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

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Vector-valued differential forms

Given a smooth real manifold $M$, and a real vector space $V$, when we talk about a $V$-valued $k$-form on $M$, do we mean an element of $\Gamma(\wedge^{*k}M)\otimes V$, where $\Gamma$ denotes ...
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40 views

Show that if $\omega$ is a 1-form differential, then $\left\vert\int_{C}\omega\right\vert\leq ML$

Show that if $\omega$ is a 1-form differential define on $U\subset\mathbb{R}^{n}$, $c:[a,b]\to U$ is a differentiable curve and $\vert\omega(c(t))\vert\leq M$, for all $t\in [a,b]$, then ...
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89 views

Exterior Differential (and its Equivalent Differential Operator) of an Integral 0-Form

I am reading Witten's 1982 paper "Supersymmetry and Morse Theory," and while I am slowly learning the material as I read through the paper, I have come across an equivalence that, while it should be ...
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1answer
63 views

The exterior derivate and pullback commute

The above question is from a past exam. I am having trouble with the fine details, ie what $F*dw$ and $dF*w$ actually look like. Can anybody show me how this question is solved? I have solved it ...
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48 views

Definitions/ intuition for differential forms

I've read about differential forms in the Princeton Companion to Mathematics by Gower and in baby Rudin and I'm having trouble reconciling the two expositions. Rudin says a k-form in $E$ ($\subset ...
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3answers
34 views

Help with the definition of a bilinear form $\omega$

According to this for $V$ a $2n$ (real) dimensional space any bilinear form $\omega: V \times V \to \mathbb{R}$ induces a linear map $\tilde{\omega}: V \to V^*$ via $$ \tilde{\omega}(v) := \omega(v, ...
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2answers
63 views

Stoke theorem and exterior derivative

$w=x \, dy\wedge dz - 2z f(y) \, dx \wedge dy + y f(y) \, dz \wedge dx$ where $f: \mathbb{R} \to \mathbb{R}$ belong $C^1$ (differentiable and derivative is continuous) with $f(1)=1$. Find $f$ so that ...
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1answer
92 views

If $d(f\omega)=0$, then $\omega \wedge d(\omega)=0$

Here's the question: Suppose that $\omega$ is a $k$-form on an open set $U$ of $\mathbb{R}^n$ and $f:U \to \mathbb{R}$ is a $C^\infty$ function such that $f(x) \neq 0$, for all $x \in U$, and ...
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34 views

Show that a derivation mapping k-forms to (k+1)-forms is zero.

This is question 7.4-6 in "Manifolds, Tensor Analysis and Applications", by Marsden, Ratiu and Abraham. It says: show that a derivation mapping $\Omega^k(M)$ to $\Omega^{k+1}(M)$ for all $k$, is ...
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1answer
61 views

Give a volume form on $\mathbb{RP}^3$ [closed]

I was asked to determine a volume form on $\mathbb{RP}^3$. I would really appreciate any help. Thanks in advance.
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1answer
52 views

Conservative Covector Fields are Exact

$\newcommand{\R}{\mathbf R}$ I am trying to understand the proof of the following: Theorem. Let $M$ be a smooth manifold and $\omega$ be a smooth covector field on $M$. Then $\omega$ is ...
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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 ...
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2answers
93 views

How to prove $d\omega=(\nabla_\mu\omega)_\nu dx^\mu\wedge dx^\nu$ without using coordinates

This is exercise 7.8 b) of Nakahara's GTaP: Let $\omega\in\Omega^1(M)$ be a 1-form on a Riemannian manifold with Levi-Civita connection $\nabla$. Prove that $$ ...
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1answer
154 views

Total confusion about differential one-forms and non-coordinate bases

I asked this question recently (Basis of differential one-form confusion), thought I understood the answer, but now realise I don't. Lee (Introduction to Smooth Manifolds) says that at a point $p$ ...
4
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1answer
57 views

Intuition behind volume form of a sphere

What is the geometric intuition behind $$\int_{S^1}x\,dy-y\,dx=2\pi,\qquad\int_{S^2}x\,dy\,dz-y\,dx\,dz+z\,dx\,dy=4\pi r^2,$$and in general, ...
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1answer
61 views

A converse to Stokes' Theorem in $\mathbb{R}^n$

In a lecture of advanced calculus, my teacher made a very interesting remark about the generalized Stokes' Theorem (actually, he left it as an exercise!), such that, if I understood it right, is ...
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26 views

