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

learn more… | top users | synonyms

1
vote
2answers
29 views

Representing $f(x,y)$ as a Sum of Partial Derivatives

I was attempting an exam question which looked like this: Given the expression: $P(x, y)\text{d}x + Q(x,y)\text{d}y = 0$ Where: $P(x,y) = 6x +9y + 11 \\ Q(x, y) = 9x - 4y +3$ Find a function $f(...
4
votes
1answer
66 views

About codifying length and energy using a one-form

Let $M^n$ be a smooth manifold and $g$ be a Riemannian metric on $M$. Is there $\omega \in \Omega^1(M)$ such that $\int_c \omega = \ell(c)$ for every curve in $M$? In general the answer seems ...
2
votes
2answers
58 views

Sum of square of function

If $f'(x) = g(x)$ and $g'(x) = - f(x)$ for all real $x$ and $f(5) =2 =f'(5)$ then we have to find $f^2$$(10) + g^2(10)$ I tried but got stuck
4
votes
1answer
59 views

Geometric Significance of some features of the Exterior Algebra

I've been tinkering with differential forms for a while now, and I've had a few questions all rolled into one trying to understand them. The exterior derivative is quite natural to me - it looks just ...
2
votes
2answers
147 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, $\...
0
votes
1answer
384 views

Harmonic functions and polar differential forms

Given a harmonic function $u$, its differential and conjugate differential are $$du = \frac{\partial u}{\partial x}dx + \frac{\partial u}{\partial y}dy,\qquad ^{*}du = -\frac{\partial u}{\partial y}dx ...
-2
votes
1answer
39 views

Integrating differential forms on curves [closed]

How can I integrate the differential form $$\omega=x\,dx+y\,dy+z\,dz$$ in $\mathbb R^3$ on the curve $$c:[0,2\pi]\to\mathbb R^3: t\mapsto (e^{t\sin t}, t^2-2\pi t, \cos \frac{t}{2})?$$ Some advice ...
1
vote
0answers
19 views

Explicit isomorphism $H (E, E^0) \cong H_{cv} (E)$

Yo! I've been trying to understand better why $$H (D, D-\{x\}) \cong H_c (D)$$ for a disk $D$ and, more generally, why $$H (E, E^0) \cong H_{cv} (E)$$. In general, the first isomorphism can be seen as ...
3
votes
2answers
96 views

What do mathematicians mean when they say “form”?

As in differential form, modular form, quadratic form? I'm sorry if this is a really silly question.
0
votes
1answer
24 views

Finding a two-dimensional chain

Let $$T=\{(x,y,z,w,)\in R^4:x^2+y^2=z^2+w^2=\frac{1}{\sqrt 2}\}$$ and $$\omega=dx\land dy + dz\land dw$$ in $\mathbb R^4$. How do I find a two-dimensional chain $C$ where $T$ is its trace? And how can ...
3
votes
0answers
33 views

How different definitions of connections fit together?

I'm working my way through Ivey and Landsberg, Cartan for Beginners, and I'm working on 2.6.13.2(b). After defining, for $X \in \Gamma(TM)$ a vector field, with section $s:M \rightarrow \mathcal{F}_{...
8
votes
1answer
270 views

What exactly does “differential forms are coordinate free” mean?

Most introductory texts on differential forms praise their property of allowing for a "coordinate free formulation". What exactly does this mean? What would be a concrete example for which a ...
1
vote
1answer
13 views

About a proof concerning the relation of Lusternik-Schnirelman-category and cup length

In the proof of the relation between Lusternik-Schnirelman-category and Cup length (of de Rham Cohomology) for smooth manifolds from this note (theorem 2) the argument goes: Let the given manifold $M$ ...
2
votes
1answer
46 views

