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

learn more… | top users | synonyms

2
votes
1answer
144 views

Total confusion about differential one-forms and non-coordinate bases

I asked this question recently (Basis of differential one-form confusion), thought I understood the answer, but now realise I don't. Lee (Introduction to Smooth Manifolds) says that at a point $p$ ...
0
votes
0answers
15 views

Determine the analytic expression of this 2-form in a chart

I don't know how to do the following exercise. I would really appreciate if anyone knows how to solve it. $$\textbf{d}\alpha: \mathfrak{X}(M)\times \mathfrak{X}(M)\ni (X,Y)\longrightarrow ...
4
votes
1answer
42 views

Intuition behind volume form of a sphere

What is the geometric intuition behind $$\int_{S^1}x\,dy-y\,dx=2\pi,\qquad\int_{S^2}x\,dy\,dz-y\,dx\,dz+z\,dx\,dy=4\pi r^2,$$and in general, ...
5
votes
1answer
58 views

A converse to Stokes' Theorem in $\mathbb{R}^n$

In a lecture of advanced calculus, my teacher made a very interesting remark about the generalized Stokes' Theorem (actually, he left it as an exercise!), such that, if I understood it right, is ...
0
votes
0answers
19 views

The orientation of an embedded submanifold of $\mathbb R^n$

Assume $M$ is an embedded submanifold of $\mathbb R^n$ with codimension 1, by a specific coordinate of $\mathbb R^n$ $\{x_1,\dots,x_n\}$ in an area $U$, the $M\cap U$ has coordinate ...
2
votes
0answers
28 views

Proving a fact about bilinear forms

The fact I'm trying to prove is that every bilinear form $\omega$ on a set $\omega\subseteq\mathbb{R}^2$ can be written in the form $\omega = f\,dx\wedge dy$, where $f: \Omega\rightarrow\mathbb{R}$ is ...
2
votes
1answer
32 views

Inclusion, pullback of differential form

Let $\omega=x\,dy\wedge dz +y\,dz\wedge dx+z\,dx\wedge dy$ or in spherical coordinates (unless I had made some mistake) $\omega=r^3\cos \theta\, d\phi\wedge d\theta$. Now I want to find $i^*\omega$ ...
1
vote
1answer
84 views

Alternative to Arnold's mathematical methods

I have difficulties understanding Arnold's book of mathematical methods of classical mechanics. Yet I should get some familiarity with the subjects found at chapters 3,4,7,8 before next semester to ...
1
vote
2answers
50 views

if $\omega$ is a $2$-form, and $\Bbb d \omega = 0$ what can we conclude?

Let $f_1, f_2, f_3$ be smooth functions on an open subset $\Omega \subset \Bbb R ^3$, which contains the standard cube $I^3$. We define the differential $2$-form $$ \omega = f_1 \Bbb d x^2 ...
0
votes
1answer
40 views

Finding a compactly supported $f$ such that $\theta = f\omega$.

Let $\Omega \subseteq \Bbb R^N$ be open, $\omega = \sum_{j=1}^n \omega_j\,{\rm d}x_j$ be a $1$-form in $\Omega$ such that $\sum_{j=1}^N|\omega_j(x)|\neq 0$ for all $x \in \Omega$, and $\theta = ...
2
votes
1answer
58 views

Differential forms, projections

I have a problem with one exercise from differential geometry. I don't even know how to start. Anyone could help with this problem? Let $M$, $N$ be manifolds, $M$ connected. Let $\pi:M\times N \to N ...
0
votes
1answer
17 views

Divergence, contraction and lie derivatives

I'm working through this question. I can show the forward direction in (a) but can't show the converse. I have $\delta/\delta t \phi^*_t \mu$ evaluated at t=0 is 0, but I can't see how I conclude ...
3
votes
1answer
32 views

Comass of a differential form

In the wikipedia article on currents https://en.wikipedia.org/wiki/Current_%28mathematics%29 it is written that If $\omega$ is an m-form, then define its comass by $||\omega|| = \sup\{|\langle ...
1
vote
1answer
71 views

Show that $\omega^{2}\wedge\cdots\wedge\omega^{2}$ n times is equal to

Consider $\mathbb{R}^{2n}$ with coordenates $x^{1},\cdots,x^{2n}$ and the following differential form of grade two $$\omega^{2}=dx^{1}\wedge dx^{n+1}+dx^{2}\wedge dx^{n+2}+\cdots+dx^{n}\wedge ...
1
vote
0answers
31 views

Is there a way to compute the Poincaré dual of the following type of degree $(2n-2)$ de Rham class?

