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

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3
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1answer
254 views

Curvature tensor of 2-sphere using exterior differential forms (tetrads)

$ds^2= r^2 (d\theta^2 + \sin^2{\theta}d\phi^2)$ The following is the tetrad basis $e^{\theta}=r d{\theta} \,\,\,\,\,\,\,\,\,\, e^{\phi}=r \sin{\theta} d{\phi}$ Hence, $de^{\theta}=0 ...
2
votes
4answers
481 views

Classic example of a non exact form

Let $\dfrac{xdy-ydx}{x^2+y^2}$ be a 1-form defined in $\mathbb{R}^2\backslash\{0\}$. Where can I find a detailed proof that it is not exact? I would prefer a proof that doesn't use results about ...
3
votes
2answers
126 views

Symplectic Form Preserved by Orthogonal Transformation

I'm trying to prove that the symplectic form $$\omega = d(\cos\theta) \wedge d\phi$$ is preserved by the action of $SO(3)$ on $S^2$ where $\phi$ and $\theta$ are spherical polars. Now $SO(3)$ simply ...
1
vote
0answers
82 views

Is Cartan's magic formula applicable to time dependent vector fields?

Cartan's magic formula states: $$\mathcal{L}_v\omega = i_v\mathrm{d}\omega + \mathrm{d}i_v\omega$$ Is this also true for time dependent vector fields? If so: How can I prove it? If not: Is there a ...
4
votes
2answers
177 views

Orientations on Manifold

This is a very basic definition of orientable and very basic example 20.5 however ı could not understand definition in an good way so ı want you to explain my green writing please :) and my example ...
0
votes
1answer
133 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 ...
3
votes
2answers
49 views

Constructing a 2 form that does not vanish on a space from a one form that does

This may be a silly question... Given a nonzero one-form $\omega$ that vanishes on a subspace $W$ (with dimension larger than $2$), is it possible to find another one form $\phi$ such that ...
0
votes
2answers
48 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 $a^{(1/n)}=R^{(1/n)}e^{(e^{i(alpha+2k(\pi))/n})}$
2
votes
2answers
69 views

Why is $\theta \not \in C^{\infty}(S^1)$?

Why is $\theta \not \in C^{\infty}(S^1)$? I know that since $\int_{S^1} d\theta = 2\pi$ then $d\theta$ is not exact. Thus since $d(\theta)=d\theta$, $\theta$ must not be $C^{\infty}$ but it seems ...
1
vote
1answer
62 views

$k$-forms on $\mathbb{R}^n$

Given an expression like $$ dx_1\wedge dx_2 \wedge dx_4 \left( \begin{bmatrix} 1\\2\\3\ \end{bmatrix} \ , \ \begin{bmatrix} 4\\5\\6 \end{bmatrix} \ , \ \begin{bmatrix} 7\\8\\9 \end{bmatrix} \right) \ ...
0
votes
1answer
77 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!
2
votes
3answers
73 views

Whats the connection between formss and vector fields?

I heard someone talking about how vector fields are the kernels of forms. Can someone give me a detailed explanation about how this works? Thanks.
8
votes
3answers
789 views

Inducing orientations on boundary manifolds

Given a $k$-manifold $M$, such that $\partial M$ is a $(k-1)$-manifold, there is a standard way in which $\partial M$ inherits the orientation of $M$. So if $M$ is oriented by the form field $\omega$, ...
6
votes
1answer
110 views

What is the relation of $\int f dx^1\wedge dx^2\wedge …\wedge dx^n=\int f dx^1…dx^n$

In a book "calculus on manifolds" it is defined that $\int f dx^1\wedge dx^2\wedge ...\wedge dx^n=\int f dx^1...dx^n$ but how it is possible the relate the integrand of a multilinear function ...
2
votes
1answer
108 views

Computing $n$-th external power of standard simplectic form

I need some help: Define a 2-form on $R^n$ by $\omega=dx_1\wedge dx_2+dx_3\wedge dx_4+...+dx_{2n-1}\wedge dx_{2n}$. How to compute $\omega^n:=\omega\wedge\omega\wedge\ldots\wedge\omega$?
2
votes
1answer
106 views

What does $\Omega^\bullet(M)$ mean?

What does $\Omega^\bullet(M)$ mean? I know that $\Omega^k(M)$ is the set of all differential k-forms. Thanks in advance!
2
votes
1answer
254 views

Differential Forms and Area

I think I'm just misunderstanding something here, but in $\mathbb{R}^2,$ there exists a $1$-form (in fact infinitely many) $\omega$ such that for any region $S,$ we have $\int_{\partial S} \omega = ...
0
votes
2answers
61 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 ...
2
votes
1answer
216 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 ...
3
votes
3answers
78 views

Integral depending on a path?

I need to check whether differential form $\omega$ has, in the domain $G$, such property that it's integral doesn't depend on path. In my exercise: $\omega = \frac{ydx -xdy}{x^2+xy+y^2}$ and $G= R^2 ...
5
votes
2answers
262 views

If $\omega$ is a 2-form on $\mathbb{R}^4$ and $\omega \wedge \omega = 0$, then $\omega$ is decomposable.

