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

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6
votes
5answers
5k 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 ...
11
votes
5answers
456 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 ...
26
votes
4answers
7k 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 ...
29
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^{*}: ...
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 $\omega$, ...
85
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 ...
8
votes
1answer
637 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
1k 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, ...
7
votes
2answers
395 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 ...
6
votes
1answer
461 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 ...
3
votes
2answers
106 views

Does there exist a $1$-form $\alpha$ with $d\alpha = \omega$?

Let $\omega := dx \wedge dy$ denote the standard area form on $\mathbb{R}^2$. As the question title suggests, does there exist a $1$-form $\alpha$ with $d\alpha = \omega$?
4
votes
3answers
1k 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 ...
1
vote
1answer
84 views

the $\partial\bar{\partial}$-lemma dilemma

In the question here Simplifying the Kahler form, user290605 asked a question about how is that when we take the differential of Kahler form:$$\mathcal{K}=\frac{\sqrt{-1}}{2\pi}g_{i\bar{j}}dz^i\wedge ...
0
votes
1answer
76 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 ...
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
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 ...
7
votes
1answer
742 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 ...
6
votes
2answers
668 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. ...
7
votes
3answers
1k 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
479 views

Lie derivative along time-dependent vector fields

In "Lectures on Symplectic Geometry" by A. C. da Silva (http://www.math.ist.utl.pt/~acannas/Books/lsg.pdf) the author gives the following definition: $$ \mathcal{L}_{v_t} := \frac{\mathrm d }{\mathrm ...
4
votes
1answer
61 views

Is there a nowhere $1$-form $\alpha$ on $M$ with $\alpha(X) = 0$ for any vector field $X$ which is a section of $\xi$?

Let $M$ be a closed $3$-manifold, and let $\xi$ be a $2$-dimensional subbundle of $TM$. Is there a nowhere zero $1$-form $\alpha$ on $M$ with $\alpha(X) = 0$ for any vector field $X$ which is a ...
4
votes
1answer
147 views

What does a standalone $dx$ mean?

Some literature uses $dx$, in the context of differential equations, in a confusing way without defining what it really stands for: $Mdx + Ndy = 0$ Does it mean one of the following or something ...
5
votes
2answers
184 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 ...
5
votes
2answers
265 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 $$ ...
4
votes
1answer
138 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 ...
4
votes
2answers
177 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 $\varphi \in H^1(\mathbb{R}^3, \mathbb{R})$, ...
3
votes
1answer
58 views

Zeroes of exact differential forms on compact manifold

Let $M$ be a $n$ dimensional compact differentiable manifold. I would like to show that any exact differential form of degree $n$ vanishes at at least one point. I think it is a generalization of the ...
2
votes
3answers
78 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
2answers
143 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 ...
1
vote
0answers
64 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 ...
4
votes
1answer
734 views

Pullback distributes over wedge product

I'm looking to prove that the pullback of a smooth function distributes over wedge product, i.e. $$\varphi^*(\omega \wedge \eta) = \varphi^* \omega \wedge \varphi^* \eta. $$ Here the question is ...
2
votes
1answer
72 views

Simplifying the Kahler form

In the link here, p.4, it says that, given a fundamental 2-from $\mathcal{K}$ $$\mathcal{K}=\frac{\sqrt{-1}}{2\pi}g_{i\bar{j}}dz^i\wedge d\bar{z}^{\bar{j}},$$ a manifold is said to be Kahler if this ...
2
votes
1answer
161 views

Manifold is not orientable

Let $M$ be a manifold of dimension $n$ such that there exist two charts $(U_a,\phi_a)$ and $(U_b,\phi_b)$ such that $U_a,U_b$ are connected and $U_a\cap U_b\ne\emptyset$. Moreover the ...
2
votes
1answer
77 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 ...
74
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 ...
19
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 ...
20
votes
2answers
631 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 ...
10
votes
1answer
532 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 $K:\mathbb{R^n} \rightarrow \mathbb{R^n}$. ...
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 ...
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 ...
7
votes
3answers
302 views

What is the intuition behind the definition of the differential of a function?

What is the intuition behind the definition of a differential of a function in differential geometry? i.e. $$df(p)(v_{p}) =v_{p} (f)(p) $$ where $v_{p} \in T_{p} M$ is a vector in the tangent space to ...
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 ...
7
votes
3answers
695 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 ...
3
votes
1answer
978 views

Integration of a differential form along a curve

Given the differential form $\alpha = x dy - \frac{1}{2}(s^2+y^2) dt$, I'd like to evaluate $\int_\gamma \alpha$ where $\gamma(s)=(\cos s,\sin s, s)$ and $0\leq s\leq \frac{\pi}{4}$. When attempting ...
7
votes
0answers
174 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 ...
4
votes
3answers
601 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 ...
3
votes
1answer
116 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 ...
12
votes
2answers
366 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 ...
11
votes
1answer
651 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 ...
8
votes
1answer
297 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 ...