The differential-forms tag has no wiki summary.
4
votes
1answer
331 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, ...
10
votes
3answers
450 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 ...
13
votes
2answers
706 views
Intuition behind $dx \wedge dy=-dy \wedge dx$
I was re-reading this old book of mine; and I noticed that in defining the rules of differential forms, it "makes sense" that we have the rule $dx \wedge dx=0$ because if $dx$ is infinitesimal, then ...
5
votes
1answer
363 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 ...
5
votes
3answers
235 views
$\Omega=x\;dy \wedge dz+y\;dz\wedge dx+z\;dx \wedge dy$ is never zero when restricted to $\mathbb{S^2}$
We define a 2-form $\Omega$ on $\mathbb{R^3}$ by $\Omega=x\;dy \wedge dz+y\;dz\wedge dx+z\;dx \wedge dy$. How can I show that $\Omega|_\mathbb{S^2}$ is nowhere zero? Before proving that how can I ...
2
votes
1answer
191 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 ...
3
votes
2answers
81 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 ...
2
votes
0answers
201 views
Differential forms and a chain rule
Let $U$ be a Riemann surface and let $z:U\longrightarrow B(0,1)$ be a diffeomorphism, where $B(0,1)$ is the open unit disc in $\mathbf{C}$. So $z$ is a coordinate around $P=z^{-1}(0)$.
Let $Q\in U$ ...
1
vote
2answers
131 views
Closed forms and a simple relation with Cauchy-Riemann
I have a very basic question, sorry for that )=. Let's fix some notation first.
Let $ dz = dx + i \; dy $ . Given $f \in C^1$, $f : D \subset \mathbb C \to \mathbb C$, we define $df = f_x \; dx + ...
1
vote
1answer
165 views
The chain rule for a function to $\mathbf{C}$
Let $f:U\longrightarrow \mathbf{C}$ be a holomorphic function, where $U$ is a Riemann surface, e.g., $U=\mathbf{C}$, $U=B(0,1)$ or $U$ is the complex upper half plane, etc.
For $a$ in $\mathbf{C}$, ...
0
votes
1answer
103 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 ...