Zeroes of $dx_1$ on $\mathbb{R}^2$ vs. zeroes of $dx_1|_{S^1}$ on $S^1$

Let us consider $S^1$ as a manifold embedded in $\mathbb{R}^2$. Let $dx_1\in\Omega^1(\mathbb{R^2})$. $$ Z_{\mathbb{R}^2}:=\{p\in\mathbb{R}^2:(dx_1)_p=0\}\\ Z_{S^1}:=\{p\in\mathbb{R}^2:(dx_1|_{S^1})_p=...
1
vote
1answer
39 views

Bott and Tu compact cohomology of the circle “differential forms in Algebraic Topology”

On page 27 of that book, it is claimed that the inclusion map $\delta$ which maps a form from the non-empty intersection of two open covers of the circle to the disjoint union of those covers has a ...
1
vote
1answer
19 views

Norm of the gradient of a vector field in Cartesian versus Cylindrical coordinates

It is well known that for a vector $\textbf{v}=R\textbf{e}_r+\Theta\textbf{e}_{\theta}+Z\textbf{e}_z$, its 2-norm is $\|\textbf{v}\|_2=\sqrt{R^2+Z^2}$ instead of $\sqrt{R^2+\Theta^2+Z^2}$. Now, for a ...
1
vote
1answer
38 views

Determining when a differential form is closed

I'm looking at the $3$-form on $\Bbb R^4 \setminus \{0\}$ defined by $$ \gamma_k = \frac{1}{\Vert x \Vert^{2k}} i_E(dV),$$ where $k \in \Bbb R$, $E$ is the Euler vector field $x^i \frac{\partial}{\...
2
votes
0answers
35 views

Zorich's Mathematical Analysis, Volume II

Springer just published a new English version of Vladmir Zorich's two-volume Mathematical Analysis. I was looking at the second volume. It seems to have sections on both Multivariable/Vector Calculus ...
1
vote
1answer
41 views

Constructing symplectic structure on $T^*M$

I read the picture below, but I don't know how to get the equation above red line. Whether by using $T_{\xi_x}(T^*M)\cong T^*_xM$ ?But which isomorphism should be choice ? Then , how to check the ...
7
votes
1answer
145 views

Why use geometric algebra and not differential forms?

This is somewhat similar to Are Clifford algebras and differential forms equivalent frameworks for differential geometry?, but I want to restrict discussion to $\mathbb{R}^n$, not arbitrary manifolds. ...
3
votes
0answers
68 views

Understanding twisted differential forms

I'm trying to understand twisted differential forms. I do know that they are like regular differential forms but under coordinate transformations they pick up an extra factor of the sign of the ...
1
vote
1answer
24 views

Calculating pullback using inclusion vs. pullback using chart

Consider the following form on $\mathbb R^{2n + 2}$: $$ \omega = \sum_{k=1}^{n+1}x_k dy_k - y_k dx_k$$ It defines a form on the sphere. Since recently I learned that in order to make it a form on ...
3
votes
2answers
125 views

Differential Forms on the Riemann Sphere

I am struggling with the following exercise of Rick Miranda's "Algebraic Curves and Riemann Surfaces" (page 111): Let $X$ be the Riemann Sphere with local coordinate $z$ in one chart and $w=1/z$ in ...
0
votes
0answers
24 views

Restriction of a differential form vs pullback on submanifold $S^1\subset \Bbb R^2$

I was working through an exercise in Tu's book on differential geometry, (ex. 19.5) and I'm trying to get the most out of the exercise, and test my understanding, given that I've already computed the ...
1
vote
2answers
48 views

Definition of a derivative of differential form

While reading a paper I encountered the following: Let $(\mathbf{q,p}) \in \mathbb{R}^{2n}$ be canonical coordinates and let $H: \mathbb{R}^{2n} \to \mathbb{R}$ be a smooth function. The ...
4
votes
4answers
1k views

Show that the form $w$ is closed but not exact

Let $w=\dfrac{-y}{x^2+y^2}dx+\dfrac{x}{x^2+y^2}dy$ Showing that $w$ is closed is easy. Just calculate $dw$ and you'll get 0. But how do I show that $w$ is not exact? In other words, I need to ...
0
votes
1answer
19 views

