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

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35 views

Why is the derivative Df(p) defined to $ \in \Lambda^1 (\mathbb{R}^n) $, or how is it a 1-form?

I know that obviously the differential operator D would be a differential form through the word differential. But in Spivak Calculus on Manifolds he defines a k-form w $ \in \Lambda^k ...
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1answer
51 views

Equality in de Rham cohomology

Let $U_1,U_2,...,U_r$ be open sets in $\mathbb{R}^n$ such that $U_i\cap U_j =\emptyset$ for all $i \neq j$. Then prove, $H^k_{dR}(\bigcup_{i=1}^{r} U_i)=\bigoplus_{i=1}^{r} H^k_{dR} (U_i)$
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1answer
53 views

De Rham cohomology group

We know $m$-th de Rham cohomology group on $U$ is defined to be, $H^{m}_{dR}(U)=ker(d^m)/im(d^{m-1})$ where $d^m:\Omega^m(U)\to \Omega^{m+1}(U)$'s are usual exterior derivative maps. Now its saying ...
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0answers
104 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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1answer
109 views

Calculating the pullback of a $2$-form

I have a $2$-form given by $\omega = dx \wedge dp + dy \wedge dq$ and a map $i : (u,v) \mapsto (u,v,f_u,-f_v)$ for a general smooth map $f : (u,v) \mapsto f(u,v)$. I want to calculate the pullback of ...
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0answers
53 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 ...
4
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3answers
449 views

Interior product between differential forms and vector fields

I don't understand what is meant when someone writes that forms (or form fields) "eat" vectors (or vector fields). For example when I have a one form field ω=3dx+5dy+3xdz and a vector field ...
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0answers
48 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 ...
3
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1answer
33 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$_2$). ...
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2answers
906 views

Relationship Between Differential Forms and Vector Fields

I am trying reach an understanding of precisely how the space of differential forms is related to the space of vector fields. These are the definitions that I understand and am using for these ...
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1answer
74 views

If $M$ is a compact manifold what does $\partial M$ mean?

In the generalized form of stokes theorem it states that the integral of the $k+1$ differential form of an operator over a compact manifold $M$ is equivalent to the integral of the $k$ differential ...
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3answers
494 views

Why are differential forms more important than symmetric tensors?

In differential geometry, differential forms are totally anti-symmetric tensors and play an important role. I am led to wonder why do we not study totally symmetric tensors as much as forms. What ...
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0answers
114 views

Differential forms and determinants

2-forms are defined as $du^{j} \wedge du^{k}(v,w) = v^{j}w^{k}-v^{k}w^{j} = \begin{vmatrix} du^{j}(v) & du^{j}(w) \\ du^{k}(v) & du^{k}(w) \end{vmatrix}$ But what if I have two concret ...
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0answers
100 views

Second fundamental form of a graph of a function using frame fields

I am trying to recover the second fundamental form $II$ for the graph $\Gamma(f)$ of a smooth real function $f$ in the Euclidean space $\mathbb{R}^3$ using first order frame fields. I have worked with ...
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3answers
1k views

Inducing orientations on boundary manifolds

Given a $k$-manifold $M$, such that $\partial M$ is a $(k-1)$-manifold, there is a standard way in which $\partial M$ inherits the orientation of $M$. So if $M$ is oriented by the form field $\omega$, ...
3
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1answer
119 views

Differential Forms on submanifolds

Say I take an embedded submanifold of $\Bbb R^n$, like the sphere. Any differential form on $\Bbb R^n$ can be restricted to the sphere. My question is this: is any differential form on the sphere (or ...
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2answers
165 views

Is a 1-form locally expressible as $dx$?

Given a 1-form $\alpha$ which is non-zero at every point of a manifold $M^n$, is it true that locally I can express the form as $dx$? (that is around each point there is a coordinate neighborhood such ...
3
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1answer
67 views

Conceptual Question on different representations of Hyperplanes, Higher Standpoint, Coordinate-free

In a vector space $V$ over some field $F$ a hyperplane is the kernel of some linear transformation $T : V \to F$, i.e. the kernel of an element of the dual space (this could be taken as the definition ...
3
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1answer
78 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, ...
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5answers
2k views

How to calculate gradient of $x^TAx$

I am watching the following video lecture: https://www.youtube.com/watch?v=G_p4QJrjdOw In there, he talks about calculating gradient of $ x^{T}Ax $ and he does that using the concept of exterior ...
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0answers
108 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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0answers
41 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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1answer
159 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 ...
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0answers
65 views

An error applying the Stokes theorem?

M is the surface $z=x^2+y^2$ with standard orientation for $x^2+y^2\leq 1$ and $\varphi = 4x^2ydy+z^2dz$ I'd like to verify that $\int_Md\varphi=\int_{\partial M}\varphi$, which I did, but ...
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0answers
52 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 ...
2
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1answer
78 views

How to make an ideal generated by differential forms into a differential ideal?

