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

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72 views

Is there a similar equivalence like the divergence theorem for surface integrals non-linear in the normal vector?

The divergence theorem can be stated as $$\bigcirc \hspace{-1.3em} \int \hspace{-.8em} \int\limits_{\partial\Omega} dA\,n_i = \iiint\limits_\Omega dV\partial_i$$ applied to an arbitrary function ...
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1answer
112 views

Ideal generated by differential forms

I have troubles picturing what elements belong to a particular ensemble. Let $\omega_1$,...,$\omega_r$ be differential 1-forms on a $C^\infty$ n-manifold that are independent at each point. ...
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0answers
59 views

derivatives of coordinates on a riemann surface

Let $X$ be a compact connected Riemann surface of genus $g>0$. Let $(\omega_1,\ldots,\omega_g)$ be a basis for $H^0(X,\Omega^1)$. Let $x$ be a point in $X$ and let $z:U\longrightarrow B(0,1)$ be a ...
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2answers
47 views

Solve the i.v.p DE

Solve the i.v.p $y^{(4)}-y'''=0 , y(0)=0, y'(0)=0, y"(0)=0, y"'(0)=0$ Would I use the formula $a^{(1/n)}=R^{(1/n)}e^{(e^{i(alpha+2k(\pi))/n})}$
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1answer
53 views

How to proceed this computation with differential forms?

I've been studying Spivak's differential geometry book and he defines the exterior derivative of $\omega \in \Omega^k(M)$ in a coordinate system $(x,U)$ by $$d\omega = d\omega_{i_1\cdots i_k}\wedge ...
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3answers
101 views

Prove $d(f\alpha)=d(f \wedge \alpha)$

I am reading the article http://en.wikipedia.org/wiki/Exterior_derivative and a definition of an exterior derivative from Axioms for the exterior derivative. How could I show that if $f$ is a function ...
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2answers
96 views

Transformations of Volume Forms Under Diffeomorphisms

I have an exercise here, and I have no idea how to do it. Let $ U $ and $ V $ be open sets in $ \mathbb{R}^{n} $ and $ f: U \to V $ an orientation-preserving diffeomorphism. Then $$ ...
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2answers
62 views

Equality involving exterior product..

suppose you have a differential form $\omega$ writting in local coordinates as $$\omega=\sum_{i=1}^ndx_i\wedge dy_i.$$ Can anyone help me showing the following equality: $$\omega^n=n!(dx_1\wedge ...
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1answer
105 views

integral of closed differential form

This is my first course in differential forms so it might be a trivial question. If $\mu$ is a $n-1$-form on $n$-dim manifold $X$ the book uses that $$\int_X{d\mu}=0.$$ Is this expression valid for ...
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1answer
145 views

Averaging differential forms

Let $M$ be a manifold with a circle action, i.e. with a 1-parameter group of diffeomorphisms $\phi_t:M\to M$ of period 1. I think of the average of a differential form $\omega \in \Omega^n(M)$ with ...
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1answer
26 views

Differentiability problem .

Hi can someone help me with the following problem. I am having difficulties evaluating : $$ \frac {d} {dt} f'(u(t)) $$ Is it just $f''(u(t))$ ? Thanks
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2answers
65 views

Writing a diffEQ as $P(x,y)dx + Q(x,y)dy = 0$ instead of in terms of $dy/dx$

I'm reading in Tenenbaum and Pollard's Ordinary Differential Equations where they introduce the concept of the differential. Suppose $y=f(x)$ is differentiable. He defines the differential by $dy(x, ...
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1answer
55 views

$V, W$ vector fields, $\omega$ one-form, $d\omega(V, W)=?$

$V, W$ vector fields, $\omega$ one-form, what is the definition of $d\omega(V, W)$? I have seen some equalities and I suppose that $d\omega(V,W)=(V^jW^i-V^iW^j) \frac{\partial f_i}{\partial x^j}$ ...
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2answers
101 views

“Ordinary” and “polar” vector fields in Euclidean $3$-space

In his book Differential Forms with Applications to the Physical Sciences, on pages 19--20, Harley Flanders writes: "a one-form $$ \omega = P\,dx+Q\,dy+R\,dz$$ may be identified with an ...
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1answer
60 views

What is a de Rham k-form?

