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

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2
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1answer
221 views

Formal finite sum for integration on k-chains

This question is related to the integration of differential forms on chains, as exposed in the document at: http://www.math.upenn.edu/~ryblair/Math%20600/papers/Lec17.pdf . The author gives the ...
2
votes
1answer
184 views

Finding the winding number of a curve

Let $\gamma(t)=(r \cos t,r \sin t)$, for some $r>0$, and let $\Gamma$ be a $C^2$-curve in $\mathbb R^2-\{\bf 0\} $, with parameter interval $[0,2 \pi]$, with $\Gamma(0)=\Gamma(2 \pi)$, such that ...
2
votes
1answer
123 views

Can exterior calculus be used to solve differential equations?

I know one can formulate partial differential equations in terms of exterior derivatives, etc but I have been wondering for a while now how one might be able to use that formalism to extrapolate ...
2
votes
1answer
311 views

Taking the exterior derivative of a 0-form

I'm attempting to show that $dg(\vec{x})=\alpha$, where $\alpha=\Sigma^n_{i=1}f_idx_i$ and $g(\vec x)={1\over{p+1}}\Sigma_{i=1}^nx_if_i(\vec x)$...and $d$ is the exterior derivative. The $f_i$ are ...
2
votes
1answer
105 views

How do you calculate an exterior derivative on forms in $\mathbb{R}^3$?

If we have a form, say, $\omega = f(x,y,z) \, dx + g(x,y,z) \, dy + h(x,y,z) \, dz$, what is the formula for the exterior derivative $d \omega$?
2
votes
1answer
36 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 ...
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 ...
2
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1answer
43 views

An Application of Stokes's Theorem

Let $D^2=\{(x,y)\in \mathbf R^2: x^2+y^2\leq 1\}$ be the unit disc in $\mathbf R^2$, and $D^3=\{(x,y,z)\in \mathbf R^3: x^2+y^2+y^2\leq 1\}$ be the unit disc in $\mathbf R^3$. Let $i_{\pm}:D^2\to ...
2
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1answer
31 views

Does the dual basis to some basis of $T^*_pM$ looks localy like a coordinate chart?

Let $M$ be a manifold and let $\{\alpha_k\}$ be a set of $1$- forms s.t. $\{\alpha_k(p)\}$ forms a basis for $T^*_pM$. Let $(x,U)$ be a chart based in $p$ and denote $\partial_i := ...
2
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1answer
43 views

A question about a part of the proof of the existence of the exterior derivate.

Let $M$ a differentiable manifold, $(U;x_1,\ldots,x_n)$ a coordinated system and $\omega$ a differential form which domain intersects $U$. In $U \cap \operatorname{Dom} \omega$, $\omega$ can be ...
2
votes
1answer
104 views

Problem integrating in problem using the Poincaré Lemma

a) It is easy to show that $d\beta=0$. b) $\begin{align}\hat{\mathbb{X}}_t &= \left(\frac{\partial}{\partial t}\hat{\Phi}_t \right) \hat{\Phi}_t^{-1} \\ &= \left(\frac{\partial}{\partial ...
2
votes
1answer
68 views

How to derive the standard symplectic form on a 2-sphere in cylindrical polar coordinates?

I have already know that at a point $p\in S^2$ with $u,v$ the tangent vectors at $p$. Then the standard symplectic form is $\omega_p(u,v) :=\langle p,\,u\times v\rangle$ where $u\times v$ is the outer ...
2
votes
2answers
59 views

Exercise on left invariant 2-forms

I'm trying to solve a problem about the Lie group of transformation over $\mathbb{R}$: \begin{equation} x\mapsto ax+b, \end{equation} where $a,b\in\mathbb{R}, a\neq 0.$ I'm asked to find the space of ...
2
votes
2answers
104 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
votes
1answer
110 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 ...
2
votes
3answers
130 views

How to deduce this formula using differential forms?

There's a formula from vector calculus that seems terrible to deduce. This formula is: $$\nabla\times (A\times B)=(B\cdot\nabla )A-(A\cdot \nabla)B+A (\nabla\cdot B)-B(\nabla\cdot A)$$ Deducing it ...
2
votes
1answer
64 views

Proving Borsuk-Ulam with Stokes

What is the easiest way to deduce the Borsuk-Ulam theorem in the case $n=2$ by using integration on manifolds and Stokes theorem? So I want to prove the following: Given a map $f\colon S^2\rightarrow ...
2
votes
2answers
76 views

If $\omega$ is closed on $\mathbb R ^2 - 0$ and $\text d \omega =0$, then $\omega = \text d g+ \lambda \text d \theta$.

