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

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3
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1answer
64 views

Integration on $\mathbb{R}^n$ in terms of differential forms

One defines integration on a smooth manifold as follows: First define $\int_M \omega$ when $\omega$ is supported on a single coordinate chart by pulling back to $\mathbb{R}^n$ an integrating there, ...
3
votes
1answer
25 views

Global n-form on Calabi-Yau

I am now reading these lectures by Stefan Vandoren on complex geometry. Everything is fine in general, hiwever I am confused with how he defines 1-form on a Calabi-Yau 1-fold (or 2-form on CY$_2$). ...
3
votes
1answer
75 views

de Rham cohomology of $\mathbb{R}P^n$ via action by $SO(m+1)$

In lecture, my teacher proved the theorem that given a smooth $G$-action by a compact, connected Lie group on a manifold $M$, the de Rham cohomology of the $G$-invariant differential forms $H^p_G(M)$ ...
2
votes
1answer
131 views

Pullback distributes over wedge product

I'm looking to prove that the pullback of a smooth function distributes over wedge product, i.e. $$\varphi^*(\omega \wedge \eta) = \varphi^* \omega \wedge \varphi^* \eta. $$ Here the question is ...
2
votes
1answer
45 views

Cohomology of volume forms

If g and h are Riemannian metrics on the same manifold, say both of volume 1, then it follows (I guess from Poincaré duality) that their volume forms dvol_g and dvol_h are cohomologous. Question: is ...
2
votes
1answer
262 views

Differential forms and a chain rule

Let $U$ be a Riemann surface and let $z:U\longrightarrow B(0,1)$ be a diffeomorphism, where $B(0,1)$ is the open unit disc in $\mathbf{C}$. So $z$ is a coordinate around $P=z^{-1}(0)$. Let $Q\in U$ ...
1
vote
1answer
72 views

A question about differential forms

I have checked Rudin's proof about Poincaré Lemma (Principles of Mathematical Analysis) and it seems to have a mistake. Through Google, I found another guy who has noted such error. More details here: ...
1
vote
1answer
68 views

Compute the differential of a form

From Munkres "Analysis on Manifolds" Consider the form $ \omega = xydx + 3dy -yzdz $. Check by direct computation that $ d(d\omega) = 0 $. Can someone show me how to do it, because I don't seem to be ...
0
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1answer
16 views

2-form corresponding to a contravariant vector and pseudo-forms

In Frankel's book he writes that in $R^{3}$ with cartesian coordinates, you can always associate to a vector $\vec{v}$ a 1-form $v^{1}dx^{1}+v^{2}dx^{2}+v^{3}dx^{3}$ and a two form $v^{1}dx^{2}\wedge ...
0
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1answer
36 views

Harmonic functions and polar differential forms

Given a harmonic function $u$, its differential and conjugate differential are $$du = \frac{\partial u}{\partial x}dx + \frac{\partial u}{\partial y}dy,\qquad ^{*}du = -\frac{\partial u}{\partial y}dx ...
0
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1answer
40 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
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1answer
61 views

How to prove matrix differential rules using differential forms rules

I am learning differential forms on my own through lectures on youtube and one of the things that I am attempting to do is to check if I can derive matrix differential rules using the rules for ...
0
votes
1answer
39 views

DPEs system which I cannot seem to solve

Consider the following DPE system: $$\left\{ \begin{array}{rcl} g_x - f_y& = &1-x^2 \\ h_x - f_z &= &3x^2 \\ h_y - g_z &=& -1 \ \end{array}\right .$$ This comes from trying ...
1
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0answers
45 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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0answers
100 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 ...
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0answers
55 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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0answers
53 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 ...
1
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0answers
60 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 $\mathbb R^3$ ...
1
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0answers
31 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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0answers
24 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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0answers
59 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 ...
1
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0answers
44 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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0answers
51 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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0answers
80 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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0answers
47 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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0answers
95 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 ...
1
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0answers
64 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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0answers
48 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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0answers
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 ...
1
vote
0answers
61 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 ...
0
votes
0answers
30 views

$(1,0)$-forms on a complex manifold

I'm reading in a paper: "Let $\theta$ be a $(1,0)$-form with respect to $I$ ($I \theta = -i \theta$)". Here $I$ is the almost complex structure. Any idea why it says $I \theta = -i \theta$ and not $I ...
0
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0answers
22 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 ...
0
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0answers
47 views

Differential forms and minor expansion, question about notation.

There are lectures by Theodore Shifrin on differential forms, and sadly one video ends suddendly where he explains some notation. I try to formulate it in my own words: When k=n, we have ...
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0answers
53 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 ...
0
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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 ...
0
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0answers
27 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 ...
0
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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: ...
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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 ...
0
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0answers
33 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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0answers
25 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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0answers
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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0answers
11 views

How should I work with antisymmetric indices on forms?

I've always tended to be confused by antisymmetrized indices on differential forms, and the only way I can really work with them is by writing out all the terms - which is just not practical beyond 3 ...
0
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0answers
63 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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0answers
66 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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0answers
156 views

Hodge dual exterior derivative

The introduction of the Hodge dual to the structure of the cotangent space requires the reference to a specific basis or an inner product. I was wondering however, if the composition of hodge dual and ...
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0answers
185 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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0answers
60 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 $$ ...
0
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0answers
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 ...
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0answers
41 views

Taking the inverse of this object

Let $\{q_i,p_i, i=1,...,n\}$ be coordinate and their conjugate momentum. Let $\xi_k, k=1,...,2n$ be generalized coordinates which equal to $\{q_i,p_i, i=1,...,n\}$ Suppose the matrix ...
0
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0answers
344 views

pullback of 1-differential form

Given a $ \phi_{0} \in \mathbb R$. Consider the function $ \displaystyle{\phi (t) = \phi_{0} + \int_0^t (ab'-a'b)(u) \text{du}}$ where $ a^2(t) + b^2 (t)=1$ Prove that: (i) $ \phi(0)= \phi_{0}$ ...