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

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4
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5answers
2k views

How to calculate gradient of $x^TAx$

I am watching the following video lecture: https://www.youtube.com/watch?v=G_p4QJrjdOw In there, he talks about calculating gradient of $ x^{T}Ax $ and he does that using the concept of exterior ...
18
votes
4answers
4k 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 ...
8
votes
1answer
539 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 ...
4
votes
1answer
994 views

Problem book on differential forms wanted

I want to get used to differential forms. Thus I would like to solve a bunch of problems, especially on integration of differential forms. So I need a collection of problems with answers/solutions, ...
4
votes
3answers
660 views

When does a null integral implies that a form is exact?

It is trivial to prove that the integration of a $(n-1)$ exact form on the boundary of a $n$-manifold is 0. What about the contraposative ? If the integration of a $(n-1)$-form on the boundary of a ...
0
votes
1answer
71 views

Deriving generators for $H^1(T)$: what are $dx$ and $dy$?

By trial and error I found that $dx,dy$ are generators of $H^1_{dR}$ of $T=S^1\times S^1$. Verifying that they generate the first cohomology group is not difficult. My problem is: I found them by ...
62
votes
6answers
5k 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 ...
15
votes
2answers
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 ...
6
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1answer
469 views

Reference for Lie-algebra valued differential forms

I am learning about vector-valued differential forms, including forms taking values in a Lie algebra. On Wikipedia there is some explanation about these Lie algebra-valued forms, including the ...
14
votes
1answer
2k 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^{*}: ...
7
votes
2answers
334 views

Deriving BAC-CAB from differential forms

I've recently begun reading up on differential forms in a physics context, and my resources said that one can often derive vector identities from differential forms. For instance, $\nabla \cdot ...
4
votes
2answers
172 views

Why do all solutions to this equation have the same form?

In this paper on page 45, the authors state that Let's assume we know that $$ w \times dw = d\varphi + \sum_{j=1}^3\alpha_jdx_j, \tag{1}$$ where $\varphi \in H^1(\mathbb{R}^3, \mathbb{R})$, ...
2
votes
3answers
72 views

Generators of $H^1(T)$

Let $T$ denote the torus. I am working towards an understanding of de Rham cohomology. I previously worked on finding generators for $H^1(\mathbb R^2 - \{(0,0)\})$ but then realised that for better ...
2
votes
1answer
101 views

Proving $[L_X,i_Y]=[i_X,L_Y]=i_{[X,Y]}$

Let $X,Y$ be vector fields. $L_X$ is the Lie derivative and $i_X$ is the contraction of a $k$-form. I am really stuck on how you could prove the identity $[L_X,i_Y]=[i_X,L_Y]=i_{[X,Y]}$. Update: I ...
1
vote
0answers
59 views

Poincaré Lemma problems and computing contractions in an economical way

Let $x=(A,B,C,D)$ be coordinates on $\mathbb{R}^4$. $\displaystyle \beta = \frac{(AdB-BdA)\wedge(dC \wedge dD)+(dA \wedge dB)\wedge(CdD-DdC)}{(A^2+B^2+C^2+D^2)^2}$ I would like to compute ...
0
votes
1answer
64 views

Understanding the definition of a pullback of a differential $k$-form and applying it in $1-d$

I am having trouble understanding the definition of a pullback of a differential k-form in a basic course in differentiable geometry. This is the definition I am given. I believe it is easier to ...
10
votes
5answers
313 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 ...
1
vote
1answer
53 views

Compute $\int_cd\omega$ and $\int_{\partial c}\omega$

Question: Let $c:I^2\rightarrow\mathbb{R}^3$ be the singular $2$-cube given by $$c(s,t)=\left(\frac{1}{2}s^2,st,\frac{1}{2}t^2\right)$$Let $x=(x,y,z)$ denote the cartesian coordinates on ...
71
votes
2answers
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 ...
16
votes
3answers
967 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 ...
17
votes
2answers
598 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 ...
10
votes
1answer
419 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 $K:\mathbb{R^n} \rightarrow \mathbb{R^n}$. ...
14
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 ...
11
votes
3answers
755 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 ...
8
votes
1answer
243 views

When can a functional be written as the integral of a 1-form?

Let a real, smooth manifold $M$ be given. Let $\Gamma$ denote the set of all path segments on $M$, namely the set of all paths of the form $\gamma:[a,b]\to M$. Let $Q:\Gamma\to\mathbb R$ be a ...
8
votes
1answer
572 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 ...
7
votes
1answer
566 views

how to understand the tensor product canonical line bundle $\otimes$ dual bundle

Suppose we have a Riemann surface $M$ together with a holomorphic vector bundle $E \to M$ of rank n. let $K$ denote the canonical line bundle and let $E^*$ denote the dual bundle I am trying to ...
6
votes
3answers
452 views

$\Omega=x\;dy \wedge dz+y\;dz\wedge dx+z\;dx \wedge dy$ is never zero when restricted to $\mathbb{S^2}$

We define a 2-form $\Omega$ on $\mathbb{R^3}$ by $\Omega=x\;dy \wedge dz+y\;dz\wedge dx+z\;dx \wedge dy$. How can I show that $\Omega|_\mathbb{S^2}$ is nowhere zero? Before proving that how can I ...
5
votes
0answers
76 views

show that $\omega$ is exact if and only if the integral of $\omega$ over every p-cycle is 0