Let $M=\mathbb{R}^4$ with standard coordinates $x_1,x_2,x_3,x_4$. Let $\alpha=x_2dx_1+x_3dx_3+dx_4$ and $\beta=2dx_2+x_1^2dx_3+x_1dx_4$ How to find a 1-form $\gamma$ such that the ideal generated ...
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2answers
106 views

understanding differential form from do carmo

I am recently read the differential form book of do carmo and found the following Here I can not understand what is $(dx_i)_p$ here?Is it the derivative map of $x_i$. And I also can not ...
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2answers
64 views

Integrating a differential form inside a cylinder

Let S be cylinder given by $x^2+y^2=1$ between $z=1$ and $z=3.$ For $\varphi=e^xdx\wedge dy+ ydz\wedge dx+xdy\wedge dz$ find $\int_S\varphi$. I managed to finish the problem, but I'm getting ...
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1answer
98 views

How to define integration over the boundary of a curve?

When learning about Stokes' theorem ($\int_{\partial \Omega} \omega=\int_{\Omega} \mathrm d \omega$), we are told that it is just a generalization of the 2nd Fundamental Theorem of Calculus $(\int_a^b ...
5
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1answer
102 views

Can differential forms be generalized to (separable) Banach spaces?

This thought occurred to me earlier and I'm surprised I hadn't considered it previously. I get the feeling that no meaningful generalization can occur in a non-separable Banach space but on the ...
6
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1answer
189 views

Evaluating differential forms.

Can someone please check my work? It's an exercise from Barret O'Neill's Elementary Differential Geometry. I want to be really sure that my understanding of this is right. I see that the forms ...
7
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2answers
192 views

Interpretation of $p$-forms

Let $M$ be a smooth manifold, let $C^{\infty}(M)$ be set of all smooth functions from $M$ to $\mathbb R$ and let $Vec(M)$ denote the set of all vector fields on $M$. A $1$-form on $M$ is a ...
0
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1answer
44 views

Let $\phi $ be an exterior $k$-form, where $k$ is an odd integer. Show that $\phi \wedge \phi =0 $

Let $\phi $ be an exterior $k$-form, where $k$ is an odd integer. Show that $\phi \wedge \phi =0 $ We know that If $\phi$ is a $k$ form and $\pi $ is a $l$ form then $\phi \wedge \pi = (-1)^{kl} ...
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2answers
263 views

Pullback of a Volume Form Under a Diffeomorphism.

I have an exercise here, which I have no idea how to do. Problem: Let $ U $ and $ V $ be open sets in $ \mathbb{R}^{n} $ and $ f: U \to V $ an orientation-preserving diffeomorphism. Then show that ...
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1answer
47 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 ...
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1answer
65 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 ...
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1answer
22 views

linear form $\phi : \mathbb{R^3} \times \mathbb{R^3} \to \mathbb{R}$ is alternate if and only if $\phi(v,v)=0$,for all $v \in \mathbb{R^3}$

Prove that a bilinear form $\phi : \mathbb{R^3} \times \mathbb{R^3} \to \mathbb{R}$ is alternate if and only if $\phi(v,v)=0$,for all $v \in \mathbb{R^3}$. My thought:- If $\phi : \mathbb{R^3} ...
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1answer
57 views

Flux Integral - where did I go wrong?

S is the graph $z=25-(x^2+y^2)$ over the disk $x^2+y^2\leq 9$ and $\varphi = z^2dx\wedge dy$. Find $\int_S \varphi$. According to the book the answer is $3843\pi$, but the answer I got is ...
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1answer
34 views

What line does ω project vectors onto?

I have just started learn differential form from the bachman book (page 29)and I found some difficulties in the following problem in 2nd part. Let $ω(<dx,dy>) = −dx + 4dy$. 1. Compute $ω(<1, ...
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0answers
85 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 ...
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2answers
75 views

The property of an integral manifold of differential form

Let's define the $1$-form $\eta$ on $\mathbb{R}^3$ by formula: $$\eta = A(x,y,z)\;\mathrm{d}x + B(x,y,z)\;\mathrm{d}y + \mathrm{d}z \text{.}$$ And let's assume that $$\mathrm{d}\eta \wedge \eta = ...
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0answers
47 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 ...
2
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2answers
102 views

Wedge product and determinants

I am attempting to self-study differential forms this summer, but I ran into this definition for the wedge product in my book, and it doesn't make any sense to me. Note that $\Bbb R^3_p$ is the ...
2
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0answers
82 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 ...
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1answer
108 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 $d$ ...
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0answers
32 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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1answer
37 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 ...
3
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1answer
117 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 ...
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0answers
91 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 ...
3
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2answers
66 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 $$\gamma=\left\{ (x,y,z):3x+y+3z=3,x\ge0,y\ge0,z\ge0\right\} $$ ...