Questions tagged [differential-forms]
For questions about differential forms, a class of objects in differential geometry and multivariable calculus that can be integrated.
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Pullback of Differential Forms
In exercise 47 from Gauge Fields, Knots and Gravity by Baez and Munain, we want to show that if $\phi:M\to N$ is a map of smooth manifolds, then there is a unique pullback map on forms $$\phi^*:\Omega(...
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Local structure of symplectic manifold
Algebraic geometry, in a naive setting, could be described as the study of spaces that locally are the solutions of systems of polynomial equations.
Similarly, locally any smooth manifolds can be ...
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Uniqueness of Differential form in Spivak
I have a question related to a proof in Spivaks "Calculus on Manifolds". On page 117 he defines what $d\omega$ ought to mean on a manifold:
$$ d\omega(x)(v_1, \ldots, v_{p+1}) = d(f^*\omega)(...
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Solve the differential equation with the corresponding replacement
So I have this problem from the Elementary Differential Equations by Kells:
Solve this differential equation with the corresponding replacement:
$$
(st+1)t\cdot ds+(2st-1)s \cdot dt=0$$
with
$$z=st$$
...
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Why is $\Omega$ used to denote the space of differential forms?
A simple curious question about notation. The space of $k$-differential forms over a manifold $M$ is typically denoted by $\Omega^k(M)$. Does anybody know why, or where this notation originated? It ...
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A proof for bivector decomposition
Let $V$ be an $n$-dimensional vector space over some field $\mathbb F$. I'm interested in the following result:
For every bivector $\alpha\in\bigwedge^2 V$, there exists a linearly independent set $S=...
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What is the difference between $dx/x$ and $d(\log x)$?
I am not comfortable with differential forms and do not understand what $d (\log x)$ represents. My understanding is that since
$$
\frac{d}{dx} \log(x) = \frac{1}{x}
$$
then we can write
$$
d \log x = ...
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Restriction of Differential Form On a Connected and Compact Submanifold
Show that, conversely, if $M \subset \mathbb{R}^3$ is a compact and connected submanifold with the proper
$$
\left.(x d x+y d y+z d z)\right|_M=0,
$$
then $M$ is one of the spheres centered at the ...
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Reconciling two characterizations of symplectic potential
The reference text is N.M.J. Woodhouse's Geometric Quantization.
Let $Q$ be a smooth manifold with coordinates $\{q_i\}$, and $M=T^*Q$ be its cotangent bundle with coordinates $(p_i, q_i)$ (so that $\{...
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Confused about "k-form"
Lee's Riemannian Manifolds: An Introduction to Curvature states the following on page 14:
We let $\Lambda ^k (V)$ denote the space of covariant alternating
$k$-tensors on $V$, also called $k$-...
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u-substitution shows 0=1
This question/observation is inspired by the integral:
$$\int_0^{\sqrt{\pi}}x\sin(x^2)\cos(x^2)dx$$
The $u$-substitution $u=\sin(x^2)$ yields $du=2x\cos(x^2)dx$ and
$$\int_0^{\sqrt{\pi}}x\sin(x^2)\cos(...
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How can partial derivatives of a function be expressed as components of a covector field independently of coordinates?
At the end of page 280 of Lee's Introduction to Smooth Manifold the author states
Although the partial derivatives of a smooth function cannot be interpreted in a
coordinate-independent way as the ...
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How can I interpret this object? Is it a 1-form?
THE PROBLEM
I am dealing with the following object:
$$I = \displaystyle\int_C f(x,y)\,dxdy,$$
where $C$ is a curve on the Cartesian plane and $f: \mathbb{R}^2\rightarrow \mathbb R$ is any Riemann ...
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Implicit function theorem / Implicit selections when Jacobian not invertible
I saw the attached result in the book by Dontchev and Rockafellar.
It requires the Jacobian to be of full rank m. I suspect this condition can be further relaxed. Assume that we know that the columns ...
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Rewriting a complex differential form in real coordinates
Considering the one-form $$\alpha=\bar{z}_1dz_1+\bar{z}_2dz_2$$ appeared in this question, I was trying to rewrite this in real coordinates by taking $z_1=x+yi$ and $z_2=z+wi$. So, we have $$\alpha=(x-...
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Rewriting FTC to look like Stoke's Thm
The Fundamental Theorem of Calculus states
$\int_{a}^{b} f' \ = f(b)-f(a)$
If I define $\frac{df}{dx}:=f'$ and $\int_{a}^{b} f \, dx \ := \int_{a}^{b} f \ $, then I can rewrite above as
$\int_{a}^{b} \...
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Differentiating a pullback along a family of curves
Say I have a family of curves $x_s : [0,1] \longrightarrow M$ where $s \in (-\epsilon, \epsilon)$ is my family's parameter, and $M$ is a manifold (which, for all purposes being, we can assume to be $\...
