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

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152 views

“Ordinary” and “polar” vector fields in Euclidean $3$-space

In his book Differential Forms with Applications to the Physical Sciences, on pages 19--20, Harley Flanders writes: "a one-form $$ \omega = P\,dx+Q\,dy+R\,dz$$ may be identified with an ...
3
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3answers
547 views

When does a null integral implies that a form is exact?

It is trivial to prove that the integration of a $(n-1)$ exact form on the boundary of a $n$-manifold is 0. What about the contraposative ? If the integration of a $(n-1)$-form on the boundary of a ...
1
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1answer
102 views

symplectic strucutre

Suppose $\omega$ is symplectic structure on $\mathbb R^n$. Let $\omega_0:=\omega|_{x=0}$. Let $\overline{\omega}= \omega_0-\omega$ and for $t\in[0,1]; \omega_t:= \omega+ t\overline{\omega}$. How ...
3
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3answers
379 views

Exterior derivative of a complicated differential form

Let $\omega$ be a $2$-form on $\mathbb{R}^3\setminus\{0\}$ defined by $$ \omega = \frac{x\,dy\wedge dz+y\,dz\wedge dx +z\,dx\wedge dy}{(x^2+y^2+z^2)^{\frac{3}{2}}} $$ Show that $\omega$ is closed but ...
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1answer
70 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 ...
8
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1answer
225 views

Maurer-Cartan 1-form

Can anyone help me with the following? Let $\rho$ be the right-invariant Maurer-Cartan 1-form $$\rho = dg\ g^{-1}$$ I want to show that the MC equation $$d\rho - \rho \wedge\rho = 0$$ holds. So ...
3
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0answers
136 views

Intuitive meaning of $L_X\omega$

Suppose $\alpha$ is a one form and $\omega$ is two form such that $$\alpha= \omega(X,.)\text{ for some } X$$ Then what does intutive meaning of the following expression $$L_X\omega= \omega$$
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1answer
609 views

Good intro to differential forms

I am looking for an intro book to learn about diff forms, maybe undergrad. Reading sentences like "Let M be a smooth manifold. A differential form of degree k is a smooth section of the kth exterior ...
8
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1answer
344 views

Visualizing Exterior Derivative

How do you visualize the exterior derivative of differential forms? I imagine differential forms to be some sort of (oriented) line segments, areas, volumes etc. That is if I imagine a two-form, I ...
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3answers
634 views

What is the motivation for differential forms?

I am that point in my mathematical career where I am learning differential forms. I am reading from M.Spivak's Calculus on Manifolds. So far I have gone over the tensor and wedge products and their ...
3
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3answers
111 views

Inner product of De Rham cohomology classes

Is there a well-defined inner product between cohomology classes? In particular, is it possible to extend the Hodge inner product? If I try, I obtain this: $$\int *(\omega + d\lambda)\wedge (\sigma + ...
3
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1answer
128 views

What do we mean when we say a differential form “descends to the quotient”?

Let $S$ be a surface and let $f:S\rightarrow S$ be a diffeomorphism. We define the mapping torus $M_f$ of the pair $(S,f)$ to be the quotient $$(S\times I) /\sim \quad \text{ where } \ (1,x) \sim ...
4
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1answer
303 views

Baby Rudin, Chapter 10, Problem 23 (d) - Differential forms.

In problems 21 and 22, Rudin defines the differential forms $\eta=\dfrac{xdy-ydx}{x^2+y^2}$ and $\zeta=\dfrac{x dy \wedge dz+ydz \wedge dx+z dx \wedge dy}{r^3}$ and the reader is asked to prove ...
3
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1answer
249 views

Lie derivative along time-dependent vector fields

In "Lectures on Symplectic Geometry" by A. C. da Silva (http://www.math.ist.utl.pt/~acannas/Books/lsg.pdf) the author gives the following definition: $$ \mathcal{L}_{v_t} := \frac{\mathrm d }{\mathrm ...
3
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2answers
636 views

Chain rule and differential forms

It is easy to show that the differential forms of order $1$ obeys a form of chain rule. To be precise, $d(f(g(x)) = f^\prime(x) d(g(x))$. This can be for example proved by fixing a co-ordinate basis ...
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3answers
810 views

Geometric understanding of differential forms.

