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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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Homogeneous 1st Order Differential Equations

I have come across two different definitions of a homogeneous 1st order ODE; that the equation can be written in the form $y' = f(\frac{y}{x})$, and that in the form $M(x, y)dx + N(x, y)dy = 0$, it is ...
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Why is $K_{\bar X}=\pi^* K_X +\sum a_i E_i$?

Let $\pi:\tilde X \to X$ a blow up with exceptional divisors $E_i$. Why is it that $$K_{\bar X}=\pi^* K_X +\sum a_i E_i?$$ I know that in the case of $X$ a complex manifold, this follows simply from $...
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Finding an explicit formula for a Hamiltonian vector field

I've been looking at this question: Existence of vector field given a smooth function That is: Given a symplectic manifold $M$ of dimension $2n$, with a symplectic form $\omega \in \Omega^2(M)$, do ...
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Exact Differential Equation Integrating Factor

Finding an integrating factor can be a genuine mathematical art. However, certain differential forms can remind us of differentiation techniques that may aid in the solution of the equation at ...
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36 views

Understanding Stokes' Theorem Proof

I'm new to differential forms and i'm having trouble understanding Generalized Stokes' Theorem. The Theorem requires a (n-1)-form to be compactly supported on $M$, so it vanishes near the boundaries ...
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Holomorphic coordinates on Riemann surfaces

I have a big problem understanding the meaning of holomorphic coordinates on Riemann surfaces, especially in relation to 1-forms. Holomorphic coordinates on a Riemann surface $X$ is an open set $U \...
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27 views

Showing a 1-form on the sphere $S^2$ can not be obtained from exterior derivative

I have the following problem: Can the vector field $X(x,y,z)=(-y,x,0)$ on $S^2$ be the gradient (on the sphere) of a function $f:S^2\rightarrow \mathbb{R}$ with respect to the standard euclidean ...
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57 views

Canonical bundle of blow up at singular point

Let $X$ be a complex variety/ manifold with one singular point $x_0\in X$. If we blow up $X$ at $x_0$, we obtain a smoot variety/manifold with exceptional divisor $Y$. How can we calculate the ...
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Relation between tame symbol and residue on a curve

For an discrete valuation field $K$ we can define the tame symbol: $$(\,,\,)_K:K^\times\times K^\times\to \overline K^\times$$ $$(a,b)\mapsto(-1)^{v(a)v(b)}\overline{a^{v(b)}b^{-v(a)}}$$ Consider ...
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38 views

Codifferential / divergence of differential form under conformal metric change

I have a question related to this and a second post. I want to calculate the codifferential under a conformal metric change, $g_\psi = e^{2\psi} g$. By Besse's book on Einstein manifolds, or an ...
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Why does ${\partial\over\partial t} w_i(tx)=\sum_{j=1}^n{\partial\over\partial x_j}w_i(tx)x_j$?

Let $w$ be a $1$-differntial form. Why does the equality holds? $${\partial\over\partial t} w_i(tx)=\sum_{j=1}^n{\partial\over\partial x_j}w_i(tx)x_j$$
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Arithmetics of '$\wedge$' and '$d$' operators

I don't find arithemtic rules of the operators $\wedge$ and $d$. For example, why does this equality hold? $$ \\ (u^2\cos^2v+u^2\sin^2v)[\cos vdu-u\sin vdv]\wedge [\sin vdu+u\cos vdv] \ \\ +u\cos v[\...
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Why does $D\pi^i(p)(v)=v^i$?

There's an equality that I don't understand from Spivak' book "Calculus on Manifolds" (p. 89). We define $$ df(p)(v_p)=Df(p)(v) $$ Let us consider in particular the 1-forms $d\pi^i$. [...] ...
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gradient of $x^tAy$ with respect of $y$ and gradient of the Euclidean norm.

I was reading this paper where they want to find the saddle point of this equation: $$\text{min }\text{max }\{x^TAy + \frac{\lambda}{2} ||By−z||^2\}$$ Where $x,y,z$ are vectors and $A,B$ are ...
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56 views

hodge star and pull back

Let $\phi:\mathbb{R}^n\to\mathbb{R}^n$ be an orthogonal linear map. Prove that $\phi^*(*\alpha) = *\phi^*(\alpha)$ for all $k$-forms $\alpha$ on $\mathbb{R}^n$. I tried to write out $\phi^*(*\alpha)$ ...
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Differentials and Integration

I have been informed that consecutive differentials in iterated integral problems are actually connected via the exterior product. So the factor $dx\ dy$ in $\int\int x^2\ dx\ dy$ is actually the ...
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Tensor and the $k$ times derivative of a smooth function $f$