The orientation of an embedded submanifold of $\mathbb R^n$

Assume $M$ is an embedded submanifold of $\mathbb R^n$ with codimension 1, by a specific coordinate of $\mathbb R^n$ $\{x_1,\dots,x_n\}$ in an area $U$, the $M\cap U$ has coordinate ...
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35 views

Proving a fact about bilinear forms

The fact I'm trying to prove is that every bilinear form $\omega$ on a set $\omega\subseteq\mathbb{R}^2$ can be written in the form $\omega = f\,dx\wedge dy$, where $f: \Omega\rightarrow\mathbb{R}$ is ...
3
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1answer
67 views

Inclusion, pullback of differential form

Let $\omega=x\,dy\wedge dz +y\,dz\wedge dx+z\,dx\wedge dy$ or in spherical coordinates (unless I had made some mistake) $\omega=r^3\cos \theta\, d\phi\wedge d\theta$. Now I want to find $i^*\omega$ ...
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1answer
109 views

Alternative to Arnold's mathematical methods

I have difficulties understanding Arnold's book of mathematical methods of classical mechanics. Yet I should get some familiarity with the subjects found at chapters 3,4,7,8 before next semester to ...
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2answers
54 views

if $\omega$ is a $2$-form, and $\Bbb d \omega = 0$ what can we conclude?

Let $f_1, f_2, f_3$ be smooth functions on an open subset $\Omega \subset \Bbb R ^3$, which contains the standard cube $I^3$. We define the differential $2$-form $$ \omega = f_1 \Bbb d x^2 ...
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1answer
42 views

Finding a compactly supported $f$ such that $\theta = f\omega$.

Let $\Omega \subseteq \Bbb R^N$ be open, $\omega = \sum_{j=1}^n \omega_j\,{\rm d}x_j$ be a $1$-form in $\Omega$ such that $\sum_{j=1}^N|\omega_j(x)|\neq 0$ for all $x \in \Omega$, and $\theta = ...
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1answer
60 views

Differential forms, projections

I have a problem with one exercise from differential geometry. I don't even know how to start. Anyone could help with this problem? Let $M$, $N$ be manifolds, $M$ connected. Let $\pi:M\times N \to N ...
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1answer
26 views

Divergence, contraction and lie derivatives

I'm working through this question. I can show the forward direction in (a) but can't show the converse. I have $\delta/\delta t \phi^*_t \mu$ evaluated at t=0 is 0, but I can't see how I conclude ...
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1answer
40 views

Comass of a differential form

In the wikipedia article on currents https://en.wikipedia.org/wiki/Current_%28mathematics%29 it is written that If $\omega$ is an m-form, then define its comass by $||\omega|| = \sup\{|\langle ...
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1answer
90 views

Show that $\omega^{2}\wedge\cdots\wedge\omega^{2}$ n times is equal to

Consider $\mathbb{R}^{2n}$ with coordenates $x^{1},\cdots,x^{2n}$ and the following differential form of grade two $$\omega^{2}=dx^{1}\wedge dx^{n+1}+dx^{2}\wedge dx^{n+2}+\cdots+dx^{n}\wedge ...
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0answers
38 views

Is there a way to compute the Poincaré dual of the following type of degree $(2n-2)$ de Rham class?

Given a closed, connected, symplectic manifold $(X^{2n},\omega)$, is there a systematic method to computing the Poincaré dual surface to degree $(2n-2)$ classes of the form $$[\omega]^{n-2}\cup B + ...
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1answer
47 views

The wedge product of a $1$-form and a $2$-form

I'm trying to understand the wedge product of a one form and a two form. My difficulty is that I end up with a $dx \wedge dx \wedge dy$, for example, and I know a wedge of itself is 0. Does this ...
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0answers
37 views

Let $f\colon \Bbb R^n \to \Bbb R^n$ be the translation, $f(x)=x+a$. Show that $\deg(f)=1$

Let $U,V$ be connected open subsets of $\Bbb R^n$ and let $f\colon U \to V$ be a $C^{\infty}$ proper map. For all $w \in \Omega_c^n(V)$, $\int_{U}f^*(w)=\gamma \int_Vw$. Now define $\deg(f)=\gamma$, ...
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1answer
77 views

Interpretation of $d\phi(z)$ in differential geometry

In "Exercises and Solutions in Mathematics", Ta-Tsien, 2nd Edition, exercise 3343. Statement of the exercise Let $(\mathbb{H}, g)$ be the two-dimensional hyperbolic space, where \begin{equation} ...
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19 views

Building Non-vanishing sections in a certain way

Assume you have a vector bundle $\Pi: E \rightarrow M$ where $E$ is the total space, $M$ is a compact manifold. Assume you know it is parallelizable. Let $\psi_i : \Pi^{-1}U_i \rightarrow U_i \times ...
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1answer
44 views

$d(\iota_v\rho)=0 \implies d(\phi\iota_v\rho)=d\phi(v)\rho$?

