Questions tagged [differential-forms]

For questions about differential forms, a class of objects in differential geometry and multivariable calculus that can be integrated.

A degree $k$ differential form on a smooth manifold $M$ is a quantity that can be integrated on $k$-dimensional submanifolds of $M$.

Formally, a degree $k$ differential form is an element of $\Omega^k(M) = \Gamma(M, \bigwedge^kT^*M)$ which is the vector space of smooth sections of the vector bundle $\pi: \bigwedge^kT^*M \to M$; a section is a map $\alpha : M \to \bigwedge^kT^*M$ such that $\pi\circ\alpha = \operatorname{id}_M$. In particular, if $\alpha \in \Omega^k(M)$, for each $x \in M$, $\alpha(x) \in \bigwedge^kT^*_xM$; that is, $\alpha(x)$ is an alternating map $(T^*_xM)^k \to \mathbb{R}$.

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functoriality of derivations

I seem to have problems understanding algebraically why given a map of manifolds $f: M \to N$ we get a bundle map $TM \to f^*TN$. Now, fiberwise it's all good. But I do not understand how to define on sections, as a map of sheaves of…
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What is a form?

I have read about differential forms, bilinear forms, quadratic forms and some other r-linear forms but I still have this shred of doubt in my mind on what exactly is a form. I have an assumption that it is to a ring what a vector is to a field. I…
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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 coordinate free formulation is superior to choosing a…
madison54
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The algebraic de Rham complex

Let $A$ be a commutative $R$-algebra (or more generally a morphism of ringed spaces). Then there is an "algebraic de Rham complex" of $R$-linear maps $A=\Omega^0_{A/R} \xrightarrow{d^0} \Omega^1_{A/R} \xrightarrow{d^1} \Omega^2_{A/R} \to \dotsc$.…
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The connection between differential forms and ODE

Is there a connection between being an exact differential equation and being an exact differential form? I always found it bothersome with basic ode that you could somehow treat dy/dx as a bona fide fraction, is what's secretly going on here…
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Show that the form $w$ is closed but not exact

Let $$w~=~\dfrac{-y}{x^2+y^2}dx+\dfrac{x}{x^2+y^2}dy, \qquad (x,y)~\in\mathbb{R}^2\backslash \{(0,0)\}.$$ 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…
wwbb90
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Differential forms without derivatives?

Recently, I've drawn increasingly attracted to using differential forms for routine calculus computations - for instance, I've come to like the equation $$dy=2x\,dx$$ which clearly states that the rate of change of $y$ is $2x$ times the rate of…
Milo Brandt
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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 On the Visualisation of Differential Forms - Dan…
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Closed not exact form on $\mathbb{R}^n\setminus\{0\}$

I'd like to construct a closed but not exact $n-1$-form $\omega$ on $\mathbb{R}^n\setminus\{0\}$ in analogy to the winding form: $$\frac{x~dy-y~dx}{x^2+y^2}$$ I think something like $$\omega=\frac{\sum_{i=1}^n x_i(\star…
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Derivation of a representation through a vector field

Question: (Exercise 3.4.12 - Sharpe) Let $H$ be a Lie group, $V$ a vector space, and $\rho: H \to Gl(V)$ a representation. Let $U$ be a manifold, $X$ a vector field on $U$, and $h: U \to H$ and $f: U \to V$ smooth functions. Show that $$X(\rho…
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Good book about differential forms

I'm a looking for a good book to self-study differential forms. Particularly, I'm looking for a book that is as similar as possible to Bert Mendelson's "Introduction to topology" (i.e. a book that procede by following a: "Definition, theorem,…
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What do $dz$ and $|dz|$ mean?

I'm having a hard time understanding complex differentials. I know that when I have a field $\mathbb K$ and a $\mathbb K-$vector space $\mathbb K^n,$ then we define $dx_i\in \mathrm{Lin}(\mathbb K^n,\mathbb K)$ on the standard basis…
user23211
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How to compute $I(d\omega)$? (Poincaré's Lemma)

Suppose I have an $\ell$-form in $\Bbb R^n$ $$\omega=\sum_{i_1<\cdots
Pedro
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Visualizing Exterior Derivative

How do you visualize the exterior derivative of differential forms? I imagine differential forms to be some sort of (oriented) line segments, areas, volumes etc. That is if I imagine a two-form, I imagine two vectors, constituting a…
Yrogirg
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A closed $1$-form on a convex open set is exact

Baby Rudin Exercise 10.24: Let $\omega = \sum a_i(\mathbf x) \, dx_i$ be a $1$-form of class $\mathscr{C}''$ in a convex open set $E \subset \mathbb{R}^n$. Assume $d \omega = 0$ and show that $\omega$ is exact in $E$, by completing the following…
MCT
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