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

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8
votes
3answers
351 views

Where can I learn about complex differential forms?

So I'm a 3rd year grad student in number theory/modular forms/algebraic geometry, and I've worked with differential forms from an algebraic point of view without ever knowing what they really are. I'd ...
1
vote
1answer
142 views

elementary questions about differential forms

QUESTION 1: So I know that if $\omega$ is an alternating $p$-form for odd $p$ on some vector space $V$, then $\omega\wedge\omega = 0$. But...isn't the same true for any $p$? Ie, take for example $p ...
5
votes
1answer
171 views

Restriction of a differential form to an isotropic submanifold

From Analysis and Algebra on Differentiable Manifolds, first edition, exercise 2.6.4., question 1 (slightly edited for this post): Let $\vartheta$ be the canonical 1-form on the cotangent bundle $T^* ...
4
votes
1answer
176 views

What is meant by the kernel of a 2-form?

I'm given a 1-form $\alpha$ on $\mathbb{R}^n$, and asked to compute the kernel of $d\alpha$. Since $d\alpha$ is a 2-form on $\mathbb{R}^n$, it would eat a vector field to give a 1-form, or it would ...
1
vote
0answers
59 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 ...
5
votes
1answer
128 views

Diffeomorphic riemannian manifolds and volume forms

Maybe the question will be stupid, but I'm a beginner in riemannian geometry... We have two riemannian manifolds $(M,g)$, $(\overline M,\overline g)$ and a diffeomorphism $F:M\rightarrow\overline M$ ...
9
votes
1answer
464 views

Hodge Star Operator

I'm trying to understand the Hodge star operation, but have come across an impasse almost immediately. I have the definition $$(\star \omega)_{a_1\dots a_{n-p}}=\frac{1}{p!}\epsilon_{a_1\dots ...
4
votes
2answers
142 views

Function on $\mathbb{R}^{2}-\{0\}$.

Does there exist any compactly supported function $f= (f_1,f_2): \mathbb R^2-\{0\}\to \mathbb R^2$ such that $$\frac{\partial}{\partial x_2}f_1=\frac{\partial}{\partial x_1}f_2.$$ Also there does not ...
7
votes
1answer
315 views

Closed not exact form on $\mathbb{R}^n\setminus\{0\}$

I'd like to construct a closed but not exact $n-1$-form $\omega$ on $\mathbb{R}^n\setminus\{0\}$ in analogy to the winding form: $$\frac{x~dy-y~dx}{x^2+y^2}$$ I think something like ...
2
votes
1answer
75 views

Independence of coordinate charts in the definition of the order of a pole of a meromorphic 1-form on a Riemann surface

I am currently reading Forster. Let $X$ be a Riemann suface, and $Y$ an open subset of $X$. Assume we have a meromorphic 1-form $\omega \in \Omega(Y \setminus \{a\})$ with a pole at $a \in Y$. One ...
0
votes
0answers
94 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 ...
3
votes
2answers
124 views

A vanishing theorem for differential forms.

I am trying to prove that for an algebraic surface $X$ (under some extra assumptions that are probably not important) there the space $H^0(X,\Omega_X^1)$ is trivial, i.e. that there exist no globally ...
4
votes
1answer
63 views

Holomorphic 1-forms in $y^2-(z-a_1)\ldots(z-a_n)$

I know that the surface $y^2-(z-a_1)\ldots(z-a_n)$ is a Riemann Surface (that is the Riemann surface of $\sqrt{P(z)}$ with $P(z)=(z-a_1)\ldots(z-a_n)$) of genus $g$ and that ...
2
votes
0answers
61 views

Hodge decomposition on a manifold with a nontrivial connection

I am familiar with the notion of Hodge decomposition of an arbitrary differential form into an exact form, a co-exact form, and a harmonic form. Given a curved space with a connection, could you ...
2
votes
1answer
80 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 ...
4
votes
2answers
138 views

Can calculus of varations be formalised with exterior calculus?