My motivation is physical, but my question is purely mathematical. Everybody knows, that the power of the electric current in a piece of wire is $$P=UI$$ where the wire is regular domain $V$ in a ...
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0answers
19 views

Contraction of a vector form

I'm trying to make sense of this definition, but I cannot see why the resulting map is in a space of dimension $k-1$, surely as it is comprised of k vectors this maps a k-form to a (k+1)-form? I ...
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1answer
36 views

Support of Pullback of Differential form

This is a dumb question, but I'm learning about differntial forms, and it seems to me that if $f:N^n\to M^n$ is a diffeomorphism and $\omega$ a smooth $n$ form on $M^n$, then $\omega$ vanishes at ...
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29 views

Using differential forms to define line integrals

I saw some similar questions and answers but they often included some information or mathematics I haven't learned/read so I'm hoping to get a somewhat simpler answer. Let $\beta:[a,b] \to ...
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1answer
37 views

$\int _c \omega$ is independent of orientation-preserving re-parameterization of c

I'm working on the following problem from Guillemin and Pollack's Differential Topology: Let $c : \left[a, b\right] \rightarrow X$ be a smooth curve, and let $f: \left[a_1, b_1\right] \rightarrow ...
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632 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 ...
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84 views

$\Delta x\approx dx$

How do you state $($small $\Delta x)\approx dx$ in terms of differential forms? $dx$ is a one-form, but I don't see why $\Delta x$ approximates a one-form if gets smaller.
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24 views

sympletic form of the hyperboloid

I'm working with the Hyperboloid of two leefs $H_2$ as a coadjoint orbit of $\mathfrak{sl}^*(2, R)$. I know that $H_2$ is a symplectic manifold by the following theorem: Given a Lie group G and ...
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0answers
19 views

Hermitian vector space and relation of associated operators.

Here what i want to do is prove proposition 1.1 in chapter 5, on Wells, Differential analysis on complex manifold, The propositions are follows For Hermitian vector space of complex dimension $n$. ...
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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 ...
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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 ...
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1answer
40 views

$M \times N$ orientable iff both $M$ and $N$ are orientable proof in terms of volume forms

I'm studying differential forms, and in my homework I'm asked to show that the product of two manifolds $M \times N$ is orientable if and only if both $M$ and $N$ are orientable. I want to show this ...
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1answer
55 views

Induced bilinear form on exterior powers - Towards a global Hodge Star Operator

In all constructions of the hodge star operator I've seen so far there was a part where an inner product on the exterior power of the tangent space was defined by the ungodly local formula: ...
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1answer
60 views

Computing the integral of a differential form in $\mathbb{R}^{2}$.

Let $D$ be the disk \begin{equation} D=\{(x,y)\in\mathbb{R}^{2}\:|\:x^{2}+y^{2}\leq 1\}, \end{equation} which is easily verified to be a compact $2$-differentiable manifold with boundary. Let ...
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Proving df is the sum of partial derivatives

Wikipedia states: Since any vector $v$ is a linear combination $\sum v_je_j$ of its components, $df$ is uniquely determined by $df_p(e_j)$ for each $j$ and each $p \in U$, which are just the ...
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1answer
85 views

Multivariable calc “second course” that does differential forms

I've worked through a computation-heavy, "standard" but quite nonrigorous treatment of multivariable calculus in the past. What book would do well as a rigorous (but not overly) "second course"? In ...
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1answer
36 views

Definition of integration of differential forms

I am trying to understand precisely the following paragraph: Question Why would he define the support $K$ of a form $\omega$ defined on an open set $U$ as a subset $K\subseteq M$ instead of a ...
3
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2answers
86 views

Computing Lie derivative

Can anyone help me with computing Lie derivative ${L}_{X}Y$ using its definition for these two vector fields: $X=y\frac{\partial}{\partial x}-x\frac{\partial}{\partial y}, ...
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1answer
153 views

What is the intuition behind differential forms?

I am comfortable with the way physicists use differentials as elements of area/volume. I know the (algebraic) formal definition of differential forms, but it makes no intuitive sense, especially since ...