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

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How to understand the notion of a differential of a function

In elementary calculus (and often in courses beyond) we are taught that a differential of a function, $df$ quantifies an infinitesimal change in that function. However, the notion of an infinitesimal ...
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1answer
20 views

Finding all $2$-forms in the right half-plane that are invariant under glide transformations

I'm trying to find all 2-forms $\omega$ that are invariant under glide transformations in the right half-plane, $X = \{ (x,y) \in \mathbb{R}^2 : x > 0\}$. To do this, we can write the vector field ...
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1answer
29 views

Unit square as union of two simplexes

? If Rudin regarded $\Phi$ as a function of $2$-forms and suppose that $\omega=f(\mathbf{x})dx_{i_1}\land dx_{i_2}$ is $2$-form on $\mathbb{R}^m$ then $$I_{\Phi}(\omega)=\int ...
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22 views

Examples of compactly supported exact differential forms [on hold]

I am having some trouble finding any examples of compactly supported exact differential forms on $\mathbb{R}^n$. I have found $e^{\frac{1}{x^2 -1}}$ when taken to be zero everywhere except on the ...
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29 views

Boundary of oriented $k$-simplex from PMA Rudin

But paragraph which I marked by red line seems to me confusing. Let $k=3$ then $\sigma=[\mathbf{p}_0,\mathbf{p}_1,\mathbf{p}_2,\mathbf{p}_3]$ and $\partial ...
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1answer
48 views

Finding A 1-form on $R^2 - {\{(0,0)}\}$

I want to find a 1-form on $R^2 - {\{(0,0)}\}$ such that $w(Y) = 0$ and $w(X) = 1$. Here, $$X = -y\frac{\partial }{\partial x} + x\frac{\partial}{\partial y}\ \text{and}\ Y = x\frac{\partial ...
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1answer
47 views

A necessary and sufficient condition for the admittance of integrating factor

Let $\omega$ be a smooth 1-form on a smooth manifold $M$. A smooth positive function $\mu$ on some open subset $U\subset M$ is called an integrating factor for $\omega$ if $\mu\omega$ is exact on ...
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2answers
37 views

Specific example of integrating a 1-form over a curve

I was given the following definition in my course but no corresponding examples: Supppose $\gamma:[a,b]\rightarrow{M}$ is a smooth curve and $\omega$ a 1-form on $M$ (so $\omega:M\rightarrow{T^*M}$). ...
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1answer
57 views

Affine chains from PMA Rudin. Confusing examples

I understood the definition of affine $k$-chain and that he defines $\int \limits_{\Gamma} \omega$ as $(82)$. But I can't understand the last two above examples. What does they mean? Can anyone ...
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24 views

Equivalent conditions for $\mathfrak{F}$ to be a differential ideal

Heres the question: Let $\mathfrak{F}$ be an ideal of forms on a manifold $M$ locally generated by $r$ independent $1$-forms. Say $\mathfrak{F}$ is generated by $\omega_1,\ldots, \omega_r$ on $U$. ...
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28 views

Let $f : \mathbb{R}^n \to \mathbb{R}^n$, suppose that $\mu = dx^1\wedge\ldots\wedge dx^n$, then $f^{\ast}\mu = \det (df)\mu$

Let $f : \mathbb{R}^n \to \mathbb{R}^n$, suppose that $\mu = dx^1\wedge\ldots\wedge dx^n$, then $f^{\ast}\mu = \det (df)\mu$. I am trying to prove this. I made several low dimensional cases and ...
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1answer
27 views

Prove the exterior derivative of the following (n-1) form is zero

Let $\omega(x)=\frac{1}{{\parallel x \parallel}^n}\displaystyle\sum_{i=1}^{n}(-1)^{i-1}x_{i} dx_{1} \wedge \dots \wedge \widehat{dx_{i}} \wedge \dots \wedge dx_{n}$ be a differential $(n-1)$ form on ...
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1answer
54 views

Confusing moment in Theorem 10.27 from PMA Rudin

Theorem 10.27 If $\sigma$ is an oriented rectilinear $k$-simplex in an open set $E\subset \mathbb{R}^n$ then $$\int \limits_{\overline{\sigma}}\omega=\varepsilon\int \limits_{\sigma}\omega \qquad ...
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17 views

How can I convert this second order equation into a first order equation? [closed]

In a previous exercise sheet, we were asked to transform the second order differential equation $$ x'' = -x + \alpha x^{3} $$ Into a first order equation. The solutions have since been released, but ...
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2answers
48 views

Example of exact form

Consider the differential 1-form $\omega = ydx+dy$. I need to show that this is not exact, and find an example of a function $G(x,y)$ such that $G\omega=G(x,y)(ydx+dy)$ is an exact form. I have done ...
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1answer
40 views

Theorem 10.22 from PMA RUdin

We know that $(dy_I)_T=dt_{i_1}\land \dots \land dt_{i_k}$ and using definition 10.18 we get $$d((dy_{I})_{T})=d1\land dt_{i_1}\land \dots \land dt_{i_k}=0$$ since $dc=0$ for any $c\in ...
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1answer
34 views