I noticed that a calculus of variations problem is just an integral over a differential form. Therefore, I would think it would be possible to formulate the Euler-Lagrange equations using exterior ...
0
votes
1answer
57 views

Proof of the naturality of integration

I have a bit of a problem with the following identity: Suppose that $U, V \subset \mathbb{R}^n$, are two open sets. Let $x^1,...,x^n$ be a system of coordinates of $U$ and $y^1,...,y^n$ one on $V$. ...
1
vote
2answers
468 views

Pullback of differential form of degree 1

Good evening, In differential forms (in the proof of the naturality of the exterior derivative), I don't get why if $h\in \Lambda^0(U)$ and $f^*$ is the pullback then, $f^*dh=d(f^*h)$. I wrote ...
8
votes
3answers
164 views

Interesting question in differential geometry

Let $ \alpha $ be a closed $ 3 $-form on $ \mathbb{R}^{4} \setminus \{ 0 \} $. Let $ i: S^{3} \hookrightarrow \mathbb{R}^{4} $ be the canonical embedding of $ S^{3} $, and suppose that $ \Omega := ...
2
votes
1answer
111 views

Integrals of Differential Forms

I am working out of Munkres Analysis on Manifolds and I see that he claims for $\eta = f dx_1 \wedge \ldots \wedge dx_k$. $$ \int_A\eta = \int_{x \in A} \eta(x)\big((x;a_1) ,\ldots, (x;a_k) \big)$$ ...
5
votes
2answers
176 views

$\alpha\wedge\beta = 0$ for all $\beta$ implies $\alpha = 0$ without using the Hodge dual

Let $\alpha$ be a differential $k$-form on an orientable smooth $n$-dimensional manifold. If $\alpha\wedge\beta = 0$ for every differential $(n - k)$-form $\beta$, then $\alpha = 0$ because we can ...
2
votes
1answer
307 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: ...
6
votes
1answer
343 views

Differential Forms in Spivak vs Rudin

Can anyone give me the gist of the difference of the treatment of Stokes' Theorem in Spivak versus baby Rudin (chapter 4 in spivak, chapter 10 in rudin)? I need to do some problems from Rudin but ...
3
votes
2answers
245 views

Differential Forms Help

I have a background in Analysis, specifically with Baby Rudin. However, as many people note, Rudin does not do a very good job discussing differential forms. Could someone please refer me to an ...
1
vote
1answer
41 views

Expressing a differential form in terms of a scalar function

We can express every k-form in the form $ \omega(x) = \sum_Id_Idx_I $ where $ I$ is k-tuple and $ d_I$ is just some scalar function of x. That's entirely understandable for me. But while reading ...
5
votes
1answer
257 views

Line integral and integration of differential forms

The definition of integral of a $k$-form $\omega$ over a parametrized manifold $ Y_\alpha $ is $ \int_{Y_\alpha}\omega = \int_A\alpha^*\omega$ where $ \alpha\colon A \to R^n $. Let $ \gamma:(a, b) ...
1
vote
1answer
66 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 ...
4
votes
2answers
214 views

algebraic manipulation of differential form

Suppose $\phi_1, \phi_2, \dots, \phi_k \in (\mathbb{R}^n)^*$, and $\mathbf{v}_1, \dots, \mathbf{v}_k \in \mathbb{R}^n$ $(\mathbb{R}^n)^*$ stands for the space of all linear transformations that goes ...
15
votes
3answers
2k views

What is a covector and what is it used for?

From what I understand, a covector is an object that takes a vector and returns a number. So given a vector $v \in V$ and a covector $\phi \in V^*$, you can act on $v$ with $\phi$ to get a real number ...
2
votes
1answer
168 views

Wedge product with a non-degenerate form

Let $\alpha$ be a non-degenerate form in $\Lambda^k(V)$ for some vector space $V$, $\dim V = n$. (Here non-degenerate means that if $x\in V$ is nonzero, then $(y_1 , ... , y_{k-1}) \mapsto \alpha(x , ...
5
votes
0answers
419 views

How to prove this formula for Lie Derivative for differential forms

The professor gave this formula without providing a proof. I would like to know how this can be derived. Let $X$ be a vector field, $w$ be a $p$-form. Then, $$L_X ...
2
votes
0answers
332 views