I generally know what a differential k-form is. But what does it mean for a k-form to be a "de Rham" k-form? Thanks in advance!
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1answer
230 views

Definition of pull back operation

Let $\varphi:U \rightarrow V$ be a differentiable map between open sets $U \subset \mathbb{R}^n$ and $V \subset \mathbb{R}^m$. Define the pull back operation $\varphi^*: \Omega^{k}(V) \rightarrow ...
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1answer
65 views

Dimension of intersection of two manifold

For any $f\in C^\infty(X)$, $X$ smooth manifold. Define $$X_{df}:=\{(x,df_x): x\in X, df_x= T^*_x X\}$$ $$X_0:=\{(x,\zeta): \zeta=0 \text{ in } T_x^*X\}$$ In the exercise we are asked for proof: If ...
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1answer
263 views

Integrating factor and differential 1-forms

I am working on the following exercise: The function $f$ is called an integrating factor for the 1-form $\omega$ if $f({\bf x}) \neq 0$ for all $\bf x$ and $d(f\omega) = 0$. If the 1-form $\omega$ ...
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1answer
127 views

Statement of the $d d^c$-Lemma

I'm looking at the definition of Green's function $g_\mu$ for the Laplacian $\Delta_\mu$ associated to a positive $(1,1)$-form $\mu$ on a Riemann Surface $X$. In specific the main request that the ...
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1answer
30 views

pullback on vector fields and 1-forrms

On a Riemannian manifold $M$ one can identify 1-forms and vector fields by $$ \alpha(p) = \langle X(p),\cdot\rangle_p $$ Since we can perform a pullback on both 1-forms and vector fields I expected ...
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1answer
101 views

Finding Reeb Vector Field Associated with a Contact Form

I would greatly appreciate it if you could help me with the following: I'm curious as to how to find the Reeb field $R_w$ associated to a specific contact form $w$; does one actually find $R_w$ as ...
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1answer
28 views

Question on 2-chain on $\mathbb{R}^3$

Let $\gamma:[0,1]\to\mathbb{R}^3\setminus\{0\}$ be a simplex, with $\gamma(0)=\gamma(1)$. How can I show that exists a $2$-chain $\sigma$ on $\mathbb{R}^3\setminus\{0\}$ such that ...
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1answer
152 views

Clarification about differential forms in polar coordinates

In my course about differential forms, we define 1-forms as follows: If $(e_1,..,e_m)$ is the standard basis of $\mathbb{R}^m$ and $\sigma$ a chart around $p\in M$ on the m-dimensional manifold $M$, ...
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1answer
51 views

Restricting the domain of an integral on a manifold

I would like to prove the following: Guess. Suppose M is an orientable smooth manifold without boundary and W is an open set in M. If $\omega$ is a smooth n-form on M such that $\operatorname{supp} ...
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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 $V$. ...
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1answer
289 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
36 views

index $ n(F;D)$ is odd integer

Let $ F: U \subset \mathbb R ^2 \to \mathbb R^2$ be a map with $ F(x,y) = (f(x,y) , g(x,y))$ ,satisfies $F(-q)=-F(q) \quad \forall q \in D \subset U$ where $D$ is a closed disk with center the origin ...
0
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1answer
79 views

degree of proper mapping for linear case

Let $U,V$ be two open sets in $R^n$ and $f:U\to V$ proper $C^{\infty}$ map (proper = preimage of compact set is compact). Then we have $$\int f^{*}\omega=\deg(f)\int \omega,$$ for $\omega \in ...
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1answer
121 views

Tangent Vectors and Differential 1-forms.

I have this 1-form on $\mathbb R^3$ given by $\omega=dz+\frac{x}{2}dy-\frac{y}{2}dx$. If $p_0=(x_0,y_0,z_0)$ and $\vec v=(u_0,v_0,w_0)$, then find the set of tangent vectors $\vec v_{p_0}$ such that ...
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0answers
23 views

differential form and preserving volume

I was reading a beamer about differential foms and I found the following problem. if $M$ is an orientable manifold and $\omega \in \Omega^n(M)$ is the volume form and Let $X$ any vector field. Then ...
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20 views

Show that $d\log f$ is a 1-current on a 1-dimensional complex manifold

I am having trouble with this problem (and it might be because I have the formulation slightly off). I need to show that $d\log f$ is a 1-current on a 1-dimensional complex manifold $M$. This means ...
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50 views

Proving $d(f^*e_i')=0$

Let $f:M\to N$ be a differentiable map between two manifolds. $e_i$ is a basis vector of $N$ with respect to some chart and $e_i'$ its dual (i.e. $e_i'(e_j)=\delta_{ij}$). How do I prove the ...
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0answers
11 views

How should I work with antisymmetric indices on forms?