Given a closed, connected, symplectic manifold $(X^{2n},\omega)$, is there a systematic method to computing the Poincaré dual surface to degree $(2n-2)$ classes of the form $$[\omega]^{n-2}\cup B + ...
1
vote
1answer
43 views

The wedge product of a $1$-form and a $2$-form

I'm trying to understand the wedge product of a one form and a two form. My difficulty is that I end up with a $dx \wedge dx \wedge dy$, for example, and I know a wedge of itself is 0. Does this ...
1
vote
0answers
37 views

Let $f\colon \Bbb R^n \to \Bbb R^n$ be the translation, $f(x)=x+a$. Show that $\deg(f)=1$

Let $U,V$ be connected open subsets of $\Bbb R^n$ and let $f\colon U \to V$ be a $C^{\infty}$ proper map. For all $w \in \Omega_c^n(V)$, $\int_{U}f^*(w)=\gamma \int_Vw$. Now define $\deg(f)=\gamma$, ...
0
votes
0answers
21 views

Existence of a function in differential forms

The book is Spivak, Calculus on manifolds, page 100. I don't understand why the existence of a unique function "$f$". I hope you can help me on this. Thank you all.
5
votes
1answer
74 views

Interpretation of $d\phi(z)$ in differential geometry

In "Exercises and Solutions in Mathematics", Ta-Tsien, 2nd Edition, exercise 3343. Statement of the exercise Let $(\mathbb{H}, g)$ be the two-dimensional hyperbolic space, where \begin{equation} ...
1
vote
0answers
16 views

Building Non-vanishing sections in a certain way

Assume you have a vector bundle $\Pi: E \rightarrow M$ where $E$ is the total space, $M$ is a compact manifold. Assume you know it is parallelizable. Let $\psi_i : \Pi^{-1}U_i \rightarrow U_i \times ...
3
votes
1answer
42 views

$d(\iota_v\rho)=0 \implies d(\phi\iota_v\rho)=d\phi(v)\rho$?

My motivation is physical, but my question is purely mathematical. Everybody knows, that the power of the electric current in a piece of wire is $$P=UI$$ where the wire is regular domain $V$ in a ...
0
votes
0answers
18 views

Contraction of a vector form

I'm trying to make sense of this definition, but I cannot see why the resulting map is in a space of dimension $k-1$, surely as it is comprised of k vectors this maps a k-form to a (k+1)-form? I ...
1
vote
1answer
27 views

Support of Pullback of Differential form

This is a dumb question, but I'm learning about differntial forms, and it seems to me that if $f:N^n\to M^n$ is a diffeomorphism and $\omega$ a smooth $n$ form on $M^n$, then $\omega$ vanishes at ...
0
votes
2answers
28 views

Using differential forms to define line integrals

I saw some similar questions and answers but they often included some information or mathematics I haven't learned/read so I'm hoping to get a somewhat simpler answer. Let $\beta:[a,b] \to ...
1
vote
1answer
34 views

$\int _c \omega$ is independent of orientation-preserving re-parameterization of c

I'm working on the following problem from Guillemin and Pollack's Differential Topology: Let $c : \left[a, b\right] \rightarrow X$ be a smooth curve, and let $f: \left[a_1, b_1\right] \rightarrow ...
20
votes
2answers
541 views

Why do we care about differential forms? (Confused over construction)

So it's said that differential forms provide a coordinate free approach to multivariable calculus. Well, in short I just don't get this, despite reading from many sources. I shall explain how it all ...
3
votes
1answer
80 views

$\Delta x\approx dx$

How do you state $($small $\Delta x)\approx dx$ in terms of differential forms? $dx$ is a one-form, but I don't see why $\Delta x$ approximates a one-form if gets smaller.
0
votes
0answers
22 views

sympletic form of the hyperboloid

I'm working with the Hyperboloid of two leefs $H_2$ as a coadjoint orbit of $\mathfrak{sl}^*(2, R)$. I know that $H_2$ is a symplectic manifold by the following theorem: Given a Lie group G and ...
2
votes
0answers
19 views

Hermitian vector space and relation of associated operators.

Here what i want to do is prove proposition 1.1 in chapter 5, on Wells, Differential analysis on complex manifold, The propositions are follows For Hermitian vector space of complex dimension $n$. ...
2
votes
1answer
67 views

Simplifying the Kahler form

In the link here, p.4, it says that, given a fundamental 2-from $\mathcal{K}$ $$\mathcal{K}=\frac{\sqrt{-1}}{2\pi}g_{i\bar{j}}dz^i\wedge d\bar{z}^{\bar{j}},$$ a manifold is said to be Kahler if this ...
7
votes
0answers
70 views

A closed $1$-form on a convex open set is exact

Baby Rudin Exercise 10.24: Let $\omega = \sum a_i(\mathbf x) \, dx_i$ be a $1$-form of class $\mathscr{C}''$ in a convex open set $E \subset \mathbb{R}^n$. Assume $d \omega = 0$ and show that ...
2
votes
1answer
34 views

$M \times N$ orientable iff both $M$ and $N$ are orientable proof in terms of volume forms

I'm studying differential forms, and in my homework I'm asked to show that the product of two manifolds $M \times N$ is orientable if and only if both $M$ and $N$ are orientable. I want to show this ...
5
votes
1answer
49 views

Induced bilinear form on exterior powers - Towards a global Hodge Star Operator

In all constructions of the hodge star operator I've seen so far there was a part where an inner product on the exterior power of the tangent space was defined by the ungodly local formula: ...
2
votes
1answer
56 views

Computing the integral of a differential form in $\mathbb{R}^{2}$.