Explicit formulation of hermitian form and corresponding alternating form

I can't understand the following basic thing: Let $X$ be a complex manifold of dimension $n$ and $T_xX$ its tangent space in $x\in X$. Then on $T_xX$ we can define a hermitian form $\sum_{i=1}^ndz_id\...
2
votes
1answer
62 views

Exterior derivative on principal bundle

In Nakahara's Geometry, Topology and Physics on page 375, he constructs a Lie-algebra-valued one-form $\omega$ on a principal bundle $P$ by "lifting" a Lie-algebra-valued one-form $\mathcal A_i$ on an ...
2
votes
0answers
22 views

The definition of the operator $\Delta_{\bar\partial}$

Preliminaries: Let $(X,h)$ be a Kahler manifold of complex dimension $d=2n$. Lets denote with $\mathscr A^{p,q}_X$ the sheaf of $C^\infty$ (complex) $(p,q)$-forms over $X$ and let $A^{p,q}(X):=\Gamma(...
0
votes
0answers
22 views

Coordinates of exterior derivative of dual basis of local frame for the tangent bundle

Let $M$ be an $n$-manifold. Let $E_1, E_2,\dots, E_n : U\subset M \to TM $ be a local frame for $TM$ with associated local dual frame $\epsilon^1, \epsilon^2,\dots, \epsilon^n : U\subset M \to T^*M $. ...
0
votes
0answers
19 views

Adjoint of the gauge covariant derivative

Suppose $A=A_1dx_1+A_2dx_2$ is a 1-form connection in $\mathbb{R}^2$ and $D_A \phi=d\phi-iA\phi$ is the gauge covariant derivative with $\phi=\phi_1+i\phi_2$ is a complex scalar field. May I ask what ...
3
votes
1answer
91 views

Need help understanding local coordinates of differential forms

I was trying to check a variation of theorem 2.2: I did the first part of the proof using the standard contact structure on $\mathbb R^{2n + 1}$ which worked out as expected. Then I wanted to do it ...
1
vote
0answers
41 views

Challenging calculation of a Jacobian for an unusual matrix coordinate transformation

I am studying a random matrix ensemble and I am having trouble performing a coordinate transformation. My question is very straightforward, but perhaps a bit technical. I have the following integral--...
0
votes
0answers
18 views

About the choice of a differential form such that the exterior derivative is a a determinant of a Jacobian of an application

If $D\subset \mathbb{R}^{2}$ is a compact domain with regular boundary and $F:D\to \mathbb{R}^{2}$ is such that $F(x)=(f(x),g(x))$ where $F \in C^{2}$ , then if i choose the 1-form $\omega=f.dg$, ...
1
vote
1answer
36 views

Let $ \omega = dx_1 \wedge dx_2 + dx_3 \wedge dx_4 + \cdots + dx_{2n-1} \wedge dx_{2n} \in \mathbb{R}^{2n}$. Find $\omega^n$ (in respect to $\wedge$) [duplicate]

Let $ \omega = dx_1 \wedge dx_2 + dx_3 \wedge dx_4 + \cdots + dx_{2n-1} \wedge dx_{2n} \in \mathbb{R}^{2n}$. Find $\omega^{n}$ (in respect to $\wedge$) When I say "$\omega^{n}$ (in respect to $\wedge$...
4
votes
0answers
52 views

Integrating factor for a non exact differential form

I can't find an integrating factor for the differential form $$ -b(x,y)\mathrm{d}x + a(x,y)\mathrm{d}y $$ where $$ a(x,y) = 5y^2 - 3x $$ and $$ b(x,y) = xy - y^3 + y $$ The problem has origin form ...
1
vote
1answer
53 views

Frobenius condition in terms of Lie brackets

Let $\alpha$ be a $1$-form and $\xi = \ker \alpha$. Frobenius theorem tells us that $\xi$ is integrable iff $\alpha\wedge{\rm d}\alpha = 0 .$ In the book "Introduction to Contact Topology" from ...
2
votes
2answers
73 views

Finding the antiderivatives for the differential form $\alpha = x_2 d x_1 \wedge d x_2$

Consider the differential form $$\alpha = x_2 d x_1 \wedge d x_2$$ on $\mathbb{R}^3$. I first want to find a $1$-form $\beta \in \Omega^1(\mathbb{R}^3)$ that is an antiderivative for $\alpha$, i.e. ...
1
vote
1answer
54 views

How to evaluate a $1$-form on a vector field?