Derivative of $0$-form

Rudin states that $1$-form $xdy$ is not the derivative of any $0$-form. By contradiction, suppose that that exists $0$-form whose derivative is $xdy$. Then $f\in C'$ and $df=xdy$. But ...
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1answer
35 views

Noncanonical isomorphism of spaces of differential forms

Let $\pi: V \to M$ be a smooth $n$-dimensional vector bundle over $M$. Are the spaces of differential forms $\Omega^i(V)$, $\Omega^i(V^*)$ noncanonically isomorphic? If so, how do I see this? Is there ...
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2answers
64 views

Frank Warner's definition of the Hodge star

Frank Warner's book, chapter 2, excercise 13 states the following: If $V$ is an oriented inner product space ($n$ dimensional) there is a linear map $\ast \colon \Lambda (V) \to \Lambda (V)$, ...
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1answer
78 views

Does such property exist for differential forms?

I am studying differential forms from Rudin's PMA. Here's the definition of $k$-form Also he proves the anticommutative relation: $dx_1 \land dx_2=-dx_2\land dx_1$ Does the following expressions ...
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1answer
41 views

Method to calculate the de Rham cohomology of $\mathbb{R}\mathrm{P}^n$

I'm trying to follow through a method to calculate the de Rham cohomology groups of $\mathbb{R}\mathrm{P}^n$ from the de Rham cohomology groups of $S^n$. I'm trying to show that differential k-forms ...
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1answer
21 views

Are curvature forms in complex line bundles symplectic

I know that the curvature form $F_\nabla$ of a connection $\nabla$ in a complex line bundle $L \to B$ is presymplectic (i.e. antisymmetric and closed). Does it also have to be non-degenerate, i.e ...
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14 views

How to show that a tensorial field,$t$, $k$-times covariant over a differential variety $M$ is differentiable by its components

How can i show that a tensorial field,$t$, $k$-times covariant over a differential variety $M$ is differentiable if and only if the following functions (The components of t) are differentiable.: ...
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39 views

vanishing of differential form on connected compact manifold

Let $M$ be a $n$ dimensional compact connected manifold. Let $\alpha$ be a differential form of degree $n$ such that $$ \int_{M} \alpha = 0 $$ then I would like to show that $M$ vanishes at at least ...
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1answer
41 views

What is the skew-symmetric part of the covariant derivative of a one-form?

This is a followup question to here. Let $E \to M$ be a vector bundle with connection $D := \nabla$. Extend $D$ to $E^*$ and $\text{Hom}(E, E)$. Let $E = TM$ here, and suppose that the torsion is ...
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32 views

Integration of forms on manifolds

If you have a $n$-form $\omega$ on $\mathbb{R}^n$, then $\omega = f \mathop{}\!\mathrm{d}x_1 \wedge \dots \wedge \mathop{}\!\mathrm{d}x_n$ locally. Integrating $\omega$ is easy now - let's assume ...
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1answer
55 views

Zeroes of exact differential forms on compact manifold

Let $M$ be a $n$ dimensional compact differentiable manifold. I would like to show that any exact differential form of degree $n$ vanishes at at least one point. I think it is a generalization of the ...
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1answer
26 views

Prove that $p-$form $\omega$ on $M\times N$ is $\delta \pi(\alpha)$ iff $i(X)\omega=0$ and $L_X\omega=0$

Let $M$ be connected and let $\pi:M\times N \rightarrow N$ be the natural projection. Prove that $p-$form $\omega$ on $M\times N$ is $\delta \pi(\alpha)$ for some $p-$form $\alpha$ on $N$ if and ...
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2answers
45 views

Integrate ${xdy+ydx} \over {x^2+y^2}$ on the circle $(x-1)^2+(y-1)^2=1$

I tried to integrate ${xdy+ydx} \over {x^2+y^2}$ on the circle $(x-1)^2+(y-1)^2=1$ counterclockwise. Used Grin's theorem, then went to polar cordinates but can't integrate the expression I got. So I ...
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1answer
29 views

Showing that the exterior derivative of a 1-form is 0.

The question is $$ Let\quad f : \Bbb R \to \Bbb R$$ $$\omega = f(||\mathbf x||)(\sum_{i=1}^n x_{i}dx_{i}) \in \mathcal A^1(\Bbb R^n) $$ $ (a) $ Assuming f is differentiable, prove that $d\omega = 0$ ...
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31 views

Elementary properties of diff. form from PMA Rudin

How he got $d\omega$ in the RHS of $(39)$? Or maybe it's a typo?
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40 views

How am I supposed to answer this question?

I've got the following question: The first part of this question I can do. I've deduced that the DE has oscillatory solutions for all $\lambda > 16$; that the Eigenvalues are given by $$ ...
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3answers
45 views

Reference request: integration of *one*-forms along curves on a differentiable manifold.

Could somebody please direct me to a book/lecture notes with an introduction to integration of one-forms along curves in a differentiable/Riemannian manifold -- preferably leaning more towards ...
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1answer
28 views

The value of the integral of the curvature of a complex line bundle

I am trying to show that if $L \to M$ is a complex line bundle endowed with a connection $\nabla$, $F$ is the curvature form, and $S \in M$ a closed surface, then $\int \limits _S F \in 2 \pi \textrm ...
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1answer
29 views

Prove the pullback of the wedge product is the wedge product of the pullbacks.

