-form and $x_1,\dots,x_k$ are vector fields. How can we see ...
3
votes
1answer
46 views

Why does a differential form represent a vector field?

I'm trying to learn the Divergence/Stoke's theorem and I can't wrap my head around the meaning of a differential form in this context. What does it mean that a differential form represents a vector ...
4
votes
1answer
63 views

Computing the sheaf of 1-forms on a toric variety

Consider projective space $P^{2}$ and its corresponding fan. We have the affine opens defined by $U_{\sigma_{0}} = Spec(\mathbb{C}[x,y])$, $U_{\sigma_{1}} = Spec(\mathbb{C}[x^{-1},x^{-1}y])$ and ...
1
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0answers
56 views

Wedge product equality of 2n-forms in 2n+2 dimension

I have a question that seems to me clear but i couldn't prove it. I have a diffeomorphism between the cotangent bundle of the n-sphere and the complex quadric as $$T^*(S^n)\to (S^n)^{\mathbb{C}},\ ...
2
votes
1answer
37 views

Integration of $V$-valued differential form

When studying fibre bundles, connections and gauge theories it is usual to consider vector-valued differential forms, like the connection one-form, or it's pull back by a local trivialization known as ...
2
votes
1answer
42 views

Differential Forms on Surfaces. Show that $N\cdot (\nabla \times V)\eta=d\phi$ on $x(D)$.

Let $M$ be an orientable surface in $\Bbb R^3$ with a unit normal vector field $N$ and let $x: D\to M$ be a patch. Let $\eta$ be a differential 2-form on $x(D)$ defined by $\eta(x_u,x_v)=\pm\|x_u ...
3
votes
1answer
29 views

Exterior derivarive dependent only on point

For any one-form (a linear form on the tangent space of each point) we have its exterior derivative $d\omega$ which is a two-form defined by $d\omega(X,Y)=D_X(\omega(Y))-D_Y(\omega(X))-\omega([X,Y])$ ...
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1answer
40 views

How do you compute the pull-back of a complex differential (1,1)-form given its potential?

Let $F: X \to Y$ be a holomorphic map. Let $\omega$ be a complex differential $(1,1)$-form on $Y$, $\omega=\partial \bar \partial f$, where $f$ is a pluri-subharmonic function. How would one ...
2
votes
1answer
52 views

Symplectic Geometry of 2-sphere in stereographic projection

I am trying to put the symplectic form of the 2-sphere defined by $\omega_u(v,w) := \langle u,v\times w\rangle,$ where $u \in \mathbb{S}^2$ and $v, w \in T_u\mathbb{S}^2$ in stereographic coordinates ...
4
votes
2answers
194 views

Integrals of Pullbacks

This is a problem from Guillemin's Differential Topology: Suppose that $f_0, f_1: X \to Y$ are homotopic maps and that the compact boundaryless manifold $X$ has dimension $k$. Prove that for all ...
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0answers
20 views

why is $\omega^n \neq 0$ for a nondegenerate 2 form $\omega$. [duplicate]

Let $\omega$ be a nondegenerate alternating $2$-form on an $2n$-dimesional Vectorspace $V$, meaning that for all nonzero $v \in V$ the map $w \mapsto \omega(v,w)$ is not identically zero. Why is the ...
3
votes
2answers
30 views

Pushforward injective

Let $f : M \rightarrow N$ be a smooth surjective map between smooth manifolds. Now, consider a 2-form $\omega$ on $T_pN$. Does it now follow that the pullback satisfies? $f^* d \omega =0 \Rightarrow ...
3
votes
2answers
39 views

Exterior derivative of a form and $d(d\omega)=0$?

We know that in differential geometry, $d^2\omega=0$, where $\omega $ is a form and $d$ is the exterior derivative. However if this form happens to be the exterior derivative of another form ...
0
votes
1answer
33 views

(Co)Tangent bundle of Cone manifold

Given a Riemannian manifold $(M,\bar{g})$, we can construct the Riemannian cone manifold $(C(M), g )$ as follows. Topologically, $C(M)$ is $M \times \mathbb{R}_{>0}$. We equip this with the ...
2
votes
1answer
26 views

exact and closed differential forms

This exercise is taken from the Meyer-Hall-Offin book on Hamiltonian systems. Let $Q(p,q)$ and $P(p,q)$ be smooth functions defined on an open set in $\mathbb{R}^2$. Consider the four differential ...
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1answer
38 views

$e^{xy}dx \wedge dy$: determine the $1$-form that it induces on $S^1$ and check if the obtained $1$-form respects or not the induced orientation

Consider the $2$-form $e^{xy}dx \wedge dy$ on $\mathbb{R}^2$. Determine the $1$-form that it induces on $S^1$, viewed as the boundary of $B_2$. Check if the obtained $1$-form respects or ...
0
votes
1answer
52 views

Construct a $k$-form on $S^k$ with nonzero integral

How to construct a $k$-form on $S^k$ with nonzero integral ? I think this can be done by Bump function $\rho$ on $R^k$ and define $w = \rho\, dx_1\,dx_2 \cdots dx_k$ on $R^k$. Now pull it back by the ...
1
vote
1answer
46 views

Differential calculation in multiple variables function (cannot understand 2nd order differential form)

This question is somehow related to this question. Consider a multiple variables function $G(u, v) = \left(\matrix{x(u,v) &=& G_x(u,v)\\y(u,v) &=& G_y(u,v)\\z(u,v) &=& ...
3
votes
1answer
50 views

Proving that the pullback map commutes with the exterior derivative

I'm trying to prove that the pullback map $\phi^{\ast}$ induced by a map $\phi:M\rightarrow N$ commutes with the exterior derivative. Here is my attempt so far: Let $\omega\;\in\Omega^{r}(N)$ and let ...