I'm trying to solve problem 4-30 from “Calculus on manifolds”, which is the one in the title, where $$\text d \theta = -\dfrac{y}{x^2+y^2}\text d x+\dfrac{x}{x^2+y^2}\text d y.$$ I think I'm on the ...
2
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1answer
75 views

Product integration of differential forms

Let $\alpha, \beta$ two forms continuous with compact supports and maximum degree on surfaces oriented $M, N $ respectively. consider $\pi_M:M\times N \rightarrow M$ and $\pi_N:M\times N \rightarrow ...
2
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1answer
132 views

Relation between de Rham cohomology and integration

This question is a follow-up to When does a null integral implies that a form is exact? . As mentionned in the selected answer, given certain conditions it is possible to find an isomorphism between ...
2
votes
1answer
92 views

Question on Differential forms

Let $\Omega\subseteq \mathbb{R}^2$ be an open set and $\omega=\omega^1dx_1+\omega^2dx_2$ an $1$-form on $\Omega$ and $$L=\omega^2\frac{\partial}{\partial x_1}-\omega_1\frac{\partial}{\partial ...
2
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1answer
268 views

Concept of integration to differential form

How to integrate differential form actually. As far as I know, a differential form is a multilinear function mapping from a vector space to a real number. Let's take $\int_c fdx+gdy$ as an example. It ...
2
votes
1answer
94 views

Is $\zeta=\frac{x dy \wedge dz+y dz \wedge dx+z dx \wedge dy}{r^3}$ exact in the complement of every line through the origin?

$r=\sqrt{x^2+y^2+z^2}$ of course. If the line is the $z$ axis, it is given in the book (Rudin) that $\zeta=d \left( -\dfrac{z}{r} \dfrac{xdy-ydx}{x^2+y^2} \right)$ I've managed to figure out 2 ...
2
votes
1answer
146 views

Integrals of Differential Forms

I am working out of Munkres Analysis on Manifolds and I see that he claims for $\eta = f dx_1 \wedge \ldots \wedge dx_k$. $$ \int_A\eta = \int_{x \in A} \eta(x)\big((x;a_1) ,\ldots, (x;a_k) \big)$$ ...
2
votes
1answer
711 views

non-vanishing k-form on a k-manifold in $\mathbb{R}^n$ implies orientability

I want to know how to prove the theorem: If M is a k-manifold in $\mathbb{R}^n$, then it is orientable if and only if there is a volume form defined globally on M. I'm currently stuck at this step: ...
2
votes
1answer
491 views

Leibniz rule for exterior derivative of a contraction

If I have a contraction of a vector field with a 1-form valued 2-form, what would be the appropiate product rule? $$d_{\left[a\right.} \left(P_{[bc]i} v^i \right)_{\left. \right]} = \, ?$$ This ...
2
votes
1answer
500 views

property to be exact a 1-form on $\mathbb R^2 -\{(0,0)\}$

(a) Let $\omega$ a $1-$form defined on the open set $ U \subset \mathbb R ^n$ and $ c:[a,b] \to U$ a $ C^1 -$differentiable curve such that $ |\omega (c(t))| \leq M \quad \forall t \in [a,b]$ Prove ...
2
votes
1answer
227 views

Differential forms on a $S^1$-manifold

I am reading about differential forms on manifolds with group actions and there is an 'obvious' formula which I don't quite understand. Let $X$ be a manifold with a smooth circle action, that is a ...
2
votes
2answers
28 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 ...
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 ...
2
votes
1answer
51 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 ...
2
votes
1answer
37 views

Pullback Solid Angle, Stereographic projection

I have an issue with a differential geometry task. Given is the solid angle form: $$\omega = \frac{\epsilon_{ijk} x^i dx^j \wedge dx^k}{2 [ (x^1)^2 + (x^2)^2 +(x^3)^2]^{3/2}}$$ The aim of the task ...
2
votes
1answer
71 views

An interpretation of $ \frac{\partial^{2}}{\partial x^{2}} $.

