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

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86
votes
7answers
9k views

Exterior Derivative vs. Covariant Derivative vs. Lie Derivative

In differential geometry, there are several notions of differentiation, namely: Exterior Derivative, $d$ Covariant Derivative/Connection, $\nabla$ Lie Derivative, $\mathcal{L}$. I have listed them ...
76
votes
3answers
3k views

Direct proof that the wedge product preserves integral cohomology classes?

Let $H^k(M,\mathbb R)$ be the De Rham cohomology of a manifold $M$. There is a canonical map $H^k(M;\mathbb Z) \to H^k(M;\mathbb R)$ from the integral cohomology to the cohomology with coefficients ...
41
votes
3answers
860 views

Is Stokes' Theorem natural in the sense of category theory?

Stokes' Theorem asserts that for a compactly-supported differential form $\omega$ of degree $n-1$ on a smooth oriented $n$-dimensional manifold $M$ we have the marvellous equation $$\int_M d\omega = \...
30
votes
1answer
4k views

How to calculate the pullback of a $k$-form explicitly

I'm having trouble doing actual computations of the pullback of a $k$-form. I know that a given differentiable map $\alpha: \mathbb{R}^{m} \rightarrow \mathbb{R}^{n}$ induces a map $\alpha^{*}: \Omega^...
28
votes
4answers
643 views

Intuition behind an integral identity

A proof for the identity $$\int_{-\infty}^{\infty} f(x)\, dx=\int_{-\infty}^{\infty} f\left(x-\frac{1}{x}\right)\, dx,$$ has been asked before (for example, here), and one answer to that question ...
26
votes
4answers
8k views

What is a covector and what is it used for?

From what I understand, a covector is an object that takes a vector and returns a number. So given a vector $v \in V$ and a covector $\phi \in V^*$, you can act on $v$ with $\phi$ to get a real number ...
21
votes
1answer
506 views

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 ...
20
votes
2answers
658 views

Why do we care about differential forms? (Confused over construction)

So it's said that differential forms provide a coordinate free approach to multivariable calculus. Well, in short I just don't get this, despite reading from many sources. I shall explain how it all ...
20
votes
2answers
1k views

Symmetric and wedge product in algebra and differential geometry

I have been struggling with this issue for a while (and asked a similar question here), but still not found a satisfying answer. The question boils down to: which is the correct identity? $dx \, dy ...
19
votes
3answers
2k views

Geometric understanding of differential forms.

I would like to understand differential forms more intuitively. I have yet to find a book which explains how the use of the exterior product in differential forms ties into the geometrical ...
19
votes
0answers
423 views

Wanted: A purely algebraic proof of the Frobenius theorem on distributions

Is there a purely algebraic proof of the Frobenius theorem? Here's a rough sketch of what i'm looking for: Let $Der(R)$ denote the $R$-module of ($R$-valued) derivations of the algebra $R$ endowed ...
18
votes
3answers
2k views

Geometric interpretation of connection forms, torsion forms, curvature forms, etc

I have just begun learning about the connection 1-forms, torsion 2-forms, and curvature 2-forms in the context of Riemannian manifolds. However, I am finding it hard to relate these notions to any ...
18
votes
0answers
277 views

Characterization of the exterior derivative $d$

In the paper Natural Operations on Differential Forms, the author R. Palais shows that the exterior derivative $d$ is characterized as the unique "natural" linear map from $\Phi^p$ to $\Phi^{p+1}$ (...
16
votes
4answers
1k views

Where can I learn about complex differential forms?

So I'm a 3rd year grad student in number theory/modular forms/algebraic geometry, and I've worked with differential forms from an algebraic point of view without ever knowing what they really are. I'd ...
15
votes
2answers
1k views

Intuition behind $dx \wedge dy=-dy \wedge dx$

I was re-reading this old book of mine; and I noticed that in defining the rules of differential forms, it "makes sense" that we have the rule $dx \wedge dx=0$ because if $dx$ is infinitesimal, then ...
15
votes
2answers
1k views

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,\...
14
votes
3answers
686 views

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} ...
14
votes
2answers
2k views

Are Clifford algebras and differential forms equivalent frameworks for differential geometry?