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Proof of fundamental lemma of Riemannian geometry in the 1-form connection formalism
I'm trying to understand the proof of this version of the fundamental lemma of Riemannian geometry.
Let $\pi : \mathcal{F}_{on}(M) \rightarrow M $ the orthonormal frame bundle of an n-dimensional ...
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Why is integration of differential forms defined in this way?
From what I can say, in most books about differential geometry and differential forms (see for example Flanders, Differential Forms with applications to the Physical Sciences), the integral of a $p$-...
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Equivalence of p-vectors and skew-symmetric tensors
It is said (e.g. Lovelock and Rund) that p-vectors are equivalent to skew-symmetric tensors. However, a skew-symmetric form, such as $A_{ijk}$, is a specific form on $\mathbb{R}^n\times\mathbb{R}^n\...
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interior product of smooth vector field and k-form is smooth
Exercise (Lee Smooth Manifolds Exercise 14.22):
Let $M$ be a smooth manifold and $X$ be a smooth vector field on $M$, $\omega$ a smooth differential $k$-form. $\omega \in \Omega^k(M)$.
Show, that $i_X\...
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Converting a differential form to a measure
so today I was looking at the Generalized Stokes' Theorem:
\begin{align}
\intop_{\Omega} d\omega=\intop_{\partial\Omega}\omega\ \ ,
\end{align}
where $\Omega$ is some region, and $\omega$ is a ...
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Harmonic L2-Forms
Let $(M,g)$ be a geodesically-complete Riemannian manifold. It is well-known that every harmonic $L^{2}$-function on $M$ is necessarily constant. The way to prove this is to show that every harmonic $...
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What is the difference/relationship between the gradient and the Jacobian?
What is the difference/relationship between the gradient and the Jacobian?
I think it has to do with vectors/covectors, tangent spaces / contangent spaces, but I'm not sure what's going on. (A gentle ...
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Exterior algebra question
I don't know how to approach this question from Flanders' Differential Forms. I see it was discussed here, but I don't believe that argument is correct and would apply to something like $2v_1 \wedge ...
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Linear dependence and nonzero p-vectors
If a, b, and c are linearly dependent, then it's easy to show that $ a\wedge b\wedge c = 0$. However, if a, b, and c are linearly independent, how do you show that $ a\wedge b\wedge c \ne 0$?
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why do pullbacks of closed forms along homotopic functions only differ by an exact one?
I am trying the get a deeper understanding of the following lemma:
Let $V \subset \mathbb{R}^{n+1}$ be open; and let futhermore $U \subset \mathbb{R}^{n}$ be open with $U \times [0,1] \subset V$. We ...
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Intuition behind curvature 2-forms
I recently asked about the intuition behind connection 1-forms on pricipal bundles (Intuition behind connection 1-forms and Ehresmann connections). Thanks to the phenomenal answer I received, I now ...
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Does derivative assigns diffrential?
So there are 3 main definitions of derivation in 3 different contexts.
Calculus of one variable real functions.
Say we have an everywhere differentiable function $f: \mathbb{R} \to \mathbb{R}$.
Then ...
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Compute the induced homomorphism on deRham cohomology
Consider the smooth map on the torus to itself $f: S^1 \times S^1 \rightarrow S^1 \times S^1$ defined by $f(z_1, z_2) = (z_1^2z_2, z_1^{-1}z_2)$ (here we identify $S^1$ as the unit circle on the ...
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Integral of an $n$-form with compact support.
Setup:
Let $M^n$ be an oriented manifold and let $$\mathcal{A} := \{(U_i,\varphi_i):i \in I\}$$ be a positively oriented atlas ($\varphi_i:U_i \to \varphi_i(U_i)$ preserves orientation). Furthermore, ...
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Commutativity relation on forms [closed]
Working through Spivak's Calculus on Manifolds on page 79, is this a typo?
$$
\omega \wedge \eta = (-1)^{kl} \, \eta \wedge \omega
$$
Working through the identity I'm getting instead:
$$
\omega \...
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Coordinate expression for the codifferential of a $p$-form
I have been having difficulty obtaining the component expression of the codifferential of a $p$-form found on Wikipedia and in the book Riemannian Geometry and Geometric Analysis by Jürgen Jost. Let ...
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Geometric meaning of the rising and the lowering of indices
I'm an undergraduate student and currently I'm approaching tensorial calculus. I was wondering: is there some geometric meaning to the operation of rising/lowering indices (and then if there was any ...
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Cyclic rule for graded differential forms
I know that if I have a zero 3 form on a manifold, say $f= dx^a dx^b dx^c f_{abc} = 0$ then it follows that its coefficients obey the cyclic permutation relation: $f_{abc} + f_{bca} + f_{cab} = 0$. ...
