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

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94 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
135 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 ...
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1answer
157 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
134 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 $F:M\rightarrow\overline M$ ...
2
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1answer
231 views

Relevance of Differential Forms

I recently started reading about differential forms, and I am trying to figure out their purpose. Lets say $\omega=y\,dx+x\,dy$, and we want to evaluate $\int_C \omega$ over the curve parametrized by ...
4
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1answer
205 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 ...
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0answers
64 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 ...
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2answers
142 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 ...
2
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1answer
87 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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1answer
398 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 ...
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0answers
96 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 ...
3
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2answers
131 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 ...
4
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1answer
65 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 ...
2
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0answers
66 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
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1answer
88 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

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 $V$. ...
3
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1answer
292 views

Jacobian when representing integral of differential form by Riemann integral?

In Terence Tao's note: If $Ω$ is any open bounded domain in $R^n$ , we then have the identity $$\int_Ω f (x)dx_1 ∧ . . . ∧ dx_n = \int_Ω f (x) dx$$ where on the left we have an integral of ...
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1answer
507 views

Is there a Stokes theorem for covariant derivatives?

A $V$-valued differential form on $M$ is a smooth map $\omega : TM \to V$ such that $\omega$ restricted to any tangent space $T_p M$ is an element of the $V$-valued exterior algebra $\Lambda^n (T_p M, ...
8
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3answers
170 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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2answers
1k 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 ...
2
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1answer
116 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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3answers
3k 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 ...
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1answer
403 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 ...
5
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2answers
187 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 ...
2
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1answer
397 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: ...
4
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2answers
287 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 ...
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1answer
43 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
311 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) ...
4
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2answers
231 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 ...
2
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1answer
187 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 , ...
2
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0answers
395 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
331 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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1answer
46 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 ...
2
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0answers
442 views

Evaluating a surface integral with differential forms

Let $\alpha=x dy-\frac{1}{2}(x^2+y^2)dz$ be a differential form in $\mathbb{R}^3=\{(x,y,z)\;|\;x,y,z\in\mathbb{R}\}$and let $Z=\{(\cos\theta,\sin\theta,s)\;|\;0\leq\theta \leq2\pi, 0\leq s\leq 1\}$ be ...
5
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2answers
511 views

Differential Forms: High Level Approach to Real Analysis?

I am currently skimming through the differential forms book by Edwards. I was wondering whether real analysis is basically just a special case of differential forms? I am learning about flows, ...
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1answer
345 views

How can I find the winding number of a curve?

I need to find the winding number of the closed curve $c(t)=(a \cos(t),b \sin(t))^T $, where $a,b > 0$ and $c:[0,2\pi) \to\mathbb{R}^2\setminus\{0\}$. I don't understand how to do this.
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1answer
318 views

Existence of orthonormal frame

Let $M$ be a surface with Riemannian metric $g$. Recall that an orthonormal framing of $M$ is an ordered pair of vector fields $(E_1,E_2)$ such that $g(E_i,E_j)=\delta_{ij}$. Prove that an orthonormal ...
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0answers
64 views

How to apply Gauss's theorem when the metric is unknown

Let $f:U \to \mathbb{R}^3$ be a surface, where $U=\{(u^1,u^2)\in \mathbb{R}^2:|u^1|<3, |u^2|<3\}.$ Consider the two closed square regions $F_1=\{(u^1,u^2)\in \mathbb{R}^2:|u^1|\leq1, ...
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1answer
352 views

The area form of a Riemannian surface

Let $(M,g)$ be an oriented Riemannian surface. Then globally $(M,g)$ has a canonical area-$2$ form $\mathrm{d}M$ defined by $$\mathrm{d}M=\sqrt{|g|} \mathrm{d}u^1 \wedge \mathrm{d}u^2$$ with respect ...
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1answer
474 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 ...
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0answers
48 views

Finding a function given a differential

I'm still pretty new to this differential business and I have a couple of question concerning this problem I've come across. We're given a differential form $\alpha=p_1 dx_1+p_2 dx_2-H(p_1,p_2)dt$ ...
2
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1answer
217 views

Basic computations with differential forms

I've never had any experience with differential forms before, so I'm trying to work through a couple of examples to see if I understand what's going on. I think I understand what I've been doing so ...
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0answers
41 views

Taking the inverse of this object

Let $\{q_i,p_i, i=1,...,n\}$ be coordinate and their conjugate momentum. Let $\xi_k, k=1,...,2n$ be generalized coordinates which equal to $\{q_i,p_i, i=1,...,n\}$ Suppose the matrix ...
4
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1answer
659 views

Definition of “interior derivative” and “exterior derivative”?

In Willie Wong's reply to one question, he used some concepts: "interior derivative" of a differential form and "exterior derivative" of a scalar function on $\mathbb{R}^3$. For "exterior ...
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359 views

Proof of residue theorem (residue formula) for differential forms on curves over an arbitrary closed field.

I have been reading the book Algebraic Geometric Codes: Basic Notions by Tsfasman, Vladut and Nogin. They give a residue formula like this: Let $\mathbb{k}$ be an algebraically closed field and $X$ ...
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1answer
253 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 ...
2
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1answer
350 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 ...
3
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1answer
559 views

Covariant derivative of tensor in terms of local coordinates

Let $f:U \to \mathbb{R}^3$ be a surface with local coordinates $f_i=\frac{\partial f}{\partial u^i}$. Let $\omega$ be a one-form. I want to express $\nabla \omega$ in terms of local coordinates and ...
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1answer
169 views

Computing a Differential Form

Apologies in advance, I don't know TeX, so this might look a bit gross... I'm given a 1-form $A=f_1dx_1+...+f_ndx_n$, infinitely differentiable and closed on $R^n$. I want to show that $dg=A$ for ...
5
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1answer
294 views

Could Residue theorem be seen as a special case of Stokes' theorem?

Residue theorem in complex analysis is seems like Stokes' theorem in real calculus, so a question arose that could Residue theorem be seen as a special case of Stokes' theorem?