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

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2
votes
0answers
32 views

Does the wedge product of bundle-valued forms induce a universal object?

Given a smooth manifold $M$ and a vector bundle $E$ over $M$, the $C^\infty(M)$-module of $E$-valued $p$-forms on $M$ is defined to be $$\Omega^p(M; E) := \Gamma_M\left( \bigwedge^p T^*M \otimes E ...
9
votes
1answer
96 views

What makes differential forms special

There are two natural structures defined on differential manifolds, namely tangent bundle and its dual cotangent bundle. An element of tangent bundle $TM$ at a base point $p\in M$ can be described by ...
3
votes
1answer
108 views

Geometric intuition about the exterior derivative

Let $M$ be a smooth manifold. One $k$-form is a section of the bundle $\bigwedge^k T^\ast M$, that is, if $p\in M$ and $\omega$ is a $k$-form then $\omega_p$ is one $k$-linear alternating real ...
1
vote
0answers
41 views

1-form integration

Let $\alpha:[-1,1]\rightarrow R^2$ be the curve segment given by $\alpha=(t,t^2)$. If $\phi=v^2du+2uvdv$, (the fist component of $R^2$ is $u$ and the second one is $v$) I have $$\int_\alpha ...
2
votes
1answer
70 views

Express a quadratic form as a sum of squares using Schur complements

So I was able to figure out the first part of this problem, but I have no concept of how it relates to Schur complements, so I'm not sure (no pun intended) how to proceed. The question is as follows: ...
8
votes
2answers
40 views

$\text{Alt}\,(\phi_1 \otimes \phi_2 \otimes \phi_3)$

How do I write out $\text{Alt}(\phi_1 \otimes \phi_2 \otimes \phi_3)$ for $\phi_1, \phi_2, \phi_3 \in V^*$?
3
votes
1answer
51 views

Integration with 2-forms

Wikipedia says: Let $$ \omega=f_{z}\, \mathrm dx \wedge \mathrm dy + f_{x}\, \mathrm dy \wedge \mathrm dz + f_{y}\, \mathrm dz \wedge \mathrm dx $$ be a 2-form on a surface with parametrization ...
0
votes
0answers
26 views

Proof of a result in differential forms

I want to prove the following result: Let $w = w_1 dx + w_2 dy + w_3 dz$ a 1-differential form, such that $ w_1,w_2,w_3$ are homogenous of order $\alpha$ show that if $w $ is closed then $ w = df$ ...
1
vote
1answer
57 views

Is $\omega = x^2\,dy\,dz +y^2\,dz\,dx + z^2\,dx\,dy$ exact?

this isn't a homework problem or anything. Basically is $\omega = x^2 \,dy\,dz +y^2\,dz\,dx + z^2\,dx\,dy$ exact? That is, is there a $\lambda$ such that $\omega=d\lambda$, if so what is it? I think ...
0
votes
1answer
67 views

What is the first arrow in the Mayer-Vietoris sequence?

On page 449 of Lee's Introduction to Smooth Manifolds (2nd Edition), the Mayer-Vietoris Theorem is given: Let $M$ be a smooth manifold. Let $U$ and $V$ be open in $M$ such that $U\cup V=M$. Then ...
3
votes
1answer
61 views

Area form a differential form?

I just read that $\omega_x( \eta, \zeta) := \langle x, \eta \times \zeta \rangle $ is the area form on the sphere, where $x \in \mathbb{S}^2$ and $\eta,\zeta \in T_xM.$ All I see is that this is ...
1
vote
1answer
47 views

Derivative for numerical models.

I am working in Mechanical engineering and Computer vision, in which I use a matlab code (or codes) to represent a specific system or process. Of course such model has an input , an implimented ...
1
vote
1answer
44 views

rank of the symplectic form

This is a general question about ranks of differential forms. I read in a book the phrase "symplectic form has constant rank..." I understand that the symplectic form is a nondegenerate differential ...
2
votes
1answer
44 views

An Application of Stokes's Theorem

Let $D^2=\{(x,y)\in \mathbf R^2: x^2+y^2\leq 1\}$ be the unit disc in $\mathbf R^2$, and $D^3=\{(x,y,z)\in \mathbf R^3: x^2+y^2+y^2\leq 1\}$ be the unit disc in $\mathbf R^3$. Let $i_{\pm}:D^2\to ...
0
votes
1answer
38 views

how to find points where a k-form is nonvanishing.

for example, if we are given 2-form $\omega=2xdx\wedge dy+2ydy\wedge dz$, what are the points where the form vanishes? I can only think of points $(0,0,z)$, is it all? Additionally, if we have a form ...
0
votes
0answers
70 views

Integration over a manifold with boundary (Check).

Assume that $ f: \Bbb{R}^{3} \to \Bbb{R} $ is a smooth function such that $ M \stackrel{\text{df}}{=} \left\{ \mathbf{x} \in \Bbb{R}^{3} ~ \middle| ~ f(\mathbf{x}) \ge 0 \right\} $ is a non-empty ...
1
vote
1answer
48 views

Is the form closed?

