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

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Flat Surfaces in $\mathbb{R}^3$ Can Be Bent Only Along Straight Lines

This is a problem out of Elementary Differential Geometry by Barrett O'Neill (Chapter 6 Section 3 Number 2). Let $M$ be a flat surface in $\mathbb{R}^3$ with principal curvatures $k_1$ and $k_2$, ...
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48 views

If differential 1-forms agree on chains with integer coefficients, are they equal?

Let $M$ be a real, smooth manifold. Let $\omega_1$ and $\omega_2$ be differential 1-forms on $M$, and let $C_1(\mathbb Z,M)$ denote the set of 1-chains with integer coefficients. If \begin{align} ...
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122 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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72 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
50 views

understanding simple multivariable integrals in terms of differential forms

I am learning a bit about differential forms: defining differential forms in terms of elementary forms, integrating forms over parametrized domains, etc. I would like to relate this to my previous ...
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500 views

Differential Forms, Exterior Derivative

I have a question regarding differential forms. Let $\omega = dx_1\wedge dx_2$. What would $d\omega$ equal? Would it be 0?
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100 views

Use the Fundamental Theorem to deduce the formula for the area of an ellipse.

Use the Fundamental Theorem (Green's Theorem) to deduce the formula for the area of an ellipse. Hint: find a 1-form whose exterior derivative is $ dxdy $.
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156 views

Confused about Wikipedia page on differential forms

I know next to nothing about differential forms, but I saw them being mentioned repeatedly on this site, so I went to Wikipedia to try to understand what a differential form is. Most of it is going ...
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49 views

For a differentiable map $f: \mathbb{R^n}\to \mathbb{R^n}$, Show that $f^*({dy_1 \wedge\cdots \wedge dy_n})=\det(df)dx_1\wedge \cdots\wedge dx_n$

Let $f: \mathbb{R^n}\to \mathbb{R^n}$ be a differentiable map given by $f(x_1,\cdots, x_n) = (y_1,\cdots,y_n)$. Show that $f^*({dy_1 \wedge\cdots \wedge dy_n})=\det(df)dx_1\wedge \cdots\wedge dx_n$ ...
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42 views

n-form associated with a vector field with general metric

With the euclidean metric I use the musical isomorphisms to obtain $1$-form associated with a vector field, so for a vector field $\vec{F}=(f_1,f_2,f_3)$ we have $ \vec{F}^{\flat}=f_1dx+f_2dy+f_3dz$ ...
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122 views

Covariant differential on p forms

On Peter Li's book Geometric Analysis on page 19 (http://www.im.ufrj.br/andrew/GR14-2/Lecture%20Notes%20on%20Geometric%20Analysis.pdf) I can't understand the following line $\begin{array}{l} ...
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21 views

contraction identity on $k$-forms

$i_\mathbb{X} \omega $ is the contraction of $\omega$ with respect to $\mathbb{X}$. In my notes it is stated that $i_\hat{\mathbb{X}} dx = dx(\hat{\mathbb{X_t}})$. I cannot see how this fits the ...
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40 views

Given an $(n-1)$-form $\varphi$ on a smooth orientable $n$-manifold, there is a vector field $v$ such that $i_v\varphi = 0$.

I am working on the following problem. Let $M$ be a smooth orientable $n$-manifold, $n \geq 2$, and let $\varphi$ be a smooth $(n-1)$-form on $M$. Show that there is a vector field $v$ on $M$ such ...
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1answer
62 views

If $M$ is a compact manifold what does $\partial M$ mean?

In the generalized form of stokes theorem it states that the integral of the $k+1$ differential form of an operator over a compact manifold $M$ is equivalent to the integral of the $k$ differential ...
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57 views

Differential forms and minor expansion, question about notation.

There are lectures by Theodore Shifrin on differential forms, and sadly one video ends suddendly where he explains some notation. I try to formulate it in my own words: When k=n, we have ...
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1answer
36 views

Integrate the differential form over a cardioid

$\omega=\dfrac{-ydx+(x-1)dy}{(x-1)^2+y^2}$ Calculate $\int_C\omega$ where $C...r=1+\cos\varphi$ (positively oriented) I'm still pretty lost when it comes to differential forms but as far as I ...
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47 views

Is it true that $0$ is the only exact $0$-form

I am totally new to the concepts of forms so sorry if my question is trivial. I came across a statement that ''there are no exact $0$-forms as there is no $-1$ form. So I revisited the definition ...
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60 views

If $\omega\wedge\beta$ is exact for every closed form $\beta$, then $\omega$ is exact.