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Preserving the symplectic 2-form vs phase space volume
Say I have a Hamiltonian system of $N$ particles in 3D-3V phase space. I'm using some sort of update scheme taking the system from $t^{n-1}$ to $t^{n}$ to $t^{n+1}$. I want to know if the update ...
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Define a differential $1$-form on the cylinder which is closed but not exact.
Problem: Define a differential $1$-form on the cylinder $$C = \{ (x,y,z) \in \mathbb{R}^3: x^2 + y^2 = 1 \} \subset \mathbb{R}^3 $$ which is closed but not exact.
My attempt (EDITED):
I am using $\...
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Is my understanding of dX accurate?
When writing $df$, for a differentiable function $f:\mathbb{R}^{n} \to \mathbb{R}$, I know rigorously speaking, it is a one form, such that $df(p)(v_{p}) = Df(p)(v)$, where $Df(p)$ is the linear ...
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How to prove the existence of differential forms on a manifold using de Rham cohomology?
Let $S^3$ be the 3-sphere, and $\Sigma$ be a 2-dimension manifold. Let $\omega$ be a 2-form on $\Sigma$. $f:S^3\rightarrow \Sigma$ is a $C^{\infty}$ map.Then there is a 1-form $\alpha$ on $S^3$ such ...
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If $\alpha\wedge d\alpha$ is a volume form, there exists a vector field $X$ such that $i_X\alpha\equiv1$ and $i_X (d\alpha)\equiv0$.
I'm currently stuck on the following problem:
Let $\alpha$ be a 1-form on a connected 3-manifold $M$ such that $\alpha\wedge d\alpha$ is a volume form. Show that there exists a vector field $X$ on $M$...
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Intuition: Topological notion of genus and Genus as the dimension of the complex vector space of holomorphic differentials
In my self-study of Lec.7 of Belyi Maps and Dessins d'Enfants, I came across the following statement
Definition 3: Genus can be defined as the dimension of the complex vector space of holomorphic ...
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Generalize the metric $g_s(r)$ to the class of surfaces $S$
Let $M=(0,1)^n$ and take $(M,g_n)$ where $g_n$ is the usual Euclidean metric. Let's assume that in dim. $n$ we want a smooth codimension one foliation of $M$ (can have exactly one singular leaf) whose ...
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How is the fundamental theorem of calculus dependent on orientation?
Using standard Lebesgue integration, we can write:
\begin{equation}
\int_{(a,b)}f'(x) d\lambda = f(b) - f(a)
\end{equation}
There's no orientation on the left hand side of the equation, yet on the ...
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Proving isotropy of a groupoid multiplication graph in a quasi-presymplectic groupoid
I'm studying quasi-presymplectic groupoids and I've come across the following proposition which I'm finding challenging to prove. Any help or guidance would be greatly appreciated.
Given a Lie ...
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On Marsden's 'Introduction to Mechanics and Symmetry' Exercise 5.2-3. (symplectic map is immersion)
I'm either confused with the definition of symplectic forms / immersions or the way exercise 5.2-3 in Marsden's 'Introduction to Mechanics and Symmetry' was stated.
It reads as follows
Exercise 5.2-3....
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Finding a holomorphic 1-form in one chart that extends to a nearby coordinate domain.
I know that for a compact Riemann surface of genus $g$, there should be $g$ different globally defined holomorphic 1-forms. However, I am having trouble finding any in my case.
Here is the set up:
Let ...
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On Marsden's 'Introduction to Mechanics and Symmetry' Exercise 4.3-3. (orientability of product manifold)
Given two differentiable manifolds $\mathcal M$ and $\mathcal N$ I needed to show that
$$\mathcal M, \; \mathcal N \mbox{ orientable } \Rightarrow \mathcal M \times \mathcal N \mbox{ orientable.}$$
...
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How to calculate $ \lim\limits_{t \to 0}\frac{1}{t^2}\int_0^t f(g_1(x,t),g_2(x,t))\frac{\partial g_1}{\partial x_1}(x,t)-samefunction(x,0)dx$
How can it calculated $ \lim\limits_{t \to 0}\frac{1}{t^2}\int_0^t f(g_1(x,t),g_2(x,t))\frac{\partial g_1}{\partial x_1}(x,t)-f(g_1(x,0),g_2(x,0))\frac{\partial g_1}{\partial x_1}(x,0)dx$ ,where $f,...
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Wedge product defined on not alternating tensors?
I am currently reading Calculus on Manifolds by Spivak. In there, it defined wedge product as follows
To determine the dimensions of $\Lambda^k(V)$, we would like a theorem analogous to Theorem 4-1. ...
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1
answer
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Confusion on the proof of showing that a differential form is not exact
I am trying to solve a classic problem of showing that the form $\omega = \frac{xdy - ydx}{x^2+y^2}$ on $\mathbb{R}^2 \setminus \{ (0,0) \}$ is not exact. However, I have some questions regarding the ...
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