The differential-forms tag has no wiki summary.
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1answer
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Are these two definitions of exterior derivative equivalent?
I saw two definition of the exterior derivative of a $k$-form $\omega$.
First definition: $$(d\omega)_p(v_0,\ldots,v_k)= \sum_{i=0}^k(-1)^id(\omega(v_0,\ldots,\hat{v_i},\ldots,v_k))_p(v_i)$$
Second ...
3
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1answer
65 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 ...
10
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2answers
123 views
The algebraic de Rham complex
Let $A$ be a commutative $R$-algebra (or more generally a morphism of ringed spaces). Then there is an "algebraic de Rham complex" of $R$-linear maps $A=\Omega^0_{A/R} \xrightarrow{d^0} \Omega^1_{A/R} ...
2
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1answer
157 views
baby rudin, chapter 10, (differential forms) theorem 10.27
I'm having difficulties with the reasoning in the proof of theorem 10.27 (regarding integration over oriented simplexes).
say ...
3
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1answer
97 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 ...
2
votes
2answers
88 views
Coordinate free definition of orientable manifold
Analysis and Algebra on Differentiable Manifolds, first edition, chapter 7.3.1, defines orientation on a vector space and orientable manifolds. There is a part of the definition that I do not ...
3
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0answers
44 views
Questions on a sub-bundle of $\mathbb R^3\setminus \{0\}$
Let us consider spherical coordinates $(r,\theta,\phi)$ on $\mathbb R^3$ and the manifold $M:=\mathbb R^3 \setminus \{0\}$.
Let us consider the 1-form on $M$
$$
\omega = zdz ...
1
vote
1answer
74 views
Why is this integral zero? (Inner product between two 1-forms on a Riemann surface)
I have a quick question regarding the proof of Proposition II.3.2 in Farkas & Kra (pg. 40). The proposition is that if $\alpha$ is a square-integrable, $C^1$ 1-form, then $\alpha$ lives in a ...
4
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1answer
86 views
Integration over a surface
Let $S$ be given by
$$S= \left[(x,y,z) \in \Bbb{R}\;|\; x^2+y^2+z^2+xy+xz+yz=\frac12 \right]$$
and $$\omega = xdy \wedge dz\, -\, \frac {2z}{y^3} \, dx\wedge dy \,+\, \frac1{y^2}dz\wedge dx $$
...
1
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0answers
74 views
Prove $\left.{\partial X\over \partial Y}\right|_Z=-\frac{\left.{\partial Z\over \partial Y}\right|_X}{ \left.{\partial Z\over \partial X}\right|_Y}$
$$\left.{\partial X\over \partial Y}\right|_Z=-\frac{\left.{\partial Z\over \partial Y}\right|_X}{ \left.{\partial Z\over \partial X}\right|_Y}$$
The above is an identity frequently used in ...
4
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1answer
80 views
Period Homomorphisms and closed 1-forms
This is from Otto Forster's "lectures on Riemann Surfaces", on integration of forms.
Let $\Gamma = \alpha_1 \mathbb{Z} + \alpha_2 \mathbb{Z}$ be a lattice in $\mathbb{C}$ (i.e. $\alpha_i \in ...
4
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1answer
106 views
Where can I learn about complex differential forms?
So I'm a 3rd year grad student in number theory/modular forms/algebraic geometry, and I've worked with differential forms from an algebraic point of view without ever knowing what they really are. I'd ...
1
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1answer
78 views
elementary questions about differential forms
QUESTION 1:
So I know that if $\omega$ is an alternating $p$-form for odd $p$ on some vector space $V$, then $\omega\wedge\omega = 0$.
But...isn't the same true for any $p$? Ie, take for example $p ...
5
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1answer
107 views
Restriction of a differential form to an isotropic submanifold
From Analysis and Algebra on Differentiable Manifolds, first edition, exercise 2.6.4., question 1 (slightly edited for this post):
Let $\vartheta$ be the canonical 1-form on the cotangent bundle $T^* ...
3
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1answer
85 views
What is meant by the kernel of a 2-form?
I'm given a 1-form $\alpha$ on $\mathbb{R}^n$, and asked to compute the kernel of $d\alpha$.
Since $d\alpha$ is a 2-form on $\mathbb{R}^n$, it would eat a vector field to give a 1-form, or it would ...
1
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0answers
51 views
Integration equivariant form
We consider the action of $S^{1}$ on $S^{2}$ by rotation respect the vertical axes. We want to integrate the $2-$ equivariant form
$$\alpha(X)= -X\cos(\phi) +\sin(\phi)\,d\phi\,d\theta.$$
We have ...