I recently discovered Clifford's geometric algebra and its application to differential geometry. Some claim that this conceptual framework subsumes and generalizes the more traditional approach based ...
14
votes
1answer
1k views

Geometric Explanation of Tamagawa Numbers

Sometimes in order to understand a concept thoroughly we need to have a algebraic view ( in terms of equations ) and corresponding geometric view. My interest always lies with understanding the ...
13
votes
3answers
1k views

What is the motivation for differential forms?

I am that point in my mathematical career where I am learning differential forms. I am reading from M.Spivak's Calculus on Manifolds. So far I have gone over the tensor and wedge products and their ...
13
votes
2answers
614 views

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 ...
13
votes
0answers
768 views

Proof of residue theorem (residue formula) for differential forms on curves over an arbitrary closed field.

I have been reading the book Algebraic Geometric Codes: Basic Notions by Tsfasman, Vladut and Nogin. They give a residue formula like this: Let $\mathbb{k}$ be an algebraically closed field and $X$ ...
12
votes
2answers
374 views

checking if a 2-form is exact

Consider the 2-form $$\sigma=\frac{x_1 dx_2 \wedge dx_3 + x_2dx_3\wedge dx_1+ x_3 dx_1 \wedge dx_2}{(x_1^2+x_2^2+x_3^2)^{3/2}}.$$ I need to show if it is exact or not. Suppose it is exact, then there ...
12
votes
3answers
2k views

Inducing orientations on boundary manifolds

Given a $k$-manifold $M$, such that $\partial M$ is a $(k-1)$-manifold, there is a standard way in which $\partial M$ inherits the orientation of $M$. So if $M$ is oriented by the form field $\omega$, ...
12
votes
1answer
574 views

Coordinate free proof that curvature is the “square” of the connection

Here's the setup. Consider a vector bundle $E$ over a manifold $M$ and let $\Omega^*(M, E)$ denote the space of $E$-valued differential forms (i.e. the space of sections of the vector bundle $\...
12
votes
2answers
805 views

Is there a Stokes theorem for covariant derivatives?

A $V$-valued differential form on $M$ is a smooth map $\omega : TM \to V$ such that $\omega$ restricted to any tangent space $T_p M$ is an element of the $V$-valued exterior algebra $\Lambda^n (T_p M, ...
12
votes
1answer
245 views

Proof of holomorphic Lefschetz fixed point formula using currents in Griffiths and Harris

I am trying to understand the proof of the Holomorphic Lefschetz fixed point formula on page 426 in Griffiths and Harris. However, I find their use of currents extremely confusing. They seem to go ...
12
votes
0answers
91 views

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 $\...
11
votes
2answers
1k views

Apparent counter example to Stoke's theorem?

I think I found an apparent contradiction to Stoke's theorem with this 2-differential form $M= \overline{B^{2}}- \{ 0 \}$, $\partial M = S^1$, $$\omega = \frac{x~dy-y~dx}{x^2+y^2}$$ defined in $\...
11
votes
2answers
3k views

Cartan's magic formula

A possible proof of Cartan's magic formula $$L_X = i_X \circ d+d \circ i_X$$ is to follow the steps: Show that two derivations on $\Omega^{\bullet}(M)$ commuting with $d$ are equal iff they agree on ...
11
votes
2answers
51 views

$\text{Alt}\,(\phi_1 \otimes \phi_2 \otimes \phi_3)$

How do I write out $\text{Alt}(\phi_1 \otimes \phi_2 \otimes \phi_3)$ for $\phi_1, \phi_2, \phi_3 \in V^*$?
11
votes
1answer
1k views

Why are de Rham cohomology and Cech cohomology of the constant sheaf the same

I am comfortable with de Rham cohomology, sheaves, sheaf cohomology and Cech cohomology. I am looking to prove the following theorem: If $M$ is a smooth manifold of dimension $m$, then we have ...
11
votes
5answers
473 views

Is line element mathematically rigorous?

I know differentials (in a way of standard analysis) are not very rigorous in mathematics, there are a lot of amazing answers here on the topic. But what about line element? $$ds^2 = dx^2 + dy^2 +dz^...
11
votes
1answer
671 views

Homework: closed 1-forms on $S^2$ are exact.

From the 2008 UCLA Geometry-Topology qualifying exam: let $\theta$ be a $1$-form on $S^2$ with $d \theta = 0$. Construct a function $f$ on $S^2$ with $d f = \theta$. I'm not very confident in my ...
10
votes
3answers
739 views

Geometric intuition behind pullback?