I am trying to prove the following from a book I am reading through. Thm: If $\omega$ is a 2-form on $\mathbb{R}^4$ and $\omega \wedge \omega = 0$, then $\omega$ is decomposable. Note ...
0
votes
1answer
313 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 ...
2
votes
1answer
68 views

Orienting curves with differential forms

Consider the circle given by the equation $x^2+y^2=1$. We can orient this curve by choosing the tangent vector field $(-y,x)^T$, which defines a direction. Supposedly we can do this with by taking an ...
1
vote
1answer
47 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 ...
1
vote
1answer
74 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 $c:[0,1]\rightarrow \Omega$ ...
4
votes
1answer
166 views

Wedge product of differential and volume form

Let $f(x)$ be a $C^1$ function defined on $\mathbb{R}^n$ and $\nabla f(x) \neq 0$ for any $x \in \mathbb{R}^n$. If $d\sigma$ is the volume form on hypersurface $f(x)=c$ induced from $\mathbb{R}^n$ ...
3
votes
1answer
173 views

Algebraic Topology Double Complexes

I am going through Bott and Tu and trying to do Exercise 9.13 which says When a homomorphism $f: K \rightarrow K'$ of double complexes induces $H_d$-isomorphism, it also induces $H_D$-isomorphism. ...
1
vote
0answers
49 views

A $k$-form is thought of as measuring the flux through an infinitesimal k-parallelepiped

On the wikipedia has written "A $k$-form is thought of as measuring the flux through an infinitesimal k-parallelepiped." How does a $k$-form do this? if this sentence is right, then the flux of which ...
4
votes
0answers
27 views

An explicit $\Lambda_R^\ell(M)$ when $M$ is not free

Let $\Lambda_R^n(M)$ be the nth exterior power of an $R$-module $M$. Let us assume $M$ is finitely generated. When $M$ if free, say, $M=R^{\oplus d}$, we know \begin{equation} \Lambda_R^n(M)\cong ...
1
vote
3answers
405 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?
2
votes
1answer
157 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 ...
1
vote
1answer
163 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 ...
1
vote
1answer
208 views

Example of differential form usage of Stoke's theorem

There are many examples that show how Kelvin-Stokes theorem is used. But I would like to see an example that uses differential form usage of Stoke's theorem and is hard or impossible to solve by ...
1
vote
1answer
49 views

Question about Interior product computation

I need to evaluate $\omega=i_X(dx\wedge dy)$ where $X$ is a vector field in $\mathbb{C}^2$ (which means $p=2)$. If I write $X(x,y)=(X_1(x,y),X_2(x,y))$, or simply $X=(X_1, X_2)$, then the interior ...
1
vote
1answer
53 views

Basis for the space of $p$-forms

According to the book I read, a general $p$-form can be written as: $$\omega=\omega_{a_1\ldots a_p} dx^{a_1}\wedge\ldots\wedge dx^{a_p},\hspace{0.5cm} a_1>a_2>\ldots>a_p$$ where I have used ...
4
votes
1answer
288 views

Baby Rudin, Chapter 10, Problem 23 (d) - Differential forms.

In problems 21 and 22, Rudin defines the differential forms $\eta=\dfrac{xdy-ydx}{x^2+y^2}$ and $\zeta=\dfrac{x dy \wedge dz+ydz \wedge dx+z dx \wedge dy}{r^3}$ and the reader is asked to prove ...
3
votes
0answers
98 views

Is this a trivial Stokes exercise?

I'm working on this exercise from an old Spivak Differential Geometry book which I must be misunderstanding. The question reads: Let $M$ and $N$ be compact $n$-dimensional manifolds with boundary and ...
3
votes
2answers
133 views

Is $\omega_n$ exact in $\mathbb R^n -\{0 \}$?

For $n \ge 2$ consider the differential form $\omega_n=r^{-n} \sum_{i=1}^n(-1)^{i-1}x_idx_1 \wedge \ldots \wedge dx_{i-1} \wedge dx_{i+1} \wedge \ldots \wedge dx_n$, defined on $\mathbb R^n \setminus ...
7
votes
2answers
177 views

Dimension of de Rham Cohomology groups?

Is there a simple way to prove that the de Rham cohomology groups of a compact manifold $M$ have finite dimension as $\mathbb{R}$-vector spaces?
1
vote
1answer
80 views

Use Fund Thm to evaluate the integral of $ ze^{x^2} dydz + 3ys dydz + (2-yz^7)dxdy $ over surface of the unit cube, except bottom face.

Use the Fundamental Theorem to evaluate the integral of $ ze^{x^2} dydz + 3ys dydz + (2-yz^7)dxdy $ over the surface of the unit cube, except the bottom face.
1
vote
1answer
98 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 $.
2
votes
1answer
84 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 ...
4
votes
1answer
176 views

Relating volume elements and metrics. Does a volume element + uniform structure induce a metric?

AFAIK a metric uniquely determines the volume element up to to sign since the volume element since a metric will determine the length of supplied vectors and angle between them, but I do not see a way ...
2
votes
1answer
152 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 ...
2
votes
1answer
225 views

Sphere volume and generalized Stokes' theorem

The area of a circle is $\pi r^2$, and the circumference is the derivative of this: $2\pi r$. The same holds in one higher dimension: the volume of a sphere is $\frac{4}{3} \pi r ^3$ and the ...
0
votes
1answer
52 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} ...
3
votes
2answers
554 views

Chain rule and differential forms

It is easy to show that the differential forms of order $1$ obeys a form of chain rule. To be precise, $d(f(g(x)) = f^\prime(x) d(g(x))$. This can be for example proved by fixing a co-ordinate basis ...
2
votes
2answers
152 views

How is differential form different from ordinary calculus objects?

I am going to learn differential form soon, but after reading some introductory parts of my texts, I couldn't get why differential form is needed and how it is different from ordinary mathematics ...
1
vote
1answer
86 views

Find a $1$-form whose exterior derivative is $2x^2y^2dydz +3xz^2dzdx - 4xy^2dxdy$

Find a $1$-form whose exterior derivative is $$2x^2y^2dydz +3xz^2dzdx - 4xy^2dxdy$$ An exterior calculus question. I am trying to learn some algebraic topology, and have hit a bump with some (I ...
0
votes
1answer
69 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 ...