Let $\omega$ be a closed $k$-form. Then: If $\omega$ is exact, for every closed form $\beta$, the form $\omega\wedge\beta$ is exact. Proof: Let $\omega=d\alpha$. Now $d(\alpha\wedge\beta) = ...
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1answer
29 views

How to show $\nu=dx_1\wedge\ldots \wedge dx_n$?

Let $\nu$ be the $n$-form in $\mathbb R^n$ satisfying $\nu(e_1, \ldots, e_n)=1$ where $\{e_1, \ldots, e_n\}$ is the canonical base of $\mathbb R^n$. Let $\displaystyle v_i=\sum_{j=1}^n a_{ij}e_i$. How ...
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76 views

The kernel of a differential one-form

I'm thinking about the kernel of a differential one-form $\theta\in\Lambda^{1}(M)$: $$ Ker(\theta):=\left\{X\in\mathfrak{X}(M) \;|\; \theta(X)=0\right\} $$ Now suppose $X\in Ker(\theta)$, then is ...
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60 views

Cartan formalism calculation

Just to test out the Cartan formalism, I decided to apply it to the sphere. So, it admits a metric, $$\mathrm{d}s^2 = \mathrm{d}r^2 + r^2 \sin^2 \phi \mathrm{d}\theta^2 + r^2 \mathrm{d}\phi^2$$ from ...
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118 views

Question about differential form

$\omega = y dx + dz$ is a differential form in $\mathbb{R}^3$, then what is ${\rm ker}(\omega)$? Is ${\rm ker}(\omega)$ integrable? Can you teach me about this question in details? Many thanks!
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82 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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208 views

Locally exact differential in a disk is exact

I'm reading through Ahlfors' Complex Analysis text for self study, and I found difficulty with a proof. In chapter 4 he defines a locally exact differential as a differential who is exact in some ...
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1answer
108 views

A differential form to compute the k-volume of a k-parallelogram in n dimensions

Computing the k-volume of a k-parallelogram (i.e. a parallelogram spanned by k n-dimensional vectors) in n dimensions is straightforward: Let $P=[\overrightarrow{v_1},...,\overrightarrow{v_k}]$, then ...
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82 views

On simply connected domains

During lecture we defined simply connected set in $\mathbb{R}^n$: $\Omega \subset \mathbb{R}^n$ is simply connected, iff it is connected and for any $C^1$ closed curve $c:[0,1]\rightarrow \Omega$ ...
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1answer
197 views

How general formulation of Stoke's theorem relate to Kelvin-Stokes theorem

I asked a similar question, but I realized the question is too vague and it's better to start a new one: We know that there are two usually used formulations of Stoke's theorem. One is vector ...
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1answer
45 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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93 views

Show that $\omega = d(I\omega)$ if $d\omega = 0$

Let $\omega = P\ dx + Q\ dy$ be a 1-form on $\mathbb{R}^2$. Also, define a 0-form $I\omega({\bf x}) = I\omega(x, y)$ by $$ I\omega({\bf x}) = \int_0^1 P(t {\bf x}) x + Q(t {\bf x}) y\ dt.$$ I would ...
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135 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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111 views

Is the differential form $\omega(x,y)=\frac{2x}{x^2+y^2-4}dx+\frac{2y}{x^2+y^2-4}dy$ exact on his natural domain?

The natural domain $D$ of the differential form $\omega$ is $D=D_1 \cup D_2$ where $D_1=B\bigl((0,0),2\bigr)$ is the open ball of center in the origin and radius $r=2$ and ...
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254 views

Recover the differential form by its pullback

I have a smooth manifold $M$ in $\mathbb{R}^n$ given by $M = \{ x \in \mathbb{R}^n \mid g(x) = 0 \}$. Its atlas consists of a single chart $(M,\varphi)$, where $$ \begin{array}{rcl} ...
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141 views

Closed differential form.

In the paper, page 5, Line 5, why the form $\eta_j$ is closed? We have to show that $d \eta_j=0$. Here $\eta_j = f(t)dt$ for some function $f$. Is it always true that $d\eta_j$ for any $f$? Thank you ...
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42 views

A question on self-dual differential 2-forms

This question is from Lemma 2 in Derdzinski's paper. Let $$\omega=e_1\wedge e_2+e_3\wedge e_4, \eta=e_1\wedge e_3+e_4\wedge e_2, \theta=e_1\wedge e_4+e_2\wedge e_3$$ be a basis for self-dual ...
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1answer
21 views

wedge product with and without a second pair of vectors

I am starting to study wedge products, and am stuck on notation. The Bachman book on differential forms says $$ \omega \wedge \nu ( v_1, v_2 ) $$ "gives the area of the parallelogram spanned by ...
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33 views

How to make an ideal generated by differential forms into a differential ideal?