I would like to understand differential forms more intuitively. I have yet to find a book which explains how the use of the exterior product in differential forms ties into the geometrical ...
3
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1answer
140 views

De Rham cohomology notation

According to http://en.wikipedia.org/wiki/De_Rham_cohomology, one defines the $k$-th de Rham cohomology group $H^{k}_{\mathrm{dR}}(M)$ to be the set of equivalence classes, that is, the set ...
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1answer
181 views

Existence of a left-invariant $n$-form on a Lie group of dimension $n$

This Do Carmo, Riemannian Geometry, Chapter 1, Exercise 7: Show that there exists a left invariant differential $n$-form $\omega$ on $G$ ($G$ is a compact connected lie group and $\dim G=n$). ...
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1answer
88 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 ...
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1answer
28 views

Question on 2-chain on $\mathbb{R}^3$

Let $\gamma:[0,1]\to\mathbb{R}^3\setminus\{0\}$ be a simplex, with $\gamma(0)=\gamma(1)$. How can I show that exists a $2$-chain $\sigma$ on $\mathbb{R}^3\setminus\{0\}$ such that ...
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1answer
216 views

Locally exact differential in a disk is exact

I'm reading through Ahlfors' Complex Analysis text for self study, and I found difficulty with a proof. In chapter 4 he defines a locally exact differential as a differential who is exact in some ...
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0answers
53 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 ...
3
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2answers
131 views

Prove that $\frac{1}{2\pi}\frac{xdy-ydx}{x^2+y^2}$ is closed

I would like to prove that $\alpha = \frac{1}{2\pi} \frac{xdy-ydx}{x^2+y^2}$ is a closed differential form on $\mathbb{R}^2-\{0\}$ . However when I apply the external derivative to this expression ...
2
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0answers
169 views

Exact and Closed forms on Manifolds with Boundary

Let $\bar{M}$ be a manifold with boundary and let $M$ be its interior. Is this statement correct? A smooth k-form $\alpha$ on $\bar{M}$ is closed (exact) if and only if its restriction to $M$, i.e. ...
2
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2answers
352 views

Hodge dual on orthonormal basis: two inconsistent answers

I'm trying to learn differential geometry using Göckeler & Schücker's book and I have some problems with the hodge star. As an example, say we have two orthonormal bases $e^i$ and ...
0
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1answer
244 views

Clarification about differential forms in polar coordinates

In my course about differential forms, we define 1-forms as follows: If $(e_1,..,e_m)$ is the standard basis of $\mathbb{R}^m$ and $\sigma$ a chart around $p\in M$ on the m-dimensional manifold $M$, ...
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1answer
111 views

A differential form to compute the k-volume of a k-parallelogram in n dimensions

Computing the k-volume of a k-parallelogram (i.e. a parallelogram spanned by k n-dimensional vectors) in n dimensions is straightforward: Let $P=[\overrightarrow{v_1},...,\overrightarrow{v_k}]$, then ...
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1answer
219 views

How general formulation of Stoke's theorem relate to Kelvin-Stokes theorem

I asked a similar question, but I realized the question is too vague and it's better to start a new one: We know that there are two usually used formulations of Stoke's theorem. One is vector ...
1
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1answer
253 views

Example of differential form usage of Stoke's theorem

There are many examples that show how Kelvin-Stokes theorem is used. But I would like to see an example that uses differential form usage of Stoke's theorem and is hard or impossible to solve by ...
4
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1answer
201 views

Why does $|dz|=-ir\frac{dz}{z}$ when $|z|=r$?

Sometimes I want to compute a line integral over some circle $|z|=r$, where I have $|dz|$ instead of $dz$ given to me. Reparametrizing with $z=re^{it}$, it follows that $dz=rie^{it}dt=izdt$. But I ...
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1answer
38 views

index $ n(F;D)$ is odd integer

Let $ F: U \subset \mathbb R ^2 \to \mathbb R^2$ be a map with $ F(x,y) = (f(x,y) , g(x,y))$ ,satisfies $F(-q)=-F(q) \quad \forall q \in D \subset U$ where $D$ is a closed disk with center the origin ...
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1answer
136 views

Tangent Vectors and Differential 1-forms.

I have this 1-form on $\mathbb R^3$ given by $\omega=dz+\frac{x}{2}dy-\frac{y}{2}dx$. If $p_0=(x_0,y_0,z_0)$ and $\vec v=(u_0,v_0,w_0)$, then find the set of tangent vectors $\vec v_{p_0}$ such that ...
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2answers
72 views

Equality involving exterior product..

suppose you have a differential form $\omega$ writting in local coordinates as $$\omega=\sum_{i=1}^ndx_i\wedge dy_i.$$ Can anyone help me showing the following equality: $$\omega^n=n!(dx_1\wedge ...
2
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2answers
69 views

Why is $\theta \not \in C^{\infty}(S^1)$?

