Tagged Questions

5
votes
0answers
64 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 ...
1
vote
1answer
57 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
35 views

Wedge product of differential and volume form

Let $f(x)$ be a $C^1$ function defined on $\mathbb{R}^n$ and $\nabla f(x) \neq 0$ for any $x \in \mathbb{R}^n$. If $d\sigma$ is the volume form on hypersurface $f(x)=c$ induced from $\mathbb{R}^n$ ...
2
votes
1answer
153 views

baby rudin, chapter 10, (differential forms) theorem 10.27

I'm having difficulties with the reasoning in the proof of theorem 10.27 (regarding integration over oriented simplexes). say ...
1
vote
1answer
34 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 ...
3
votes
1answer
118 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
0answers
44 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
81 views

For a $1$-form $h$, why does $\int_\Gamma \varphi^*h=\int_{\varphi\circ\Gamma}h$?

I'm trying to understand why for a differentiable arc $\Gamma:[a,b]\to\Omega$ and a $1$-form $h=fdx+gdy$, then $$ \int_\Gamma\varphi^*h=\int_{\varphi\circ\Gamma}h? $$ For background, $\Omega$ is an ...
7
votes
1answer
86 views

Why does $d(\varphi^*f)=\varphi^*df$?

I'm trying to learn a bit about differential forms to supplement my study in analysis, but I'm having a hard time with some of the basic manipulations. Anyway, suppose $\Omega$ is an open set in ...
0
votes
0answers
82 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 ...
2
votes
1answer
77 views

Equality of integrals of differential forms

I have two $(n-1)$-forms $\omega_{1}$ and $\omega_{2}$ on $\mathbb{R}^n$ and a smooth function $g(x) \colon \mathbb{R}^n \to \mathbb{R}$ ($dg$ doesn't vanish anywhere) such that $dg \wedge \omega_1 = ...