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

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1answer
105 views

Differential forms - looking for 3 definitions!

I am sorry for this type of question, but I currently have to deal with differential forms although I have not heard so far what they actually are, so I have just a few very particular questions about ...
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0answers
49 views

How can I prove $dz=dx+idy$?

Let's see $\Bbb C$ as an $\Bbb R$-vector space. Hence it is isomorphic to $\Bbb R^2$ and it has dimension $2$. If $v_1,v_2$ is a basis for $\Bbb R^2$, every its element can be written as $xv_1+yv_2$; ...
3
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0answers
44 views

Differential Form Pullback Definition

I'm having some difficulty following how Spivak (Calculus on Manifolds) has induced the pullback on page 89. From reading elsewhere online it seems convention is to define the induced map of the ...
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0answers
26 views

existence of integrating factor

Is there always an integrating factor to turn an incomplete differential $M(x,y)dx+N(x,y)dy$ in to a complete differential. Does the answer depend on dimensionality? for example what is the answer for ...
2
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0answers
39 views

Product of Two Orientable Manifolds is Orientable

I am trying to show that following: Let $M$ be an oriented smooth manifold of dimension $m$, and $N$ be an oriented smooth manifold of dimension $n$. Then $M\times N$ is orientable. Let ...
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0answers
27 views

Integration of a 1-form over a “split curve”.

Bit of a strange question I can't really get my head around so apologies if it is ill-posed. Suppose we take a closed curve $\gamma: S^1 \to C \subset M$ in a Riemannian manifold $M$ and integrate ...
35
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3answers
569 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 = ...
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1answer
39 views

Winding number of differential curve

Consider the one-form $\omega$ on $\textbf{R}^2$\ {(0,0)} defined by $\omega$ = $\frac{xdy-ydx}{x^2+y^2}$ Let K $\subset$$\textbf{R}^2$\ {(0,0)} denote the positive x-axis. Let $\gamma$ : $[a,b]$ ...
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0answers
28 views

One form and Vector fields on a manifold in terms of local coordinates.

Prove : $d$$\omega$$(V,W)$=$V \omega (W) - W \omega(V) -\omega([V,W])$ in local coordinates where $\omega$ is a one-form and V,W are vector fields on a smooth manifold M. I do not know how to ...
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2answers
122 views

Differential Forms Notation is Wrong? Confirm or deny? [closed]

Being an engineering student that just happens to have a large interest in math, I have always felt that appealing to definitions instead of intuitively understanding a concept is a mistake. A while ...
0
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1answer
35 views

Three-Dimensional Metrics as Deformations of a Constant Curvature Metric?

I read the following paper Three-Dimensional Metrics as Deformations of a Constant Curvature Metric and discovered the following result: I have three questions: (1) Is $h$ also a conformally flat ...
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0answers
53 views

Connection on Tangent Bundle of Group Manifolds

The thing puzzling me is about a transformation rule of connection on tangent bundle of group manifolds: Assume one has a compact and simply connected Lie group $K$. One can give a metric $g$ on $K$ ...
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2answers
119 views

Integration over manifolds

S is subset of $\Bbb R^3 $ consisting of the union of 1) $z$ axis 2) the unit circle $x^2+y^2=1,z=0$ 3) the points $(0,y,0)$ with $y \ge1 $ Let $A$ be the open set $\Bbb R^3-S$. Let $C_1, C_2, ...
1
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1answer
99 views

Integration of forms on non-simply connected manifolds

What I know is that closed forms are not exact on non-simply connected manifolds, so for instance, if $E$ is a closed form, then $dE = 0$ but $\int_\gamma E \neq 0$, where $\gamma$ is a ...
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0answers
42 views

Integrating factor for non-exact differential

Given $\frac{xdy+ydx}{x^2+y^2}$ I took partials of either coefficient wrt to other variable I get $\partial P/\partial y$ as $\frac{x^2-y^2}{(x^2+y^2)^2}$ and $\partial Q/\partial x$ as ...
2
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1answer
29 views

Does the dual basis to some basis of $T^*_pM$ looks localy like a coordinate chart?

Let $M$ be a manifold and let $\{\alpha_k\}$ be a set of $1$- forms s.t. $\{\alpha_k(p)\}$ forms a basis for $T^*_pM$. Let $(x,U)$ be a chart based in $p$ and denote $\partial_i := ...
8
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1answer
116 views

Explain densities to me please!