I am having hard time with forming a geometric intuition of pullback and pushforward. The definition the book gives is like this: There are two open sets, $A$ and $B$. There is a dual transformation ...
10
votes
1answer
539 views

Writing Integrals using Differential Forms

Consider some smooth curve $C \subset \mathbb{R^n}$ and $\gamma:[a,b] \subset\mathbb{R}\rightarrow C$ a parametrisation of $C$ and a continuous vector field $K:\mathbb{R^n} \rightarrow \mathbb{R^n}$. ...
10
votes
2answers
497 views

Integral of wedge product of two one forms on a Riemann surface

I'm having trouble verifying an elementary assertion made in this answer on MathOverflow. It seems more like a math.stackexchange question, so I'm asking it here. Anyway, the assertion is as follows ...
10
votes
1answer
1k views

Good intro to differential forms

I am looking for an intro book to learn about diff forms, maybe undergrad. Reading sentences like "Let M be a smooth manifold. A differential form of degree k is a smooth section of the kth exterior ...
10
votes
1answer
986 views

Hodge Star Operator

I'm trying to understand the Hodge star operation, but have come across an impasse almost immediately. I have the definition $$(\star \omega)_{a_1\dots a_{n-p}}=\frac{1}{p!}\epsilon_{a_1\dots a_{n-1}...
10
votes
1answer
130 views

Dimension of diffeomorphism groups preserving some $2$-tensor.

For a finite-dimensional smooth manifold $M$, let $\mathrm{Diff}(M)$ be its diffeomorphism group. Suppose we are given a $2$-tensor $\mathcal{K}$ on $M$, and let $$\mathrm{Diff}_{~\mathcal{K}}(M) = \{...
9
votes
3answers
3k views

Interior product of differential forms

The interior product of a 2-form $\beta$ and vector field $X$ is defined by $(i_x\beta)(Y)=\beta(X,Y)$ where $Y$ is a vector field. This is the definition of a 2-form (and it's similar for a p-form,...
9
votes
3answers
796 views

Why are differential forms more important than symmetric tensors?

In differential geometry, differential forms are totally anti-symmetric tensors and play an important role. I am led to wonder why do we not study totally symmetric tensors as much as forms. What ...
9
votes
1answer
1k views

What's the geometrical intuition behind differential forms?

This question can look like a duplicate of this one, but it's kind of different. I'm trying to relate some geometrical meanings I've seem in some books to the definition of differential forms in $\...
9
votes
1answer
647 views

Information captured by differential forms

My advanced calculus class is currently doing differential forms and I have a hard time really understanding what they are all about. I can read the proofs of the theorems given in Rudin's PMA chapter ...
9
votes
1answer
144 views

What makes differential forms special

There are two natural structures defined on differential manifolds, namely tangent bundle and its dual cotangent bundle. An element of tangent bundle $TM$ at a base point $p\in M$ can be described by ...
9
votes
1answer
1k views

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{\...
9
votes
3answers
214 views

What is the relation between dx in elementary calculus and dx in differential geometry?

I've recently started studying differential geometry and was really hoping that in doing so I'd finally have an answer to something that's been bugging me since I first learnt calculus - what is $dx$?!...
9
votes
1answer
377 views

Maurer-Cartan 1-form

Can anyone help me with the following? Let $\rho$ be the right-invariant Maurer-Cartan 1-form $$\rho = dg\ g^{-1}$$ I want to show that the MC equation $$d\rho - \rho \wedge\rho = 0$$ holds. So ...
9
votes
0answers
185 views

algebraic $1$-forms vs analytic $1$-forms

First let's fix some definitions: Definitions: Complex manifold (of dimension n): Is a locally ringed space $(X,\mathscr F)$, where there is an open cover $\bigcup_{i\in I} U_i=X$ such that $(U_i,\...
8
votes
3answers
197 views

Interesting question in differential geometry

Let $ \alpha $ be a closed $ 3 $-form on $ \mathbb{R}^{4} \setminus \{ 0 \} $. Let $ i: S^{3} \hookrightarrow \mathbb{R}^{4} $ be the canonical embedding of $ S^{3} $, and suppose that $ \Omega := {i^{...