5
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1answer
87 views
Diffeomorphic riemannian manifolds and volume forms
Maybe the question will be stupid, but I'm a beginner in riemannian geometry...
We have two riemannian manifolds $(M,g)$, $(\overline M,\overline g)$ and a diffeomorphism $F:M\rightarrow\overline M$ ...
6
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1answer
224 views
Hodge Star Operator
I'm trying to understand the Hodge star operation, but have come across an impasse almost immediately.
I have the definition
$$(\star \omega)_{a_1\dots a_{n-p}}=\frac{1}{p!}\epsilon_{a_1\dots ...
4
votes
2answers
137 views
Function on $\mathbb{R}^{2}-\{0\}$.
Does there exist any compactly supported function $f= (f_1,f_2): \mathbb R^2-\{0\}\to \mathbb R^2$ such that $$\frac{\partial}{\partial x_2}f_1=\frac{\partial}{\partial x_1}f_2.$$
Also there does not ...
5
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1answer
137 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 ...
2
votes
1answer
51 views
Independence of coordinate charts in the definition of the order of a pole of a meromorphic 1-form on a Riemann surface
I am currently reading Forster. Let $X$ be a Riemann suface, and $Y$ an open subset of $X$. Assume we have a meromorphic 1-form $\omega \in \Omega(Y \setminus \{a\})$ with a pole at $a \in Y$. One ...
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0answers
76 views
Existence of Spin Group
"In mathematics the spin group Spin(n) 1[2] is the double cover of the special orthogonal group SO(n), such that there exists a short exact sequence of Lie groups
As a Lie group Spin(n) therefore ...
0
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0answers
68 views
contraction with the metric tensor
What does mean "$\wedge_0^4V $ is the space of 4-vectors whose contraction with the metric tensor of the space $V$ vanishes" how can we formulate this set?
this means $i_gT=0$ for tensor $T$?
3
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2answers
94 views
A vanishing theorem for differential forms.
I am trying to prove that for an algebraic surface $X$ (under some extra assumptions that are probably not important) there the space $H^0(X,\Omega_X^1)$ is trivial, i.e. that there exist no globally ...
3
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1answer
43 views
Holomorphic 1-forms in $y^2-(z-a_1)\ldots(z-a_n)$
I know that the surface $y^2-(z-a_1)\ldots(z-a_n)$ is a Riemann Surface (that is the Riemann surface of $\sqrt{P(z)}$ with $P(z)=(z-a_1)\ldots(z-a_n)$) of genus $g$ and that ...
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0answers
38 views
Hodge decomposition on a manifold with a nontrivial connection
I am familiar with the notion of Hodge decomposition of an arbitrary differential form into an exact form, a co-exact form, and a harmonic form. Given a curved space with a connection, could you ...
2
votes
1answer
60 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 ...
3
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0answers
48 views
Can calculus of varations be formalised with exterior calculus?
I noticed that a calculus of variations problem is just an integral over a differential form. Therefore, I would think it would be possible to formulate the Euler-Lagrange equations using exterior ...
0
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1answer
50 views
Proof of the naturality of integration
I have a bit of a problem with the following identity:
Suppose that $U, V \subset \mathbb{R}^n$, are two open sets.
Let $x^1,...,x^n$ be a system of coordinates of $U$ and $y^1,...,y^n$ one on $V$. ...
1
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2answers
128 views
Pullback of differential form of degree 1
Good evening,
In differential forms (in the proof of the naturality of the exterior derivative), I don't get why if $h\in \Lambda^0(U)$ and $f^*$ is the pullback then,
$f^*dh=d(f^*h)$.
I wrote ...
-3
votes
1answer
251 views
Spivak Calculus on Manifolds, Problems # 5-13 c) [closed]
This regards Problems #5-13c) in Spivak's "Calculus on Manifolds":
Given a differentiable map $g:\mathbb{R}^n {\rightarrow} \mathbb{R}^p$ with $n \ge p$ such that $g'(x)$ as rank $p$ whenever ...
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3answers
132 views
Interesting question in differential geometry
Let $ \alpha $ be a closed $ 3 $-form on $ \mathbb{R}^{4} \setminus \{ 0 \} $. Let $ i: S^{3} \hookrightarrow \mathbb{R}^{4} $ be the canonical embedding of $ S^{3} $, and suppose that $ \Omega := ...
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0answers
42 views
Problem on parametrization/induced orientation/integration.
