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

learn more… | top users | synonyms

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 $X$ ...
44
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
228 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
254 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 ...
2
votes
1answer
314 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
539 views

Covariant derivative of tensor in terms of local coordinates

Let $f:U \to \mathbb{R}^3$ be a surface with local coordinates $f_i=\frac{\partial f}{\partial u^i}$. Let $\omega$ be a one-form. I want to express $\nabla \omega$ in terms of local coordinates and ...
4
votes
2answers
113 views

elementary question regarding differential forms

Is it possible to give a high level explanation why changing the order of differentials will give rise to a minus sign ? I.e. why do we have $$ dx\,dt = - dt\,dx $$ (I am going to take a course on ...
1
vote
1answer
161 views

Computing a Differential Form

Apologies in advance, I don't know TeX, so this might look a bit gross... I'm given a 1-form $A=f_1dx_1+...+f_ndx_n$, infinitely differentiable and closed on $R^n$. I want to show that $dg=A$ for ...
1
vote
1answer
87 views

Show that $\omega = d(I\omega)$ if $d\omega = 0$

Let $\omega = P\ dx + Q\ dy$ be a 1-form on $\mathbb{R}^2$. Also, define a 0-form $I\omega({\bf x}) = I\omega(x, y)$ by $$ I\omega({\bf x}) = \int_0^1 P(t {\bf x}) x + Q(t {\bf x}) y\ dt.$$ I would ...
5
votes
1answer
258 views

Could Residue theorem be seen as a special case of Stokes' theorem?

Residue theorem in complex analysis is seems like Stokes' theorem in real calculus, so a question arose that could Residue theorem be seen as a special case of Stokes' theorem?
1
vote
1answer
185 views

Proving Fréchet differentiability

Am learning about Fréchet differentials and was wondering if for a real matrix $X$ and positive semidefinite real matrices $A,B$ the function $f(X)=TrX^TAX-X^TBX$ is twice Fréchet differentiable or ...
4
votes
1answer
123 views

Highest DeRahm Cohomology

Let $X$ be a $C^\infty$ manifold, compact oriented and connected of dimension $n$. How do you prove that the integration map $$\int_X: \omega \mapsto \int_X \omega $$ from $H^n_{DR}(X)$ to ...
4
votes
0answers
71 views

Explicit quasi-inverse of Künneth-isomorphism?

With $A_X$ the complex of $\mathbb{R}$-differential forms on $X$, the Künneth theorem states that \begin{align*} A_X \otimes A_Y &\to A_{X \times Y}, \\ (\omega,\eta) &\mapsto {\rm ...
3
votes
1answer
65 views

insertion operator $i_X$ on $1$-form

Let $\omega $ be a $1$-form on $M$. For $f:M\to \mathbb R$, and $X\in \mathfrak{X}(M)$, Does the following true? $f. i_X \omega = i_{f.x}\omega$ According to me proof is the following: Please ...
1
vote
0answers
72 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 ...
4
votes
1answer
471 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 ...
5
votes
1answer
209 views

Pullback of differential form on the double covering

On a double covering there is a differential form $\omega$ arises by the pullback of a differential form under the projection iff it is the pullback of $\omega$ under the map $i$, where $i$ is the map ...
0
votes
1answer
145 views

Averaging differential forms

Let $M$ be a manifold with a circle action, i.e. with a 1-parameter group of diffeomorphisms $\phi_t:M\to M$ of period 1. I think of the average of a differential form $\omega \in \Omega^n(M)$ with ...
5
votes
1answer
408 views

Riemannian Volume Form

There is a following exercise in my text: Let $S^n$ be $n-$ dim sphere in $R^{n+1}$ with inclusion function $i:S^{n}\to R^{n+1}$. Let $$\omega=\sum_{i=1}^{n+1}(-1)^{i-1} x_i dx_1 \wedge... ...
3
votes
2answers
178 views

Riemannian volume form on surface of a smooth function

It should be easy calculation exercise in my text, but I am afraid I am a little bit stuck on the concept of the question. Let $f:\Bbb R^n\to \Bbb R$ be a smooth function. Consider graph $X$ of $f$ ...
5
votes
0answers
243 views

understanding this differential operator on a tensor product

I am currently trying to read the T. Kotake's paper "An Analytical Proof of the Classical Riemann Roch Theorem", in which he defines a differential operator which acts on smooth sections of a tensored ...
3
votes
1answer
130 views

What is $d(y dx)$?