$ \left( \dfrac{\partial}{\partial x} \right)_{p} $ is both an element of the tangent space $ {T_{p}}(M) $ and a linear functional on $ {C^{1}}(M) $, while $ (\mathrm{d}{x})_{p} $ is an element of the ...
2
votes
1answer
62 views

Tangent space and differential forms of quotient groups

I have difficulty understanding the following argument because it seems that many details are swept under the rug and I am looking for a detailed rigorous exposition on this (I'll try to make clear ...
2
votes
1answer
63 views

Relationship between differential forms and coordinates

For the purpose of this question let us restrict our considerations to smooth $3$-manifolds. So the manifold $M$ we consider here is endowed with smooth coordinate charts $(x,y,z)$. What I have ...
2
votes
2answers
61 views

Given a nowhere zero vector field $Z$, does there exist a one-form $\gamma$ such that $\gamma(Z) = 1$?

Take $M$ a smooth manifold, and $Z$ a vector field on $M$ such that $Z(p)\neq0$ for all $p\in M$. Is there a one form $\gamma \in \Omega^1(M)$ such that $\gamma(Z)=1$? I started to work locally, but ...
2
votes
1answer
78 views

Constructing functions such that integral along any closed curve is non-zero

Consider smooth maps $f: \mathbb R^2 \setminus \{(0,0)\} \to \mathbb R$. How can I construct such an $f$ with the property that $$ \oint_C f \neq 0$$ for any closed curve $C$ around the origin? ...
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votes
2answers
71 views

When are the exterior derivative and contraction of forms inverses?

I am trying to get a better feel for both the exterior derivative of a form and the contraction of a form by a vector field $X$. Basically, when are these inverses? If I have a one-form $\omega$ and ...
2
votes
2answers
101 views

Differential forms on $S^1$

I'm reading this old question and there are some things I don't understand. For example, why in the case of $S^1$ can every $1$-form be written in the form $f(\theta)d\theta=c d\theta+dg(\theta)$ ...
2
votes
1answer
53 views

exterior product of forms is exact.

I don't know what to do to prove the following statement: Let $U \subset \mathbb R^n$ be an open set and let $\alpha$ be a $k$-form on $U$ and $\beta$ be an $l$-form on $U$. Suppose both $\alpha, ...
2
votes
1answer
96 views

Identities for differential forms and vectorfields (reference request)

Recently I found the slides of a talk of J. E. Marsden, (Differential Forms and Stokes' Theorem). These slides introduce the required objects and summarize the basics of the corresponding theory. In ...
2
votes
1answer
82 views

Pullback of a form under the retraction $r\colon \mathbb{R}^n\setminus\{0\}\to S^{n-1}$.

The following is from Spivak's DG Lemma 7 in Chapter 8, but I'm muddled in a computation. Define two $(n-1)$-forms on $\mathbb{R}^n\setminus\{0\}$ by $$ ...
2
votes
1answer
45 views

differential form identity and permutations

If $t^1,...,t^k$ are the coordinates of a k-cube. Then apparently $$dt^{\sigma(1)} \wedge \ldots \wedge dt^{\sigma(k)}= (\operatorname{sgn} (\sigma)) dt^1 \wedge dt^k $$ I cannot see how this ...
2
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1answer
65 views

Differential Form Over $S^2$

I was checking problems on differential forms and I found the following one. Consider the sphere $S^2 \subseteq R^3$ and the map $\omega_p : T_pS^2 \times T_pS^2 \rightarrow \mathbb{R}$ given by ...
2
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1answer
81 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 ...
2
votes
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 ...
2
votes
2answers
131 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 $d$ ...
2
votes
1answer
26 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 ...
2
votes
1answer
86 views

Why generalize vector calculus with $k$-forms instead of $k$-vectors?

The motivation usually given to differential forms is that they generalize vector calculus nicely. That's true, but there are also $k$-vectors, i.e., objects from $\Lambda^k(V)$ instead of ...
2
votes
2answers
262 views

Integration of a 2-form

$\textit{What is}$ $\int_C{\omega}$ $\textit{where}$ $\omega=\frac{dx \wedge dy}{x^2+y^2}$ $\textit{and}$ $C(t_1,t_2)=(t_1+1)(\cos(2\pi t_2),\sin(2\pi t_2)) : I_2 \rightarrow \mathbb{R}^2 - ...