I am currently taking an university analysis class. I learned yesterday in the class that (1) if $f: \mathbb{R^n} \rightarrow \mathbb{R}$ is smooth, then $D^kf: \mathbb{R^n} \rightarrow T^k(\mathbb{R^...
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Partial Derivative Being Treated As Full Derivative

In this Khan Academy video (https://youtu.be/YT6XwkcPcsw?t=138), Sal takes partial derivatives of several dependent variables. When taking the full derivative of $Q(x,\ y,\ z(x,\ y))$ with respect to ...
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45 views

What is really an exact differential and how does it relate to conservative fields

If there is a differential form $ A(x,y,z) dx + B(x,y,z) dy + C(x,y,z) dz$ where there exists some function $\psi(x,y,z)$ Let $ \psi = \psi (x,y,z)$ Then the total differential is $ d \psi = \left(\...
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42 views

condition for one-form to be exact differential

Simple question I suppose: Having a (smooth) one-form $$\omega=\omega_i dx^i\in \Lambda(M)$$ Is there a test to find out if $\omega$ is exact differential? Or more precisely that there exists a ...
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Singularity of the derivative of a local homeomorphism

If I have a local homeomorphism $D$ defined in the universal covering of the deleted disk $\mathbb{D}^*\subset\mathbb{C}$ has finite degree in a fundamental domain, then 0 is a pole of $f = D'$. I ...
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$F = \nabla f $ for some smooth $f: B \rightarrow \mathbb{R}$

I am currently taking an university analysis class, and the professor of my class proved the theorem below yesterday. If $B \subset \mathbb{R}^2$ open ball, $F: B \rightarrow \mathbb{R}^2$ is smooth, ...
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Questions about compact manifolds

I have two question. Let $M$ and $N$ two compact manifolds. 1) It is true that $C^{\infty}(M\times N)\cong C^{\infty}(M)\otimes C^{\infty}(N)$??. 2) Taking $f$ $\in$ $\Omega^1(M)=\Gamma(T^{\ast}M)$ ...
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Generalized Poincaré Lemma

I'm reading the proof of an improved version of Poincaré's Lemma on Ana Cannas da Silva's Lectures on Symplectic Geometry, page 40. I am terribly confused. Here's the setup: $U_0$ is a tubular ...
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Volume of a 3D torus as a form on a quotient space

I'm trying to calculate the volume form $dx\wedge dy\wedge dz$ on $\mathbb{T}^3=\mathbb{R}^3/\mathbb{Z}^3$: $\int_{\mathbb{T}^3}dx\wedge dy\wedge dz$ I've been told that it's simply the volume of ...
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36 views

Characterizations of Riemannian Volume Form

I'm trying to understand how some characterizations of the Riemannian volume form $dV$ are equivalent on an oriented Riemannian manifold of dimension $n$. I'm a bit new to Riemannian geometry (and ...
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What manifold structure is natural for the codomain of a differential form defined on a manifold?

It is well known that a differential form $\omega$, of degree $r$, defined in a manifold $M^m$ is an application that at each point $p\in M^m $ associates an alternating $r$-linear form $\omega(p)\in ...
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Differential 1-form of line

This problem is from V.I Arnold's book Mathematics of Classical Mechanics. Q) Show that every differential 1-form on line is differential of some function Relevant equations The differential of any ...
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(Rudin PMA 10.16)

I was studying Chapter 10 in Rudin's PMA. This is my first time stduying about differential forms, so my question may be not so good. My question is about the two red-underlined statements here. ...
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Why are these three equivalence relations special? [closed]

Consider the set of all possible bivectors $\mathfrak{B}$ in $\mathbb{R}^3$. Then there are three possible equivalence relations. Equipollence: The equivalence relation $(\mathfrak{B}, \sim)$ such ...
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Interpretation of stokes theorem

I recently solved the following task: Let $A = [0,1]^3$ and $\omega = \dfrac{x_1^2 x_2^3}{1+x_3^2} \ dx_1 \wedge dx_3$ Show that this fulfills stokes theorem by showing that $\displaystyle \int_A \ d\...
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Maxwell-Chern-Simons equations: Translating from form language to component form

I am trying to solve the scalar-coupled Maxwell-CS equations, which is written in this form in the differential form language \begin{eqnarray} 0=d(Q_{IJ}\star F^J)+\frac{1}{4}C_{IJK}F^J\wedge F^K \...
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What's going on when we compute $d(\gamma(z)) = \frac{1}{|cz+d|^2}dz$, where $\gamma \in \operatorname{SL}_2(\mathbb Z)$

Let $\gamma = \begin{pmatrix} a & b \\ c & d \end{pmatrix} \in \operatorname{SL}_2(\mathbb Z)$. Consider the space $\Omega^1(\mathbb H)$ of smooth complex $1$-forms on $\mathbb H$. These ...
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How to relate two definitions of space of 1-forms?