Why is $\theta \not \in C^{\infty}(S^1)$? I know that since $\int_{S^1} d\theta = 2\pi$ then $d\theta$ is not exact. Thus since $d(\theta)=d\theta$, $\theta$ must not be $C^{\infty}$ but it seems ...
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1answer
195 views

How to prove $(0,1)$ form is not $\overline\partial$-exact

On a complex manifold, if we are dealing with the $d$ operator, there's a pretty easy way of showing some form is not $d$-exact, simply by integrating in a closed loop. If you can find a loop that is ...
4
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1answer
70 views

Smooth homotopic maps and closed forms..

does anyone have any idea for showing the following: Let $f_0, f_1:M\rightarrow N$ smooth homotopic maps between the manifolds $M$ and $N$. Suppose $M$ is compact with no boundary. Show that for every ...
3
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1answer
315 views

Curvature tensor of 2-sphere using exterior differential forms (tetrads)

$ds^2= r^2 (d\theta^2 + \sin^2{\theta}d\phi^2)$ The following is the tetrad basis $e^{\theta}=r d{\theta} \,\,\,\,\,\,\,\,\,\, e^{\phi}=r \sin{\theta} d{\phi}$ Hence, $de^{\theta}=0 ...
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4answers
556 views

Classic example of a non exact form

Let $\dfrac{xdy-ydx}{x^2+y^2}$ be a 1-form defined in $\mathbb{R}^2\backslash\{0\}$. Where can I find a detailed proof that it is not exact? I would prefer a proof that doesn't use results about ...
2
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1answer
73 views

Orienting curves with differential forms

Consider the circle given by the equation $x^2+y^2=1$. We can orient this curve by choosing the tangent vector field $(-y,x)^T$, which defines a direction. Supposedly we can do this with by taking an ...
6
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1answer
114 views

What is the relation of $\int f dx^1\wedge dx^2\wedge …\wedge dx^n=\int f dx^1…dx^n$

In a book "calculus on manifolds" it is defined that $\int f dx^1\wedge dx^2\wedge ...\wedge dx^n=\int f dx^1...dx^n$ but how it is possible the relate the integrand of a multilinear function ...
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2answers
134 views

Symplectic Form Preserved by Orthogonal Transformation

I'm trying to prove that the symplectic form $$\omega = d(\cos\theta) \wedge d\phi$$ is preserved by the action of $SO(3)$ on $S^2$ where $\phi$ and $\theta$ are spherical polars. Now $SO(3)$ simply ...
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0answers
86 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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2answers
194 views

Orientations on Manifold

This is a very basic definition of orientable and very basic example 20.5 however ı could not understand definition in an good way so ı want you to explain my green writing please :) and my example ...
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1answer
107 views

How to prove that this kind of differential form exists on an algebraic curve?

The following is a problem in Miranda's Algebraic Curves and Riemann Surfaces. Given any algebraic curve $X$ and a point $p \in X$, show that there is a meromorphic $1$-form $\omega$ on $X$ whose ...
0
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1answer
143 views

integral of closed differential form

This is my first course in differential forms so it might be a trivial question. If $\mu$ is a $n-1$-form on $n$-dim manifold $X$ the book uses that $$\int_X{d\mu}=0.$$ Is this expression valid for ...
2
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3answers
73 views

Whats the connection between formss and vector fields?

I heard someone talking about how vector fields are the kernels of forms. Can someone give me a detailed explanation about how this works? Thanks.
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2answers
49 views

Constructing a 2 form that does not vanish on a space from a one form that does

This may be a silly question... Given a nonzero one-form $\omega$ that vanishes on a subspace $W$ (with dimension larger than $2$), is it possible to find another one form $\phi$ such that ...
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1answer
190 views

Exterior Derivative question

In http://mathworld.wolfram.com/ExteriorDerivative.html, there is a section that starts from: Define the exterior derivative by $Dt ≡ \frac{\partial}{\partial x} \wedge t$ First of all, what ...
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2answers
50 views

Solve the i.v.p DE

Solve the i.v.p $y^{(4)}-y'''=0 , y(0)=0, y'(0)=0, y"(0)=0, y"'(0)=0$ Would I use the formula $a^{(1/n)}=R^{(1/n)}e^{(e^{i(alpha+2k(\pi))/n})}$
1
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1answer
65 views

$k$-forms on $\mathbb{R}^n$

Given an expression like $$ dx_1\wedge dx_2 \wedge dx_4 \left( \begin{bmatrix} 1\\2\\3\ \end{bmatrix} \ , \ \begin{bmatrix} 4\\5\\6 \end{bmatrix} \ , \ \begin{bmatrix} 7\\8\\9 \end{bmatrix} \right) \ ...