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

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37 views

Integration of $V$-valued differential form

When studying fibre bundles, connections and gauge theories it is usual to consider vector-valued differential forms, like the connection one-form, or it's pull back by a local trivialization known as ...
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1answer
42 views

Differential Forms on Surfaces. Show that $N\cdot (\nabla \times V)\eta=d\phi$ on $x(D)$.

Let $M$ be an orientable surface in $\Bbb R^3$ with a unit normal vector field $N$ and let $x: D\to M$ be a patch. Let $\eta$ be a differential 2-form on $x(D)$ defined by $\eta(x_u,x_v)=\pm\|x_u ...
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1answer
59 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 ...
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3answers
75 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 ...
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2answers
75 views

Wedge product computation

Let $\omega \in \Omega^{2}(\mathbb{R}^{2n})$ be the $2$-form $\omega=dx^{1} \wedge dx^{2} + dx^{3} \wedge dx^{4} + \dots + dx^{2n-1} \wedge dx^{2n}$. I want to compute the wedge product of $\omega$ ...
2
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1answer
65 views

Bogus proof that the Liouville Form on the cotangent bundle is nondegenerate.

Suppose we have a manifold $M$ of dimension $n$ and its cotangent bundle $T^*M$. The Liouville form $\lambda$ on $T^*M$ is defined as $\lambda_{\omega_p} = \pi^*(\omega_p)$ where $\pi$ is the standard ...
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1answer
58 views

Flux Integral - where did I go wrong?

S is the graph $z=25-(x^2+y^2)$ over the disk $x^2+y^2\leq 9$ and $\varphi = z^2dx\wedge dy$. Find $\int_S \varphi$. According to the book the answer is $3843\pi$, but the answer I got is ...
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2answers
49 views

An object is travelling in a straight line. Its distance, s meters, from a fixed point at time t seconds is given by the expression

$$s=t^3−t^2−6t$$ a) Find ds/dt when t=3 and interpret this result. b) Find d^2s/dt^2 when t=3 and interpret this result. c) Find the time in seconds when the velocity is 2m/s (d) Using the ...
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1answer
137 views

Change of Coordinate Formula for Differential Forms

Let $M$ be a manifold, $x$ local coordinates on an open set $U$, $y$ local coordinates on an open set $V$. In addition, let $(x, \alpha)$ and $(y, \beta)$ be two induced bases for the common part of ...
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1answer
104 views

De Rham cohomology for $\mathbb{R^2}$

De Rham cohomology groups for $\mathbb{R^2}$. $H^{0}_{dR}(\mathbb{R}^{2})=\mathbb{R}$ since $Z^{0}(\mathbb{R}^{2})$ is the one dimensional space of locally constant functions on $\mathbb{R}^{2}$ and ...
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1answer
143 views

Exercice on a differential form

Let $\omega$ be a $q$-form on $\mathbb{R}^2$ and let $Z_{\mathbb{R}^2}(dx_1)=\{p \in \mathbb{R}^2 \colon (dx_1)_{|p}=0\}$ $Z_{S^1}(dx_1)=\{p \in S^1 \colon (dx1_{|S^1})_{|p}=0\}$ where ...
2
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1answer
128 views

$df$ vanish in a compact manifold in at least 2 points

I need to prove that if $M$ is a compact manifold and $f$ is a smooth function in $M$, then $df$ vanish in at least 2 different points of $M$. I don't know where to start. Any suggestion will be ...
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3answers
164 views

On the scalar product of a vector field and the exterior derivative of a smooth map on a surface

I am reading Roger Penrose's wonderful book 'The Road to Reality: A Complete Guide to the Laws of the Universe'. I need some help in understanding the scalar product of a vector field and the ...
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1answer
116 views

Differential of a differential form without local coordinates

Given a differential form $w$ on a manifold, I know how to calculate $dw$ in local coordinates. But is there any way to define $dw$ independent of local coordinates?
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1answer
226 views

Formal finite sum for integration on k-chains

This question is related to the integration of differential forms on chains, as exposed in the document at: http://www.math.upenn.edu/~ryblair/Math%20600/papers/Lec17.pdf . The author gives the ...
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1answer
185 views

Finding the winding number of a curve

Let $\gamma(t)=(r \cos t,r \sin t)$, for some $r>0$, and let $\Gamma$ be a $C^2$-curve in $\mathbb R^2-\{\bf 0\} $, with parameter interval $[0,2 \pi]$, with $\Gamma(0)=\Gamma(2 \pi)$, such that ...
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1answer
124 views

Can exterior calculus be used to solve differential equations?

