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

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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
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1answer
86 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 ...
4
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1answer
62 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 ...
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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 ...
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2answers
97 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.
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34 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}_{...
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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 $M$ ...
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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 ...
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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 ...
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1answer
40 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 ...
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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 ...
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1answer
39 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}{\...
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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 ...
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2answers
59 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
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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 ...
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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 ...
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1answer
146 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. ...
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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
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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 ...
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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 ...
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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 ...
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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\...
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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(...
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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 $. ...
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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 ...
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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--...
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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 $\omega=f.dg$, ...
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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 $\wedge$...
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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 ...
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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 ...
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1answer
56 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 ...
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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\...
2
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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. ...
3
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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 ...
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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 ...
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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 $M$ ...
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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 $...
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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 ...
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1answer
92 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 ...
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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 ...
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1answer
35 views

Finding a base representation for a differential form that's given through the determinant of a matrix

Let $U \subseteq \mathbb{R}^n$ be open. I first want to show/acknowledge that the mapping $$\omega_p(v_1, ..., v_{n-1}) := det(p, v_1, ..., v_{n-1}), p \in U, v_i \in \mathbb{R}^n$$ defines a ...
2
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1answer
67 views

Isomorphisms between $\mathbb{R}^3$ and $\left( \mathbb{R}^3\right)^*$

Consider the oriented euclidean space $\mathbb{R}^3$. Every vector $A \in \mathbb{R}^3$ determines a $1$-form $ω^1_A$ by $ω^1_A(ξ)=(A,ξ), ξ\in \mathbb{R}^3$ (scalar product) and a $2$-form $ω^2_A$ by \...
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2answers
51 views

Exterior Product of 2-Forms in $\mathbb{R}^n$

So I am studying the book of Arnold, "Mathematical Methods of Classical Mechanics" and I am trying to understand the differential forms. The exterior product is defined for $1$-forms $ω_1$ and $ω_2$ ...
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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 $\int_{\gamma} \omega$ ...
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1answer
22 views

Differential forms vector space over function field?

Let $V$ be a vector space and $V^*$ be its dual space. Then I know that $V^*$ is considered a vector space because we can scale the basis covectors by real numbers and add them together and all of ...
3
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2answers
104 views

Kernel of $\omega^\#$ is $k$-dimensional

Let $M$ be a smooth manifold with coordinates $\{q^i\}_{i=1}^n$ .The variables $(q^i,p_i)$ are coordinates on the cotangent space $\Omega=T^*M$. Any cotangent space carries a natural one-form $\tilde{\...
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28 views

Inner products on $C_c^\infty$ corresponding to differential forms

Let $M$ be a compact, oriented Riemannian manifold of dimension $n$. We can locally identify smooth $k$-forms by smooth functions $\mathbb{R}^n \to \mathbb{C}^{m}$, where $m = \binom{n}{k}$ (via a ...
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0answers
62 views

Integral of an interior product is 0

Here is a question about problem 3.12 from Do Carmo's Riemannian Geometry. $M$ is a compact orientable and connected Riemannian $n$ manifold. $f$ is a differentiable function on $M$ such that $\Delta ...
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2answers
49 views

Stokes' Theorem and path integrals in $\mathbb{C}$

I have seen very short proofs of the Lemma of Goursat and Cauchy's Theorem using Stokes' Theorem. I have learnt Stokes' Theorem in the setting of (not complex) smooth manifolds as described in https://...
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1answer
55 views

Restriction of an $n$-form to a submanifold

I am currently studying exterior calculus on manifolds and am not sure if I understand things correctly. In my textbook (R.W.R. Darling, Differential Forms and Connections) there is an example of a $2$...