Let $x$, $y$ be 0-forms, thus $dx$, $dy$ are 1-forms. Since 1-forms compose an algebra over 0-forms ring, expressions like $$y dx$$ make perfect sense. Now I ask what is $$d(y dx)$$ I suggest it ...
6
votes
1answer
374 views

how to understand the tensor product canonical line bundle $\otimes$ dual bundle

Suppose we have a Riemann surface $M$ together with a holomorphic vector bundle $E \to M$ of rank n. let $K$ denote the canonical line bundle and let $E^*$ denote the dual bundle I am trying to ...
0
votes
1answer
140 views

Coordinate change in a differential form

The task is to rewrite the following differential form in polar coordinates: $$w = \sqrt{x^2 + y^2} \, dx \land dy $$ I did it by a direct substitution: $$\begin{cases}x = r \cos \varphi \\ y = r ...
1
vote
1answer
112 views

Ideal generated by differential forms

I have troubles picturing what elements belong to a particular ensemble. Let $\omega_1$,...,$\omega_r$ be differential 1-forms on a $C^\infty$ n-manifold that are independent at each point. ...
3
votes
1answer
87 views

Properties of Pull Backs of forms

Say I have an embedding: $f: X\to Y$ and a non-vanishing form $\omega$ on $Y$. Are there conditions on $f$ so that the pull-back $f^* \omega$ vanishes everywhere? Or in a specific case, say $\omega$ ...
4
votes
2answers
140 views

zeroes of forms on Riemann surfaces

Let $P$ be a point on a Riemann surface. Does there exist a non-trivial differential form $\omega$ on $X$ such that $\omega$ vanishes at $P$? Does there exist a non-constant rational function $f$ on ...
5
votes
2answers
415 views

Poincaré Lemma Contractible Hypothesis

Poincaré's Lemma is often stated as saying that a closed differential form on a star-shaped domain is exact. More generally, it is true that a closed differential form on a contractible domain is ...
4
votes
1answer
643 views

Problem book on differential forms wanted

I want to get used to differential forms. Thus I would like to solve a bunch of problems, especially on integration of differential forms. So I need a collection of problems with answers/solutions, ...
0
votes
1answer
164 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 ...
7
votes
0answers
97 views

Dimension of diffeomorphism groups preserving some $2$-tensor.

For a finite-dimensional smooth manifold $M$, let $\mathrm{Diff}(M)$ be its diffeomorphism group. Suppose we are given a $2$-tensor $\mathcal{K}$ on $M$, and let $$\mathrm{Diff}_{~\mathcal{K}}(M) = ...
2
votes
1answer
229 views

Relevance of Differential Forms

I recently started reading about differential forms, and I am trying to figure out their purpose. Lets say $\omega=y\,dx+x\,dy$, and we want to evaluate $\int_C \omega$ over the curve parametrized by ...
0
votes
0answers
156 views

differential form vanish somewhere on compact orientable without boundary manifold [duplicate]

(a)Let $ M =M^n$ a $n-$dimensional compact, orientable, differential manifold without boundary ( $\partial M = \emptyset$). Define $\omega$ a differential $(n-1)-$ form on $M$. Prove that there ...
0
votes
1answer
36 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 ...
1
vote
1answer
70 views