Let $M$ be the $2-$manifold in $\mathbb{R}^3$ consisting of all points $x$ such that
$$4x^2+y^2+4z^2 =4$$
and $y\ge 0$\
Then $\partial M$ is the circle consisting of all points such that
...
2
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1answer
88 views
Integrals of Differential Forms
I am working out of Munkres Analysis on Manifolds and I see that he claims for $\eta = f dx_1 \wedge \ldots \wedge dx_k$.
$$ \int_A\eta = \int_{x \in A} \eta(x)\big((x;a_1) ,\ldots, (x;a_k) \big)$$
...
3
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2answers
136 views
$\alpha\wedge\beta = 0$ for all $\beta$ implies $\alpha = 0$ without using the Hodge dual
Let $\alpha$ be a differential $k$-form on an orientable smooth $n$-dimensional manifold. If $\alpha\wedge\beta = 0$ for every differential $(n - k)$-form $\beta$, then $\alpha = 0$ because we can ...
1
vote
1answer
132 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:
...
5
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1answer
165 views
Differential Forms in Spivak vs Rudin
Can anyone give me the gist of the difference of the treatment of Stokes' Theorem in Spivak versus baby Rudin (chapter 4 in spivak, chapter 10 in rudin)? I need to do some problems from Rudin but ...
3
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2answers
121 views
Differential Forms Help
I have a background in Analysis, specifically with Baby Rudin. However, as many people note, Rudin does not do a very good job discussing differential forms. Could someone please refer me to an ...
1
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1answer
34 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 ...
3
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1answer
121 views
Line integral and integration of differential forms
The definition of integral of a $k$-form $\omega$ over a parametrized manifold $ Y_\alpha $ is $ \int_{Y_\alpha}\omega = \int_A\alpha^*\omega$ where $ \alpha\colon A \to R^n $.
Let $ \gamma:(a, b) ...
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Resources for Measure Theory and Differential Forms
What are the best books/resources for measure theory and a rigorous introduction to the integration of differential forms? They do not have to be in the same book and I have a Baby-Rudin level (well, ...
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0answers
44 views
Compute the differential of a form
From Munkres "Analysis on Manifolds"
Consider the form $ \omega = xydx + 3dy -yzdz $. Check by direct computation that $ d(d\omega) = 0 $. Can someone show me how to do it, because I don't seem to be ...
3
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2answers
142 views
algebraic manipulation of differential form
Suppose $\phi_1, \phi_2, \dots, \phi_k \in (\mathbb{R}^n)^*$, and $\mathbf{v}_1, \dots, \mathbf{v}_k \in \mathbb{R}^n$
$(\mathbb{R}^n)^*$ stands for the space of all linear transformations that goes ...
10
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3answers
450 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 ...
2
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1answer
68 views
Wedge product with a non-degenerate form
Let $\alpha$ be a non-degenerate form in $\Lambda^k(V)$ for some vector space $V$, $\dim V = n$. (Here non-degenerate means that if $x\in V$ is nonzero, then $(y_1 , ... , y_{k-1}) \mapsto \alpha(x , ...
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0answers
213 views
How to prove this formula for Lie Derivative for differential forms
The professor gave this formula without providing a proof. I would like to know how this can be derived.
Let $X$ be a vector field, $w$ be a $p$-form. Then,
$$L_X ...
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0answers
122 views
Top deRham cohomology group of a compact orientable manifold is 1-dimensional
Let $M$ be a compact smooth orientable manifold of dimension $n$. I am looking for a simple proof that $H_{dR}^n(M) \cong \mathbb R$. Equivalently, an $n$-form which integrates to 0 is exact. I can ...
5
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1answer
136 views
Integrating a 3-form over the 3-sphere
Consider the 1-form $\alpha = xdz + ydw -(x^2 + y^2 + z^2 + w^2)dt$ on $\mathbb{R}^5$.
I'm trying to find $\int_S d\alpha \wedge d\alpha$, where $S \subset \mathbb{R}^5$ is given by $x^2 + y^2 + z^2 ...
2
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2answers
218 views
Hodge dual on orthonormal basis: two inconsistent answers
I'm trying to learn differential geometry using Göckeler & Schücker's book and I have some problems with the hodge star. As an example, say we have two orthonormal bases $e^i$ and ...
2
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1answer
40 views
Problem with integration of $1$-form on surface
I have some problem with integration of differential forms on algebraic surfaces (I'm reading Cartan's book on analytic functions). Let $X \subseteq \mathbb{C}^2$ be an algebraic curve given by ...