Let $M$ be an oriented smooth manifold and $\omega$ a closed $p$-form on $M$. Show that $\omega$ is exact if and only if the integral of $\omega$ over every $p$-cycle is $0$. In particular, how to ...
3
votes
3answers
399 views

Exterior derivative of a complicated differential form

Let $\omega$ be a $2$-form on $\mathbb{R}^3\setminus\{0\}$ defined by $$ \omega = \frac{x\,dy\wedge dz+y\,dz\wedge dx +z\,dx\wedge dy}{(x^2+y^2+z^2)^{\frac{3}{2}}} $$ Show that $\omega$ is closed but ...
11
votes
2answers
612 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, ...
7
votes
2answers
188 views

Interpretation of $p$-forms

Let $M$ be a smooth manifold, let $C^{\infty}(M)$ be set of all smooth functions from $M$ to $\mathbb R$ and let $Vec(M)$ denote the set of all vector fields on $M$. A $1$-form on $M$ is a ...
6
votes
3answers
797 views

Differential Forms and Vector Fields correspondence

Barrett O'Neill's Differential Geometry book says that Classical vector analysis avoids the use of differential forms on $\mathbb{R}^3$ by converting 1-forms and 2-forms into vector fields via the ...
5
votes
1answer
300 views

Integral Definition of Exterior Derivative?

Is there a rigorous integral definition of the exterior derivative analogous to the way the gradient, divergence & curl in vector analysis can be defined in integral form? Furthermore can it be ...
5
votes
2answers
364 views

Why is arc length not a differential form?

I read that the arc length is not a differential form. But I don't understand why it isn't. I understand that differential forms are integrands and arc length is an expression which is integrable. ...
5
votes
2answers
876 views

Relationship Between Differential Forms and Vector Fields

I am trying reach an understanding of precisely how the space of differential forms is related to the space of vector fields. These are the definitions that I understand and am using for these ...
4
votes
2answers
225 views

Orientations on Manifold

This is a very basic definition of orientable and very basic example 20.5 however ı could not understand definition in an good way so ı want you to explain my green writing please :) and my example ...
2
votes
1answer
81 views

Why generalize vector calculus with $k$-forms instead of $k$-vectors?

The motivation usually given to differential forms is that they generalize vector calculus nicely. That's true, but there are also $k$-vectors, i.e., objects from $\Lambda^k(V)$ instead of ...
10
votes
2answers
286 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 ...
9
votes
1answer
334 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 ...
6
votes
1answer
178 views

Poincare dual of unit circle

I'm trying to self-study Differential Forms in Algebraic Topology by Bott and Tu. I've come across this exercise: Show that the closed Poincare dual of the unit circle in $ R^2-\{0 \} $ is zero, ...
4
votes
1answer
72 views

Is there an intuitve motivation for the wedge product in differential geometry?

I've recently started studying differential forms and have been looking at differential forms. I'm struggling to understand the motivation for introducing the notion of the wedge product. Does it ...
7
votes
2answers
1k 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 ...
4
votes
1answer
79 views

How “far” a differential form is from an exterior product

Consider two differential manifolds $X$ and $Y$. Consider now a differential form (of any order) $\omega$ on $X\times Y$. The easiest example is taking $\omega=\xi\wedge\eta$, where $\xi$ is a ...
4
votes
1answer
150 views

What do we mean when we say a differential form “descends to the quotient”?

Let $S$ be a surface and let $f:S\rightarrow S$ be a diffeomorphism. We define the mapping torus $M_f$ of the pair $(S,f)$ to be the quotient $$(S\times I) /\sim \quad \text{ where } \ (1,x) \sim ...
3
votes
1answer
114 views

Independence of $H(f)=\int_M \alpha \wedge f^* \beta$ on choice of $d\alpha=f^*\beta$?

I came across the following UCLA qual question while studying for my upcoming qual: Let $f: M^{4n-1} \to N^{2n}$ be a smooth map between closed connected oriented manifolds of the indicated ...
3
votes
2answers
172 views

Inverse Functions and $u$-Substitution

Back in my undergrad days I wrote a false proof of the following. Problem. Prove that $\displaystyle\int_0^{2\pi}\frac{dx}{1+e^{\sin{x}}}=\pi$ Proof. Integrating by parts gives $$ ...
3
votes
1answer
215 views

Computing the restriction of a differential form

Define $\omega$ on $\mathbb{R}^3$ by $\omega = x\,dy\wedge dz + y\,dz\wedge dx + z\,dx\wedge dy$. Thus far I have computed $\omega$ in spherical coordinates $(\rho,\phi,\theta)$, as well as computed ...
2
votes
2answers
90 views

Wedge product and determinants

I am attempting to self-study differential forms this summer, but I ran into this definition for the wedge product in my book, and it doesn't make any sense to me. Note that $\Bbb R^3_p$ is the ...
0
votes
2answers
68 views

Relationship between divergence operators defined with respect to two different volume forms.

Let us assume that you have a volume form $\mu$ defined on a manifold $\mathcal{M}$. Then you can define the divergence operator with respect to this metric, such that the following relationship holds ...