1-form is exact

Let $\omega = f_1 dx_1 +f_2dx_2 + \cdots + f_ndx_n$ be a closed $ C^{\infty}$ $1-$form on $ \mathbb R ^n$. Define a function $g$ by $\displaystyle{ g(x_1, x_2,\cdots, x_n) = \int_{0}^{x_1} f_1(t,x_2 ...
2
votes
1answer
304 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 ...
3
votes
1answer
101 views

differential form is exact on $ U \cup V$

Let $ U ,V \subset \mathbb R ^n$ two simply connected open sets such that $ \displaystyle{ U \cap V}$ is a connected set. If $\omega$ is a closed 1-form wich is exact in $U$ and $V$ prove that: ...
3
votes
2answers
287 views

wedge product with the exterior derivative of the form $ \omega:= dz +x_1 \, dy_1+ x_2 \, dy_2 + \cdots + x_n \, dy_n $.

Write the coordinates on $ \mathbb {R} ^{2n+1}$ as $ \displaystyle{ (x_1 , y_1, x_2, y_2, \cdots ,x_n, y_n ,z)}$. Define the 1-form $ \displaystyle{ \omega:= dz +x_1 \, dy_1+ x_2 \, dy_2 + \cdots + ...
0
votes
1answer
79 views

degree of proper mapping for linear case

Let $U,V$ be two open sets in $R^n$ and $f:U\to V$ proper $C^{\infty}$ map (proper = preimage of compact set is compact). Then we have $$\int f^{*}\omega=\deg(f)\int \omega,$$ for $\omega \in ...
4
votes
0answers
69 views

Coboundary of Thom class and Thom class of boundary

In Griffiths and Morgan's book "Rational Homotopy Theory and differential forms", pages 154-158, they give an example of a computation in de Rham cohomology of the minimal model of a DGA using Massey ...
2
votes
0answers
139 views

Calulation of pullback of form

If $M$ is $2n+1$ dimensional manifold, and $M'= M\times \mathbb R$ Let $x_1,y_1,... x_n, y_n,t', t$ be coordiante of $M'$. With $t$ for coordinate for $\mathbb R$. Let $$ \omega= \sum_{i=1}^n ...
6
votes
1answer
547 views

wedge product of differential form

If $\alpha $ is one form over some manifold $M$ $2n-1$ dimensional real, and $X= M\times (0,\infty)$. $r$ is the coordinate for the second factor. Define two form on $X$: $$\omega= d(r^2\alpha)$$ ...
5
votes
1answer
266 views

differential form

one form $\alpha$ over a smooth manifold is non vanishing means for every $p\in M$, $\alpha_p\neq 0$. But $\alpha_p$ is linear map $T_M\to \mathbb R$, hence $\alpha_p(0)=0$. So confusion arises ...
3
votes
2answers
98 views

Proving that a particular submanifold of the cotangent space is Lagrangian

I have the following problem in my differential topology class: Let $M$ be an $n$-dimensional manifold and let $\omega$ denote the standard symplectic form on the cotangent space $T^*M$. Let $f \in ...
1
vote
1answer
139 views

Confused about Wikipedia page on differential forms

I know next to nothing about differential forms, but I saw them being mentioned repeatedly on this site, so I went to Wikipedia to try to understand what a differential form is. Most of it is going ...
0
votes
0answers
318 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}$ ...
8
votes
1answer
172 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
votes
1answer
243 views

$4$-form $ \omega \wedge \omega$ vanishes on $S^4$

If $\omega$ is a closed $2$-form on $S^4$, how can I show the $4$-form $ \omega \wedge \omega$ vanishes somewhere on $S^4$? I am guessing that the fact we're talking about the $2$-form being ...
2
votes
1answer
89 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
296 views

Pullback on differential forms are linear

Let $T: \mathbb{R}^n \rightarrow \mathbb{R}^m$ be a smooth transformation. Define the pullback $T^*: C^k (\mathbb{R}^m) \rightarrow C^k (\mathbb{R}^n)$ (With $C^k(\mathbb{R}^n)$ being the set of ...