$S$ is an n dimensional unit sphere such that $S^n=(x\in \Bbb R^{n+1}: |x|=1)$ with some fixed orientation and $\omega$ is a volume form on $S$. Prove that $\omega$ is closed. Prove that $\omega$ ...
0
votes
1answer
32 views

Compute the wedge product n times

Let $\omega$ be a 2-differential form in $\mathbb{R}^{2n}$ given by $$\displaystyle \omega=dx^1\wedge dx^2+dx^3\wedge dx^4 + \cdots + dx^{2n-1}\wedge dx^{2n}$$ Compute: $$\displaystyle ...
0
votes
0answers
15 views

Prove a certain property of the Hodge double star operator

I want to solve the following problem Show that $\ast\ast\omega = (-1)^{k(n-k)}\omega$ where $ \displaystyle \ast\omega =\sum_I \text{sgn}(I,J)\omega_I dx^J$ and $\omega$ is a k-form in ...
0
votes
0answers
24 views

Use a Lie series in order to find the solution to initial value problem

We were presented with a fairly difficult bonus question on my multivariable calculus exam today. I was hoping you all could hope me crack it. The question is as follows: Use a Lie series to find, ...
1
vote
0answers
49 views

Compute the double Hodge star operator

I am taking a course in Multivariable Analysis and I am asked to do the following problem: Show that $\ast\ast\omega = (-1)^{k(n-k)}\omega$ So I start as follows: We know that $\displaystyle ...
2
votes
0answers
77 views

How to define the boundary operator using the exterior derivative?

I am looking for a way to define the boundary operator $\partial : M^n \to N^{n-1}$ from an $n$-dimensional manifold $M$ to its boundary $N$ using the the expression \begin{equation*} \int_M d \alpha ...
1
vote
1answer
26 views

Direct differentaition

In a STEP problem I found this: the vector "$\vec r$" is given as below ($a$ and $L$ are constants) and then it said to perform direct differentation to get the second equation. $$\vec r = a (\sin ...
0
votes
1answer
26 views

Basic question that has to do with exterior derivative.

Basic question: If we have $$Y=d\left(\frac{1}{\alpha} +\frac{1}{\bar{\alpha}}\right)$$ where $d$ is exterior derivative, i.e, $Y$ is a $1$-form. Now we could write that as ...
0
votes
0answers
50 views

Relationship between Cartan and Fréchet derivative

Let $f: X \rightarrow \mathbb{R}$ be smooth, then the Fréchet derivative is a map $Df: X \rightarrow L(X, \mathbb{R}).$ But if $f: M \rightarrow \mathbb{R}$ is smooth and $M$ a manifold, then the ...
1
vote
0answers
25 views

An expression of covectors acting on vectors on the tangent space of a manifold

Let $M$ be a smooth manifold. Take $p\in M$ and $(U,\varphi)$, $\varphi:U\rightarrow \mathbb{R^n}$, a chart around $p$. Let $\mathbb{R}^n\left[\frac{\partial}{\partial x_i}\right]$ and ...
3
votes
2answers
310 views

Differentiation Tricks

Since most derivatives are trivial to take, it's understandable why integrals get most of the mathematical tricksters' attention. However, not all derivatives are trivial to take and I think it's good ...
1
vote
0answers
75 views

Integrability problem in Cauchy Integral Formula

This is problem on the integrability of a 2-form in the nhbd of its singularity. I was looking at the general Cauchy formula (general because it works for $\mathcal C^1$ function, and makes the case ...
0
votes
0answers
22 views

How to obtain the line element in cylindrical coordinates, using definition of differential forms

In general, a volume element is a k-form on an K-dimensional manifold. a k-form w on $\mathbb{R}^{n}$ is defined as $w(x) = \sum_{i_{1}<i_{2}<...<i_{k}} w_{i_{1}i_{2}...i_{k}}(x) ...
1
vote
0answers
32 views

Cohomology of two pieces of torus

In an exercise from an old exam, I found myself confronted with $M=\{(\sqrt{x^2+y^2}-2)^2+z^2=1\}$, $U=M\cap\{x\neq0\vee y>0\}$, $V=M\cap\{x\neq0\vee y<0\}$ and $U\cap V$, all subsets of ...
0
votes
0answers
36 views

Green's theorem via Stokes's theorem

I am considering the following form of Stokes's theorem: Let $\omega$ be an $n-1$ differential form with compact support on an oriented manifold of dimension $n$. Let us consider the boundary ...
1
vote
0answers
40 views

Pullback 1-differential form

Let $(x_0,x'_0) \in \mathbb{R}^2$ be initial data for the Euler Lagrange equation with some given Lagrangian $L: \mathbb{R}^2 \rightarrow \mathbb{R}.$ Then $F^t$ is the flow that maps the initial ...
5
votes
1answer
209 views

Different definitions of differential forms?