I know one can formulate partial differential equations in terms of exterior derivatives, etc but I have been wondering for a while now how one might be able to use that formalism to extrapolate ...
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1answer
57 views

Expressing a differential form in terms of a scalar function

We can express every k-form in the form $ \omega(x) = \sum_Id_Idx_I $ where $ I$ is k-tuple and $ d_I$ is just some scalar function of x. That's entirely understandable for me. But while reading ...
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1answer
317 views

Taking the exterior derivative of a 0-form

I'm attempting to show that $dg(\vec{x})=\alpha$, where $\alpha=\Sigma^n_{i=1}f_idx_i$ and $g(\vec x)={1\over{p+1}}\Sigma_{i=1}^nx_if_i(\vec x)$...and $d$ is the exterior derivative. The $f_i$ are ...
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1answer
105 views

How do you calculate an exterior derivative on forms in $\mathbb{R}^3$?

If we have a form, say, $\omega = f(x,y,z) \, dx + g(x,y,z) \, dy + h(x,y,z) \, dz$, what is the formula for the exterior derivative $d \omega$?
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1answer
48 views

Computing Rham Cohomology

Suppose that we have a $C^{\infty}$ manifold $X$ with and atlas $\mathcal{A}=$($U_{\alpha},\varphi_{\alpha}$) such that for every two intersecting open sets $U,V \in \mathcal{A}$ the intersection is ...
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2answers
40 views

Volume forms and volume of a smooth manifold

Choose a volume form $\omega$ on $M$, oriented manifold. For every $F\in C^{\infty}_c(M)$, we define $$ \int_M F:=\int_M F\omega $$ where in the right hand term $M$ is taken wit positive orientation ...
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1answer
87 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 ...
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1answer
38 views

Linear Maps from $V$ to $V^*$ defined by a 2-form

I came across this idea at the very beginning of a book and I don't quite seem to grasp it. It states given $\omega \in \bigwedge ^2 (V) $ you can define a linear map $ \omega^\#: V \to V^* $ by ...
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1answer
27 views

exact and closed differential forms

This exercise is taken from the Meyer-Hall-Offin book on Hamiltonian systems. Let $Q(p,q)$ and $P(p,q)$ be smooth functions defined on an open set in $\mathbb{R}^2$. Consider the four differential ...
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1answer
44 views

An Application of Stokes's Theorem

Let $D^2=\{(x,y)\in \mathbf R^2: x^2+y^2\leq 1\}$ be the unit disc in $\mathbf R^2$, and $D^3=\{(x,y,z)\in \mathbf R^3: x^2+y^2+y^2\leq 1\}$ be the unit disc in $\mathbf R^3$. Let $i_{\pm}:D^2\to ...
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1answer
97 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 $xv_1+yv_2$; ...
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1answer
32 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 := ...
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1answer
44 views

A question about a part of the proof of the existence of the exterior derivate.

Let $M$ a differentiable manifold, $(U;x_1,\ldots,x_n)$ a coordinated system and $\omega$ a differential form which domain intersects $U$. In $U \cap \operatorname{Dom} \omega$, $\omega$ can be ...
2
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1answer
104 views

Problem integrating in problem using the Poincaré Lemma

a) It is easy to show that $d\beta=0$. b) $\begin{align}\hat{\mathbb{X}}_t &= \left(\frac{\partial}{\partial t}\hat{\Phi}_t \right) \hat{\Phi}_t^{-1} \\ &= \left(\frac{\partial}{\partial ...
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1answer
70 views

How to derive the standard symplectic form on a 2-sphere in cylindrical polar coordinates?

I have already know that at a point $p\in S^2$ with $u,v$ the tangent vectors at $p$. Then the standard symplectic form is $\omega_p(u,v) :=\langle p,\,u\times v\rangle$ where $u\times v$ is the outer ...
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2answers
61 views

Exercise on left invariant 2-forms

I'm trying to solve a problem about the Lie group of transformation over $\mathbb{R}$: \begin{equation} x\mapsto ax+b, \end{equation} where $a,b\in\mathbb{R}, a\neq 0.$ I'm asked to find the space of ...
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2answers
109 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 ...
2
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1answer
119 views

Product rule for differential forms?

How would I work out $d(\omega F)$ where $\omega$ is a 1-form and $F$ is a vector valued function? I know that $d(f \omega) = df \wedge \omega + f d\omega$, for a smooth function $f$. I suppose this ...
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3answers
130 views

How to deduce this formula using differential forms?