I am trying to understand how to connect following quote from S. Carroll's "Spacetime and Geometry" with another definition of space of 1-forms from this book. Roughly speaking, the space of one-...
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Algebraic forms on an elliptic curve

On an elliptic curve defined by the equation, $$E:y^2=x^3+a x +b$$ The algebraic form $dx/y$ is defined on the elliptic curve and it is a non-vanishing section of the (trivial) canonical bundle. From ...
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A computation with derivations and skew-derivations

On a smooth manifold $M$ a derivation $D$ of degree $k$ is an endomorphism of the algebra of forms defined on $M$, which is denoted by $\Omega(M) = \bigoplus_{k \geq 0} \Omega^k(M)$, such that $D\...
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Local frame inducing a map of principal bundles

Let $V \rightarrow M$ a vector bundle. $P \rightarrow M$ a principal $G$-bundle. Let $\phi:G \rightarrow GL(V)$ be a representation. A local section $s$ for $P$, frame bundle for $V \rightarrow M$ , ...
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Maurer Cartan $1$-form explanation of notation

This is the construction I am given: Let $G$ be a Lie group, $\mathfrak{g}:=T_eG$ its Lie algebra. We define $1$-form $w_G \in \Omega^1(G) \otimes \mathfrak{g}$, i.e. closed $1$-forms with value in ...
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Rigorous proof of the Cauchy-Goursat theorem using Green's theorem

A special case of the Cauchy-Goursat theorem says that if $f: \Omega \to \mathbb{C}$ is continuously holomorphic and $\gamma \subseteq \Omega$ is a closed curve whose interior is contained in $\Omega$,...
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Worked examples of Lie derivatives

I'm trying to find the Lie derivative of a 2-form $\sin(\theta)d\theta \wedge d\phi$ with respect to a vector field given in a differential basis $a \partial/ \partial \phi$ and I think the way to go ...
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“Dividing through by Dp” in Joule Thomson effect

In the wikipedia page for the Joule Thompson Effect, it says: I understand that the first formula is a 1-form on a 2-manifold. I don't understand why "Dividing by dP" is legal in this case. Is there ...
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On the total derivative of a differential form.

I look for an example of differential form $\omega$ of degree $r$ defined on a surface $ M \subset \mathbb{R}^n $ of dimension $m$ such that its derivative $D\omega$ can not be defined on the entire ...
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differential forms- $\omega $ closed but not exact

let be $$ \omega= |x|^{-3} \left(x_1 dx_2 \wedge dx_3+x_2dx_3 \wedge dx_1 + x_3dx_1 \wedge dx_2\right) $$ and $G:= \mathbb{R}^3 \backslash \{ 0 \} $ I want to prove, that $ \omega$ is closed, but ...
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Differential form ''equation''

I am having a bit of a trouble with the following. I'm working in the homogeneous Lie Group $\mathbb{R}\ltimes \mathbb{R}^3$ with an specific bracket an it give me de following system to integrate and ...
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Building Intuition for Differential forms, exterior derivative, wedge

I think I understood 1-forms fairly well with the help of these two sources. They are dual to vectors, so they measure them which can be visualized with planes the vectors pierce. Gravitation 1973 ...
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Multivariable integration $\int_{y_0}^{y} \alpha (x,y) dy + \int_{x_0}^{x} \beta (y,z) dx = 0$

We got two functions given as: $\alpha(x,y)=a_1 + a_2\Delta y + a_3\Delta x$ $\beta (y,z)= b_1 + b_2\Delta y +b_3\Delta z$ and I need to figure out if the coefficients $a_i$ and $b_i$ ...
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A manifold admits a nowhere vanishing volume form if and only if it is orientable?

on this wikipedia article, it is said : A manifold admits a nowhere vanishing volume form if and only if it is orientable I don't really understand why. Isn't $dx^1 \wedge ... \wedge dx^n$ ...
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50 views

How to prove that the definition of exterior product of differential forms is not ambiguous?

In page 91 of book A Visual Introduction to Differential Forms and Calculus on Manifolds the exterior product of two differential forms $\alpha \in \bigwedge^{r}(\mathbb{R}^n)$ and $\beta \in \...
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$d(I\omega) + I(d\omega) = \omega$ for differential forms

Let $U$ be a convex connected and open set in $\mathbb{R}^n$, such that $0\in U$. For every $k$-differential form $\omega$, $$\omega =\sum_{i_1\lt\dots\lt i_k} c_{i_1,\dots,i_k}(x)dx{_{i_1}}\wedge\...
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How to determine an integral of a differential form?

Let $$ \eta = x^2 \, dy \wedge dz + yx\,dz \wedge dx + z^3 \, dx \wedge dy $$ Can you show me how to calculate : $$ \int_{\Phi} \eta, $$ where $ \Phi $ is supposed to be the parametrization of ...