Let $D$ be the disk \begin{equation} D=\{(x,y)\in\mathbb{R}^{2}\:|\:x^{2}+y^{2}\leq 1\}, \end{equation} which is easily verified to be a compact $2$-differentiable manifold with boundary. Let ...
1
vote
2answers
37 views

Proving df is the sum of partial derivatives

Wikipedia states: Since any vector $v$ is a linear combination $\sum v_je_j$ of its components, $df$ is uniquely determined by $df_p(e_j)$ for each $j$ and each $p \in U$, which are just the ...
0
votes
1answer
76 views

Multivariable calc “second course” that does differential forms

I've worked through a computation-heavy, "standard" but quite nonrigorous treatment of multivariable calculus in the past. What book would do well as a rigorous (but not overly) "second course"? In ...
3
votes
1answer
31 views

Definition of integration of differential forms

I am trying to understand precisely the following paragraph: Question Why would he define the support $K$ of a form $\omega$ defined on an open set $U$ as a subset $K\subseteq M$ instead of a ...
2
votes
2answers
68 views

Computing Lie derivative

Can anyone help me with computing Lie derivative ${L}_{X}Y$ using its definition for these two vector fields: $X=y\frac{\partial}{\partial x}-x\frac{\partial}{\partial y}, ...
3
votes
1answer
93 views

What is the intuition behind differential forms?

I am comfortable with the way physicists use differentials as elements of area/volume. I know the (algebraic) formal definition of differential forms, but it makes no intuitive sense, especially since ...
0
votes
1answer
54 views

How to write differential forms on manifolds?

In the book "Differential Forms and applications" by Manfredo do Carmo, he says that a differential $k$-form on a $n$-dimensional smooth manifold $M$ is determined by a choice, for each ...
0
votes
1answer
21 views

De Rham Differential for Vector Valued Forms?

Let $M$ be a smooth manifold of dimension $n$ and let $V$ be a $\mathbb R$-vector space of finite dimension $\ell$. A $k$-form on $M$ with values on $V$ is a map $\omega$ on $M$ such that: ...
1
vote
2answers
35 views

Grammatically confused: $\omega=4dV$ for 3-form $\omega$ and volume in $\Bbb R^4$?

Background: Against the advice I should have been given but wasn't, I'm taking a Lie theory course with no background in differential geometry. We finally made it into the part of the course where we ...
2
votes
1answer
37 views

Volume Forms Induced by Embedding

Let $(M, g)$ be a Riemannian Manifold of dimension $d$, $g$ naturally gives rise to an invariant volume form $V_M \in \Omega^d(M)$. Let $\Sigma$ be a smooth embedded submanifold of dimension $d-1$ in ...
3
votes
1answer
157 views

Why is Schouten-Nijnhuis bracket trivial on Poisson cohomology?

For a commutative algebra $A$, let a biderivation $P$ be called a Poisson structure if $[[P,P]]=0$ (the bracket is Schouten-Nijenhuis). Then one obtains a complex of multiderivations with $[[P,{}]]$ ...
0
votes
1answer
41 views

The formula for the differential of a vector-valued function

If we have a vector, $\,U=U\left(x_1,x_2,x_3\right)$, in the coordinate axis $\left(x_1,x_2,x_3\right)$, then why does the following differential relation hold? $$ dU= \left(\frac{\partial ...
2
votes
0answers
24 views

Is there a commutative algebra for which multiderivations are not generated by order 1?

By multiderivations (of order $k$) I mean polylinear skew-commutative operations with values in my algebra which satisfy Leibniz rule in each of the arguments. (That is, the dual module to ...
0
votes
1answer
52 views

Problem proving Cartan's identity

There is a famous identity stating that, if $X$ is a field and $\omega$ a form, then: $$\mathcal{L}_X\omega=\iota_Xd\omega+d\iota_X\omega.$$ I'm trying to prove it. Thanks to Anthony Carapetis, I ...
0
votes
0answers
13 views

Find curves that generate a dual basis to a space of one-forms

Let $V$ be the vector space of one-forms on the plane that have quadratic functions as coefficients of $dx$ and $dy$, with basis $\{x^2dx,xy\;dx,y^2dx,x^2dy,xy\;dy,y^2dy\}$. For any curve $\Gamma$ ...
1
vote
1answer
64 views

Interior product and exterior product

I have seen around the internet that this should hold: $$\iota_X(\alpha\wedge\beta)=\iota_X\alpha\wedge\beta+(-1)^k\alpha\wedge\iota_X\beta,$$ where $X$ is a vector field, $\alpha$ a $k$-form, ...
2
votes
1answer
26 views

primitive of closed forms along a continuous curve

Here is the theorem: Suppose $\omega$ is a closed differential form in open domain $\Omega$. There is $\gamma:[a,b]\to\Omega$, that defines a continuos curve. Then $\omega$ has a primitive along ...