I am a physicist and was reading about differential forms in Classical Mechanics. Now, I thought that a two-form is a smooth map $\omega : M \rightarrow \Lambda(T^*M)$ so that a point $p$ on the ...
1
vote
0answers
34 views

Bianchi Identity - Gauge Theory

I am reading through some lecture notes (found here) and following a proof of the Bianchi identity in the context of principal bundles. That is, $h^*\Omega = 0$, where $\Omega$ is the curvature ...
0
votes
1answer
123 views

Differential forms - looking for 3 definitions!

I am sorry for this type of question, but I currently have to deal with differential forms although I have not heard so far what they actually are, so I have just a few very particular questions about ...
2
votes
1answer
97 views

How can I prove $dz=dx+idy$?

Let's see $\Bbb C$ as an $\Bbb R$-vector space. Hence it is isomorphic to $\Bbb R^2$ and it has dimension $2$. If $v_1,v_2$ is a basis for $\Bbb R^2$, every its element can be written as $xv_1+yv_2$; ...
3
votes
0answers
62 views

Differential Form Pullback Definition

I'm having some difficulty following how Spivak (Calculus on Manifolds) has induced the pullback on page 89. From reading elsewhere online it seems convention is to define the induced map of the ...
0
votes
0answers
30 views

existence of integrating factor [duplicate]

Is there always an integrating factor to turn an incomplete differential $M(x,y)dx+N(x,y)dy$ in to a complete differential. Does the answer depend on dimensionality? for example what is the answer for ...
3
votes
0answers
64 views

Product of Two Orientable Manifolds is Orientable

I am trying to show that following: Let $M$ be an oriented smooth manifold of dimension $m$, and $N$ be an oriented smooth manifold of dimension $n$. Then $M\times N$ is orientable. Let ...
0
votes
0answers
28 views

Integration of a 1-form over a “split curve”.

Bit of a strange question I can't really get my head around so apologies if it is ill-posed. Suppose we take a closed curve $\gamma: S^1 \to C \subset M$ in a Riemannian manifold $M$ and integrate ...
37
votes
3answers
646 views

Is Stokes' Theorem natural in the sense of category theory?

Stokes' Theorem asserts that for a compactly-supported differential form $\omega$ of degree $n-1$ on a smooth oriented $n$-dimensional manifold $M$ we have the marvellous equation $$\int_M d\omega = ...
1
vote
1answer
40 views

Winding number of differential curve

Consider the one-form $\omega$ on $\textbf{R}^2$\ {(0,0)} defined by $\omega$ = $\frac{xdy-ydx}{x^2+y^2}$ Let K $\subset$$\textbf{R}^2$\ {(0,0)} denote the positive x-axis. Let $\gamma$ : $[a,b]$ ...
0
votes
0answers
32 views

One form and Vector fields on a manifold in terms of local coordinates.

Prove : $d$$\omega$$(V,W)$=$V \omega (W) - W \omega(V) -\omega([V,W])$ in local coordinates where $\omega$ is a one-form and V,W are vector fields on a smooth manifold M. I do not know how to ...
-4
votes
2answers
142 views

Differential Forms Notation is Wrong? Confirm or deny? [closed]

Being an engineering student that just happens to have a large interest in math, I have always felt that appealing to definitions instead of intuitively understanding a concept is a mistake. A while ...
0
votes
1answer
40 views

Three-Dimensional Metrics as Deformations of a Constant Curvature Metric?

I read the following paper Three-Dimensional Metrics as Deformations of a Constant Curvature Metric and discovered the following result: I have three questions: (1) Is $h$ also a conformally flat ...
1
vote
0answers
59 views

Connection on Tangent Bundle of Group Manifolds

The thing puzzling me is about a transformation rule of connection on tangent bundle of group manifolds: Assume one has a compact and simply connected Lie group $K$. One can give a metric $g$ on $K$ ...
0
votes
1answer
131 views

Integration over manifolds

S is subset of $\Bbb R^3 $ consisting of the union of 1) $z$ axis 2) the unit circle $x^2+y^2=1,z=0$ 3) the points $(0,y,0)$ with $y \ge1 $ Let $A$ be the open set $\Bbb R^3-S$. Let $C_1, C_2, ...
1
vote
1answer
116 views

Integration of forms on non-simply connected manifolds

What I know is that closed forms are not exact on non-simply connected manifolds, so for instance, if $E$ is a closed form, then $dE = 0$ but $\int_\gamma E \neq 0$, where $\gamma$ is a ...
1
vote
0answers
46 views

Integrating factor for non-exact differential

Given $\frac{xdy+ydx}{x^2+y^2}$ I took partials of either coefficient wrt to other variable I get $\partial P/\partial y$ as $\frac{x^2-y^2}{(x^2+y^2)^2}$ and $\partial Q/\partial x$ as ...
2
votes
1answer
32 views

Does the dual basis to some basis of $T^*_pM$ looks localy like a coordinate chart?

Let $M$ be a manifold and let $\{\alpha_k\}$ be a set of $1$- forms s.t. $\{\alpha_k(p)\}$ forms a basis for $T^*_pM$. Let $(x,U)$ be a chart based in $p$ and denote $\partial_i := ...