Top deRham cohomology group of a compact orientable manifold is 1-dimensional

Let $M$ be a compact smooth orientable manifold of dimension $n$. I am looking for a simple proof that $H_{dR}^n(M) \cong \mathbb R$. Equivalently, an $n$-form which integrates to 0 is exact. I can ...
5
votes
1answer
287 views

Integrating a 3-form over the 3-sphere

Consider the 1-form $\alpha = xdz + ydw -(x^2 + y^2 + z^2 + w^2)dt$ on $\mathbb{R}^5$. I'm trying to find $\int_S d\alpha \wedge d\alpha$, where $S \subset \mathbb{R}^5$ is given by $x^2 + y^2 + z^2 ...
2
votes
2answers
320 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 ...
2
votes
1answer
45 views

Problem with integration of $1$-form on surface

I have some problem with integration of differential forms on algebraic surfaces (I'm reading Cartan's book on analytic functions). Let $X \subseteq \mathbb{C}^2$ be an algebraic curve given by ...
3
votes
3answers
303 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 ...
2
votes
0answers
391 views

Evaluating a surface integral with differential forms

Let $\alpha=x dy-\frac{1}{2}(x^2+y^2)dz$ be a differential form in $\mathbb{R}^3=\{(x,y,z)\;|\;x,y,z\in\mathbb{R}\}$and let $Z=\{(\cos\theta,\sin\theta,s)\;|\;0\leq\theta \leq2\pi, 0\leq s\leq 1\}$ be ...
0
votes
1answer
288 views

How can I find the winding number of a curve?

I need to find the winding number of the closed curve $c(t)=(a \cos(t),b \sin(t))^T $, where $a,b > 0$ and $c:[0,2\pi) \to\mathbb{R}^2\setminus\{0\}$. I don't understand how to do this.
1
vote
1answer
267 views

Existence of orthonormal frame

Let $M$ be a surface with Riemannian metric $g$. Recall that an orthonormal framing of $M$ is an ordered pair of vector fields $(E_1,E_2)$ such that $g(E_i,E_j)=\delta_{ij}$. Prove that an orthonormal ...
2
votes
0answers
63 views

How to apply Gauss's theorem when the metric is unknown

Let $f:U \to \mathbb{R}^3$ be a surface, where $U=\{(u^1,u^2)\in \mathbb{R}^2:|u^1|<3, |u^2|<3\}.$ Consider the two closed square regions $F_1=\{(u^1,u^2)\in \mathbb{R}^2:|u^1|\leq1, ...
1
vote
1answer
387 views

Integration of a differential form along a curve

Given the differential form $\alpha = x dy - \frac{1}{2}(s^2+y^2) dt$, I'd like to evaluate $\int_\gamma \alpha$ where $\gamma(s)=(\cos s,\sin s, s)$ and $0\leq s\leq \frac{\pi}{4}$. When attempting ...
3
votes
1answer
283 views

The area form of a Riemannian surface

Let $(M,g)$ be an oriented Riemannian surface. Then globally $(M,g)$ has a canonical area-$2$ form $\mathrm{d}M$ defined by $$\mathrm{d}M=\sqrt{|g|} \mathrm{d}u^1 \wedge \mathrm{d}u^2$$ with respect ...
1
vote
0answers
46 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 $\alpha=p_1 dx_1+p_2 dx_2-H(p_1,p_2)dt$ ...
2
votes
1answer
188 views

Basic computations with differential forms

I've never had any experience with differential forms before, so I'm trying to work through a couple of examples to see if I understand what's going on. I think I understand what I've been doing so ...
0
votes
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
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 ...
8
votes
0answers
307 views

Proof of residue theorem (residue formula) for differential forms on curves over an arbitrary closed field.

I have been reading the book Algebraic Geometric Codes: Basic Notions by Tsfasman, Vladut and Nogin. They give a residue formula like this: Let $\mathbb{k}$ be an algebraically closed field and $X$ ...
43
votes
5answers
3k views

Exterior Derivative vs. Covariant Derivative vs. Lie Derivative

In differential geometry, there are several notions of differentiation, namely: Exterior Derivative, $d$ Covariant Derivative/Connection, $\nabla$ Lie Derivative, $\mathcal{L}$. I have listed them ...
2
votes
1answer
226 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 ...
7
votes
1answer
252 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 ...