There's a formula from vector calculus that seems terrible to deduce. This formula is: $$\nabla\times (A\times B)=(B\cdot\nabla )A-(A\cdot \nabla)B+A (\nabla\cdot B)-B(\nabla\cdot A)$$ Deducing it ...
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1answer
67 views

Proving Borsuk-Ulam with Stokes

What is the easiest way to deduce the Borsuk-Ulam theorem in the case $n=2$ by using integration on manifolds and Stokes theorem? So I want to prove the following: Given a map $f\colon S^2\rightarrow ...
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2answers
76 views

If $\omega$ is closed on $\mathbb R ^2 - 0$ and $\text d \omega =0$, then $\omega = \text d g+ \lambda \text d \theta$.

I'm trying to solve problem 4-30 from “Calculus on manifolds”, which is the one in the title, where $$\text d \theta = -\dfrac{y}{x^2+y^2}\text d x+\dfrac{x}{x^2+y^2}\text d y.$$ I think I'm on the ...
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1answer
75 views

Product integration of differential forms

Let $\alpha, \beta$ two forms continuous with compact supports and maximum degree on surfaces oriented $M, N $ respectively. consider $\pi_M:M\times N \rightarrow M$ and $\pi_N:M\times N \rightarrow ...
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1answer
133 views

Relation between de Rham cohomology and integration

This question is a follow-up to When does a null integral implies that a form is exact? . As mentionned in the selected answer, given certain conditions it is possible to find an isomorphism between ...
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1answer
92 views

Question on Differential forms

Let $\Omega\subseteq \mathbb{R}^2$ be an open set and $\omega=\omega^1dx_1+\omega^2dx_2$ an $1$-form on $\Omega$ and $$L=\omega^2\frac{\partial}{\partial x_1}-\omega_1\frac{\partial}{\partial ...
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1answer
271 views

Concept of integration to differential form

How to integrate differential form actually. As far as I know, a differential form is a multilinear function mapping from a vector space to a real number. Let's take $\int_c fdx+gdy$ as an example. It ...
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1answer
94 views

Is $\zeta=\frac{x dy \wedge dz+y dz \wedge dx+z dx \wedge dy}{r^3}$ exact in the complement of every line through the origin?

$r=\sqrt{x^2+y^2+z^2}$ of course. If the line is the $z$ axis, it is given in the book (Rudin) that $\zeta=d \left( -\dfrac{z}{r} \dfrac{xdy-ydx}{x^2+y^2} \right)$ I've managed to figure out 2 ...
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1answer
146 views

Integrals of Differential Forms

I am working out of Munkres Analysis on Manifolds and I see that he claims for $\eta = f dx_1 \wedge \ldots \wedge dx_k$. $$ \int_A\eta = \int_{x \in A} \eta(x)\big((x;a_1) ,\ldots, (x;a_k) \big)$$ ...
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1answer
723 views

non-vanishing k-form on a k-manifold in $\mathbb{R}^n$ implies orientability

I want to know how to prove the theorem: If M is a k-manifold in $\mathbb{R}^n$, then it is orientable if and only if there is a volume form defined globally on M. I'm currently stuck at this step: ...
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1answer
509 views

Leibniz rule for exterior derivative of a contraction

If I have a contraction of a vector field with a 1-form valued 2-form, what would be the appropiate product rule? $$d_{\left[a\right.} \left(P_{[bc]i} v^i \right)_{\left. \right]} = \, ?$$ This ...
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1answer
507 views

property to be exact a 1-form on $\mathbb R^2 -\{(0,0)\}$

(a) Let $\omega$ a $1-$form defined on the open set $ U \subset \mathbb R ^n$ and $ c:[a,b] \to U$ a $ C^1 -$differentiable curve such that $ |\omega (c(t))| \leq M \quad \forall t \in [a,b]$ Prove ...
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1answer
230 views

Differential forms on a $S^1$-manifold

I am reading about differential forms on manifolds with group actions and there is an 'obvious' formula which I don't quite understand. Let $X$ be a manifold with a smooth circle action, that is a ...
2
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1answer
58 views

Good book about differential forms

I'm a looking for a good book to self-study differential forms. Particularly, I'm looking for a book that is as similar as possible to Bert Mendelson's "Introduction to topology" (i.e. a book that ...
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2answers
32 views

Exercise about differential forms and pull-back: $\omega_p=0$ if and only if $(\omega|_U)_p=0$ [duplicate]

Let $M$ be a manifold, $\omega\in\Omega^q(M)$. Let $U\subset M$ be an open subset embedded as manifold in $M$. Let $p\in U$. Show that $\omega_p=0$ if and only if $(\omega|_U)_p=0$. Notation: Let ...
2
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2answers
33 views

Line integrals in differential form

I'm a bit confused as to the format of line integrals in differential form (i.e. the form in which Green's theorem is often presented). For example: $$ \oint\limits_\mathcal{C} \left( y^2 \mathrm{d}x ...