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

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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
107 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
104 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
246 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
212 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
256 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
304 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
45 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
73 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
160 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
48 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
374 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
161 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
206 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
174 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
79 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
97 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
169 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
150 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
223 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
540 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
85 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 ...
7
votes
1answer
683 views

Are Clifford algebras and differential forms equivalent frameworks for differential geometry?

I recently discovered Clifford's geometric algebra and its application to differential geometry. Some claim that this conceptual framework subsumes and generalizes the more traditional approach based ...
4
votes
1answer
228 views

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 ...
6
votes
1answer
294 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 ...
12
votes
2answers
354 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} ...
3
votes
1answer
282 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 ...
7
votes
2answers
269 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
137 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
votes
0answers
51 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
95 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
votes
1answer
118 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
vote
0answers
95 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
votes
1answer
137 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 ...
9
votes
3answers
445 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 ...