When it comes to integration on manifolds, I speak two languages. The first is of course the language of differential forms, which is something I am relatively well acquinted with. The second ...
0
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0answers
25 views

Topology of “line integral convergence” on the space of curves

Let $C^1(I,\mathbb{R}^n)$ be the space of $C^1$ curves. Give it the topology that satisfies that convergence of a sequence of curves $\gamma_n \to \gamma$ occurs iff these conditions hold: a. ...
4
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1answer
72 views

How should we think about equations like $dy = 2x \cdot dx$ from the viewpoint of modern geometry?

We've just started learning about (smooth) manifolds at uni, and I'm kind of hoping this will finally help me get a handle on the dreaded Leibniz notation. Now I've read that expressions $dy$ like can ...
4
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0answers
66 views

Can a differential k-form be integrated on a manifold that is not k-dimensional?

For example, can you integrate a 2-form on some curve, a 1-dimensional manifold, or some 3-dimensional manifold? I know that Stokes's Theorem states that if you integrate $\omega \in \mathcal ...
7
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3answers
211 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 ...
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2answers
64 views

The Operator '$d$' Apparently Having two Different Meanings in Differential Geometry.

Given a smooth map $f:M\to N$ between smooth manifolds $M$ and $N$, we denote the global differential of $f$ by $df$. Also, the letter '$d$' is used for denoting exterior derivative of a differential ...
3
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0answers
38 views

Can We Write the Differential in Terms of Covectors?

Let $f:\mathbf R^n\to \mathbf R$ be a smooth map. We can write $df:T\mathbf R^n\to \mathbf R$ neatly as $$ df = \sum_{i=1}^n(\partial f/\partial x_i) dx_i $$ For a function $f:M\to \mathbf R$ defined ...
4
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1answer
76 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 ...
8
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2answers
129 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 ...
0
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2answers
40 views

Differential forms and simplification

Let us suppose we have $$(\partial_x \alpha - \partial_x \bar{\alpha} )(df+\lambda)\wedge dx+$$$$(\partial_y \alpha - \partial_y \bar{\alpha} )(df+\lambda) \wedge dy +$$$$(\partial_z \alpha - ...
1
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1answer
31 views

Exterior derivatives involving representations

I have two questions regarding the exterior derivative of vector valued forms when representations are involved: Suppose $V$ is a vector space, $M$ a smooth manifold and $\omega$ is a $V$ valued ...
6
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3answers
393 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 ...
4
votes
1answer
51 views

Ways of thinking about vector-valued differential forms

I am trying to get a better intuition of vector-valued differential forms. Let $V$ be a vector space and $M$ a smooth manifold. Consider the space $\Omega^k(M;V)=\Gamma((M\times V)\otimes ...
0
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1answer
57 views

Differential operator computation

I need to show that if $\omega=xydx+3dy-yzdz$ $\nu=xdx-yz^2dy+2xdz$ then $d(\omega\land\nu)=(d\omega)\land\nu-\omega\land(d\nu)$ I really do not understand how this differential operators work. I ...
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2answers
172 views

Sign problems in complex computations

$\newcommand{\dd}{\mathrm{d}} \newcommand{\eg}{\epsilon} \newcommand{\mg}{\mu} \newcommand{\ng}{\nu} \newcommand{\rg}{\rho} \newcommand{\et}{\wedge} \newcommand{\lbar}{\overline} ...
3
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1answer
55 views

Associated bundles: isomorphism between spaces of differential forms.

I think this will be an easy question for numerous people. Let $\pi:P\rightarrow M$ be a principal bundle and $\rho:G\rightarrow GL(V)$ a representation. The space of $k$ forms on $M$ with values in ...
3
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1answer
47 views

Volume form for a product manifold.