I have the one form : $dz + x\, dy$. If I want to evaluate this on a vector field, say $-\partial_{z}$, how do I do it? I don't understand what 1-forms defined with an exterior derivative do to ...
1
vote
0answers
39 views

Computing $\alpha\wedge d\alpha$ where $\alpha = dz + x\,dy$

Let $\alpha = dz + xdy$ be a one-form on $\mathbb R^3$. I would like to compute the wedge product $\alpha\wedge d\alpha$ as explicitly as possible but I am not sure if I am doing this correctly. $$d\...
3
votes
2answers
119 views

The integral of a function on manifold and differential form

When we want to integrate a function f over a manifold M, we may meet some problems, for example, the problem showed in the picture below: Then people used differential form to integrate. But it ...
0
votes
1answer
64 views

A question about pullback of the K-form

Let $M$ be an oriented $m$-dimensional manifold. Suppose the support of $\omega$ is in an open subset $U$ of $M$, and $\phi \colon U \to R^m$, $\psi \colon U \to R^m$ are two different charts on $M$ ...
3
votes
0answers
50 views

How to define integration along surfaces in $\Bbb R^4$?

In $\Bbb R^4$ we have curves, bi-dimensional surfaces and hypersurfaces. We integrate vector fields along curves and in hypersurfaces using a normal direction. I know that the adequate object to ...
1
vote
0answers
99 views

Rudin's proof on Poincare Lemma.

Two identical questions are here: A question about differential forms and here: http://mathforum.org/kb/thread.jspa?forumID=13&threadID=2141549&messageID=7208112 In his proof, Rudin says ...
4
votes
1answer
69 views

Study materials to help understand the generalized Stokes' theorem both intuitively and rigorously?

Dear MSE: My goal is to understand the generalized Stokes' theorem both intuitively and rigorously. Could someone give advice or recommend study materials to help understand the generalized Stokes' ...
0
votes
1answer
62 views

Poincare's lemma from PMA Rudin

It's the so called Poincare's theorem from Rudin's book. I read this theorem fully but I have 2 questions: 1) How did he conclude that equations (120) holds? What did he use in his reasoning? This ...
2
votes
2answers
79 views

Help Debugging a Bogus Proof

We want to prove the standard fact that a smooth function $u :R^2 \to R$ with $ \nabla u = 0$ everywhere in some connected open set $ \Omega $ is constant in that set. I'm comfortable with the usual ...
1
vote
1answer
28 views

Apply connections to a gauge transformation?

I'm reading Donaldson's book, Floer homology groups in Yang-Mills theory. On page 82, he considers a trivial bundle $P$ over a $4$-manifold $X$ with tubular ends which is equipped with a connection $...
0
votes
1answer
30 views

Domain simply connected for a differential form $\omega$

I have the domain $\Omega=\{(x,y): 2y+x>0\}$ and the differential form $$\omega=\frac{y}{2y+x}dx+\left(\log (2y+x)+\frac{2y}{2y+x}\right)dy.$$ I would like to evaluate $\int_{\gamma} \omega$ ...
1
vote
1answer
22 views

Proving a formula for the exterior derivative of a specific $k$-form, given in base representation

Let $U \subseteq\mathbb{R}^n$ be open, and let $\omega$ be an $(n-1)$-form that's given by $$\omega = \sum_{i=1}^n (-1)^{i-1} F_i\, dx_1 \wedge\dots\wedge dx_{i-1} \wedge dx_{i+1} \wedge\dots\wedge ...