Let $M=\mathbb{R}^4$ with standard coordinates $x_1,x_2,x_3,x_4$. Let $\alpha=x_2dx_1+x_3dx_3+dx_4$ and $\beta=2dx_2+x_1^2dx_3+x_1dx_4$ How to find a 1-form $\gamma$ such that the ideal generated ...
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49 views

Integrating a differential form inside a cylinder

Let S be cylinder given by $x^2+y^2=1$ between $z=1$ and $z=3.$ For $\varphi=e^xdx\wedge dy+ ydz\wedge dx+xdy\wedge dz$ find $\int_S\varphi$. I managed to finish the problem, but I'm getting ...
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66 views

The property of an integral manifold of differential form

Let's define the $1$-form $\eta$ on $\mathbb{R}^3$ by formula: $$\eta = A(x,y,z)\;\mathrm{d}x + B(x,y,z)\;\mathrm{d}y + \mathrm{d}z \text{.}$$ And let's assume that $$\mathrm{d}\eta \wedge \eta = ...
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66 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 ...
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65 views

A connection to Stoke's Theorem (I think)

This is homework. I just finished a question regarding double integration over the unit sphere involving pullbacks of differential forms to provide context (course is advanced Calculus). The question ...
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1answer
224 views

Line integral - should I parametrize the square?

I have the following $1-\text{form}$ defined: $$\omega = \displaystyle\frac{2xy}{(1-x^2)^2+y^2}\mathrm{dx}+\displaystyle\frac{1-x^2}{(1-x^2)^2+y^2}\mathrm{dy}$$ I'd like to find ...
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54 views

Show a 2-form is exact finding a primitive.

I have to show that $\omega=-4xy\:\mathrm{d}x\wedge \mathrm{d}y-2xz\:\mathrm{d}z\wedge \mathrm{d}x +2yz\:\mathrm{d}y\wedge \mathrm{d}z$ is exact finding a primitve of $\omega$ (by Poincare lemma I ...
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1answer
59 views

Winding number of a linear transformation?

I know that I am computing something incorrectly. I am trying to compute the index of a positive determinant linear bijection. The form I am using is $\omega = \frac{-y dx + x dy}{x^2 + y^2}$. I ...
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1answer
29 views

If $\omega$ is compactly supported form then so is $d\omega$?

If $\omega$ is a compactly supported differential form then so is $d\omega$. Is it true?
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127 views

What is the one form given its value for a vector field?

I read an article on vector fields. the author defined a 1-form on a manifold $M$ as $u(X)=\rho$ when $X$ is a given vector field and $\rho$ is a given real valued function defined on $M$. can we ...
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68 views

Differential closed form

Im trying to go alone through Fultons, Introduction to algebraic topology. He asks whether there is a function $g$ on a region such that $dg$ is the form: $$\omega =\dfrac{-ydx+xdy}{x^2+y^2}$$ in some ...
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76 views

Visualizing Non-Zero Closed-Loop Line Integrals Via Sheets?

How do I visualize $\dfrac{xdy-ydx}{x^2+y^2}$? In other words, if I visualize a differential forms in terms of sheets: and am aware of the subtleties of this geometric interpretation as regards ...
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88 views

A (not so?) simple question about differential forms

Let $M^n$ be a compact orientable manifold and let $\omega$ be a $(n-1)$-form in $M^n$. I want to show that there is $p\in M$ such that $(d\omega)_p=0$. Can somebody help me, please ? Thanks :)
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50 views

Is $\omega = dU = sin(x+y)dx+cos(x+y)dy$ an exact form?

In my thermodynamics homework I should prove that $dU = sin(x+y)dx+cos(x+y)dy$ is a function of state. Which means it's integration over any path be constant or in other word $dU$ should be an exact ...
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70 views

How to show $[\omega]=0$ implies $[\omega^n]=0$?

I'm trying to prove the following: If $(M, \omega)$ is a symplectic manifold and $[\omega]=0$ then $[\omega^n]=0$, where $[\omega]$ is the De Rham cohomology class of $\omega$. Well what I've done ...