Given two Riemannian manifolds $(M,g^M)$ and $(N,g^N)$, we can construct a product Riemannian manifold $(M\times N,g^{M \times N})$ as described in Product of Riemannian manifolds? . Is there a ...
2
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2answers
78 views

Notations of the coordinate functions in differential forms

I'm reading a book on differential forms, and it says: A basis for $(\mathbb{R}_p^3)^*$ is obtained by taking $(dx_i)_p, i=1,2,3$, where $x_i:\mathbb{R^3}\to\mathbb{R}$ is the map which assigns to ...
0
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1answer
44 views

2-forms, Edwards, Differential forms approach

Find the value of the 2-form $dxdy+3dxdz$ on the oriented triangle with $(0,0,0)$ $(1,2,3)$ $(1,4,0)$ in that order. I have tried various subtractions and plugging in values and have been unable to ...
2
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0answers
49 views

Clarification of vector valued forms (sections and tensor products)

I am seeking a bit of clarification on vector valued forms. Intuitively, a vector valued differential form is a differential form on $M$ whose values lie in some vector space. More accurately, Let ...
0
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1answer
41 views

Global differentials on projective line

Given a field $k$, it is well known that the global differentials $\Omega_{\mathbb{P}_k^1/k}$ of the projective line $\mathbb{P}_{k}^1$ are $\mathcal{O}(-2)$. This is usually proved by observing that ...
0
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1answer
23 views

Relationship between $\operatorname{supp} w$ and $\operatorname{supp} dw$

I would like to show that $\operatorname{supp} dw\subset \operatorname{supp}w$, where differential $m$-form in a $\Sigma$ surface of dimension $m+1$. If $$w(u)=\Sigma_i (-1)^ia_i(u) du_1\wedge ...
1
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1answer
33 views

If differential forms agree on one chart do they agree everywhere?

Let $\alpha,\beta$ be two $k$-forms on a manifold $M$. If there exists some chart $(U,x^1,\dots,x^n)$ on which $\alpha=\beta$ does it follow that $\alpha$ and $\beta$ are the same forms? In different ...
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0answers
9 views

Model of continuum mechanics or thermodynamics derived from potential and paradoxe on state variables

(1) In material sciences based on thermodynamics and continuum mechanics, one usually introduces potentials to summarize the constitutive model. All constitutive relations derive from the introduced ...
10
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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 ...
0
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0answers
30 views

Hodge star on this expression

if we have $\alpha$ a complex function, and we want to take the Hodge dual of $$d\bigg(\frac{1}{|\alpha|^2}\bigg)d(\bar{\alpha})$$ what will that give us? Can we take ...
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1answer
94 views

Exterior derivative of forms derived from a metric

Let $(M,g)$ be a Riemannian manifold. From $g$ and a fixed vector field $V$ we can derive the following two differential forms: A $1$- form $\alpha(X) = g(V,X)$, i.e. $\alpha = \iota_Vg$. A $2$-form ...
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0answers
60 views

Closed 1-forms on Simply Connected Manifold

Is it true that closed 1-forms on a simply connected differentiable manifold are exact. If so, could you explain why? Thanks very much
0
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0answers
27 views

Help in the proof of Poincaré Theorem to differential forms

I'm revising the proof of Poincaré Theorem, but I don't understand a pass of proof. Let be $E$ and $F$, Banach spaces and $U\subset E$ open set. Consider $\omega\in\Omega_p^n(U;F)$ a p-differential ...
0
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2answers
63 views

Self-studying differential forms and tensors

I am interested in understanding the generalized Stokes' Theorem. From my understanding, this theorem involves differential forms and exterior algebra, and tensors (to some extent). I'm not ...
2
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1answer
60 views

Tangent space and differential forms of quotient groups

I have difficulty understanding the following argument because it seems that many details are swept under the rug and I am looking for a detailed rigorous exposition on this (I'll try to make clear ...
5
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1answer
51 views

divisor of a section of the sheaf of logarithmic differentials

Let $S=\{0, 1, \infty\} \subset \mathbb{P}^1$ and let $\Omega^1_{\mathbb{P}^1}(\log S)$ be the line bundle of logarithmic differentials along $S$. Consider the form $$ ...
0
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1answer
33 views

$\bigwedge^nT^∗M$ is trivial $\Leftrightarrow M$ is orientable

I can't figure out how to prove the following: Let $M$ be an $n$-manifold. Then $\bigwedge^nT^∗M$ is trivial $\Leftrightarrow M$ is orientable Thank you
1
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1answer
54 views

for a vector field $v$, what is the differential 1-form obtained from $v$ by the canonical isomorphism induced by the inner product.

I am studying analysis on manifolds and am trying some problems and I am conceptually stuck on one. It is from Do Carmo's book Differential Forms and Applications, chapter one problem 11 b. I'll ...