Let $F:V \rightarrow W$ be a linear map. Show that $F^{\ast}(\omega \wedge \eta)=(F^{\ast}\omega) \wedge (F^{\ast}\eta)$ for all $\omega \in \Lambda^{p}(W) , \eta \in \Lambda^{q}(W)$. Where ...
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1answer
36 views

Integral of Differential 1-form

Let $\omega$ be the closed $1$-form $\omega = \frac{xdy - ydx}{x^2 + y^2}$ and let $S$ be the unit circle and let $C = \{(x,y) : (x-3)^2 +y^2 = 1\}$. I'm trying to find $\int _S \omega$ and $\int ...
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21 views

Integrate 2-Form over surface

Problem: Calculate $\int_S dx \wedge dy + dy \wedge dz$, where $S$ is the surface given by $S = \{(x,y,z) : x = z^2 +y^2 -1, x < 0\}$. Wikipedia says: Let $$ \omega=f_{z}\, \mathrm dx \wedge ...
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1answer
24 views

Proof Verification: A differentiable aplication $\psi : M\to N$ is differentiable if and only if: $\psi^{*}f\in C^{\infty}(M)$

Show that a differentiable aplication $\psi$ over $M$ to a differentiable variety $N$ is differentiable if and only if: $$\psi^{*}f\in C^{\infty}(M)$$ For: $f\in C^{\infty}(N)$ Where $\psi^{*}f$ is ...
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36 views

Exterior derivative of a two-form with conditions

With the use of this formula $$d\omega(X_1, \dots, X_{r+1}) = ...
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25 views

Confusion about holomorphic differential on elliptic curve

Let $(a:b)\in\mathbb{C}P^1$ and look at the elliptic curve $C$ given by $y^2=x^3+a^4x+b^6$. It is well known that on this elliptic curve we have the holomorphic differential $dx/y$. I have two ...
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56 views

Nowhere $0$ form on the sphere?

Consider the differential form on $\mathbb R^3$ given by $ x dy \wedge dz + y dz \wedge dx + z dx \wedge dy$. I converted this to spherical coordinates using a laborious calculation, and when I'm ...
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33 views

Proof that exact form are path independent seems to imply the same for merely closed forms

A singular $k$-cube on some set $A \subseteq \mathbb R^n$ is a continuous map $c : [0,1]^k \to A$. Consider the following exercise: Let $c_1, c_2$ be singular $1$-cubes in $\mathbb R^2$ with ...
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1answer
39 views

Integral of form pulled back from torus to sphere is zero

I'm trying to show that when we pull back (with any map $f: S^2 \to T^2$ any 2-form on the 2-torus to the 2-sphere it's integral is zero. I understand we can choose coordinates $(\theta_1,\theta_2)$ ...
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16 views

Prove that the Differential of a Function is Equal to a Closed 1-Form

Let $\omega$ be a smooth $1$-form on $\mathbb{R}^n$ such that $d\omega=0$. Define a function $f: \mathbb{R}^n \to \mathbb{R}$ by the equation $$f(\vec{x})=\int_{\ell_{(\vec{0}, \vec{x})}} \omega$$ ...
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23 views

Computing contraction of a 3-form with vector field

I am trying to understand the idea of contraction by computing the contraction of $$dx\wedge dy \wedge dz$$ over the vector field $$x\frac{\partial }{\partial y}-z\frac{\partial }{\partial x}$$ I ...
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15 views

Global Clebsch potentials

For an aribitrary vector field $\mathbf{v}$ on $\mathbb{R}^3$, it always can locally be written as $$ \mathbf{v}=\nabla f+g\nabla{h} $$ where $f$, $g$, $h$ are called Clebsch potentials. My question ...
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1answer
49 views

Doesn't this article about $1$-forms contradict itself?

I am studying the first page of this article here. The article defines a differential $1$-form to be a smooth map $\alpha : TM \to \mathbb R$ ($TM$ here is the tangent bundle) such that for $m \in ...
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42 views

If $\int_M \omega=0\Rightarrow \omega=d\varphi$, then $H^n_c(M)\simeq\mathbb{R}$? ($M$ is a connected orientable manifold)

I'm reading a book in wich the author uses this argumet the whole time. For example, he assumes that $\int_\mathbb{R}\omega=0$ then $\omega =df$ and then he concludes that that ...
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1answer
25 views

Finding a basis for the cohomology vector space of 1-forms in the 2-torus, $H^1 (T^2)$

I would like help in understanding where I am going wrong here: If I consider the 2-torus $T^2 = S^1 \times S^1$ with an atlas $(\theta_1,\theta_2)$, I can define 2 closed 1-forms $\omega_1 = ...
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1answer
70 views

Is this picture of a differential $1$-form correct?

Consider the following picture of $x dy$: (this picture appears in this article). I believe these lines should be vertical: At each point in $\mathbb R^2$ a basis for the tangent space is given ...