I've always tended to be confused by antisymmetrized indices on differential forms, and the only way I can really work with them is by writing out all the terms - which is just not practical beyond 3 ...
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1answer
64 views

Condition for Differential Forms to Pass to the Quotient

everyone: I was reading this question : What do we mean when we say a differential form "descends to the quotient"? which is related to mine. But the reply given did not answer my question ...
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0answers
56 views

Is a Differential 1-form ( on M^3)Dual to a Contact Vector Field a Symplectic Form?

say $w$ is a global contact 1-form on a 3-manifold $M^3$ , meaning $w \wedge dw \neq 0$ at any point in the manifold , and let $X$ be the vector field dual ( under, say, a choice of Riemannian metric ...
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0answers
64 views

differentiable equal r-forms

Let $\alpha$, $\beta$ be two $r$-forms continuous in $U\subseteq \mathbb{R}^n$ open. If $\int_M \alpha =\int_M \beta $ for all surface $M\subseteq U$   dimension $r$, compact, with boundary, then ...
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1answer
33 views

Prove $\Lambda^kf:\Lambda^kW^* \to \Lambda^kV^*$ well-defined and linear

Let $V$ and $W$ be finite dimensional vector spaces, and $f : V \to W$ a homomorphism. Show that $\Lambda^kf:\Lambda^kW^* \to \Lambda^kV^*$, defined by $\Lambda^k\alpha(v_1, ..., v_k)=\alpha(f(v_1), ...
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0answers
111 views

Hodge dual exterior derivative

The introduction of the Hodge dual to the structure of the cotangent space requires the reference to a specific basis or an inner product. I was wondering however, if the composition of hodge dual and ...
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0answers
152 views

$\alpha \wedge \beta = 0$ iff $\beta = \alpha \wedge \gamma$

I have been given the following problem: Let $\alpha$ be a nowhere-zero 1-form. Prove that for a (p+1)-form $\beta$ $(p\geq0)$, one has $\alpha \wedge \beta = 0$ if and only if $\beta = \alpha ...
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2answers
41 views

Decomposing a 2-form into a product of two 1-forms

I'm trying to decompose the 2-form $\omega = dx \wedge dy + 4dx \wedge dz + 3dz \wedge dy$ (in $\mathbb{R}^{3}$) as the product of two 1-forms, but get stuck. Is it posible to do this?
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1answer
58 views

How to prove matrix differential rules using differential forms rules

I am learning differential forms on my own through lectures on youtube and one of the things that I am attempting to do is to check if I can derive matrix differential rules using the rules for ...
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0answers
58 views

Show that $\int_Md\omega=0$.

Let $\omega$ be a continiously differentiable $(k-1)$-form in the open set $U\subset\mathbb{R}^n$ and $M\subseteq U$ an orientated compact k-dimensional manifold. Show that $$ ...
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2answers
56 views

Finding a particular form that orients a k-manifold

Suppose one has a $k$-manifold given by $f^{-1}(0)$ for some $C^1$ map $f: U\to \mathbb{R}^{n-k}$ (where $[D f(x)]$ is surjective). How can one construct a form-field $\omega$ that orients the ...
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0answers
94 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 ...
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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 ...
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1answer
39 views

DPEs system which I cannot seem to solve

Consider the following DPE system: $$\left\{ \begin{array}{rcl} g_x - f_y& = &1-x^2 \\ h_x - f_z &= &3x^2 \\ h_y - g_z &=& -1 \ \end{array}\right .$$ This comes from trying ...
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1answer
135 views

Coordinate change in a differential form

The task is to rewrite the following differential form in polar coordinates: $$w = \sqrt{x^2 + y^2} \, dx \land dy $$ I did it by a direct substitution: $$\begin{cases}x = r \cos \varphi \\ y = r ...
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0answers
156 views

differential form vanish somewhere on compact orientable without boundary manifold [duplicate]

(a)Let $ M =M^n$ a $n-$dimensional compact, orientable, differential manifold without boundary ( $\partial M = \emptyset$). Define $\omega$ a differential $(n-1)-$ form on $M$. Prove that there ...
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0answers
315 views

pullback of 1-differential form

Given a $ \phi_{0} \in \mathbb R$. Consider the function $ \displaystyle{\phi (t) = \phi_{0} + \int_0^t (ab'-a'b)(u) \text{du}}$ where $ a^2(t) + b^2 (t)=1$ Prove that: (i) $ \phi(0)= \phi_{0}$ ...
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1answer
164 views

Exterior Derivative question

In http://mathworld.wolfram.com/ExteriorDerivative.html, there is a section that starts from: Define the exterior derivative by $Dt ≡ \frac{\partial}{\partial x} \wedge t$ First of all, what ...