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

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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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42 views

Exterior derivative of a two-form with conditions

With the use of this formula $$d\omega(X_1, \dots, X_{r+1}) = \sum_{i=1}^{r}(-1)^{i+1}X_i\omega(x_1,\dots,\hat{X}_i,\dots,X_{r+1})+\sum_{i<j}(-1)^{i+j}\omega([X_i,X_j],X_1,\dots,\hat{X}_i,\dots,\...
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1answer
26 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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3answers
58 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 done,...
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50 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 $c_1(...
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50 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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17 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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25 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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17 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 $m \in M$...
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2answers
44 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 $H^n_c(\mathbb{R})\...
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1answer
28 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 = d\...
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1answer
72 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 ...
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1answer
92 views

I have no idea what Differential Forms are… [closed]

So in my Calc 3 class we use Shifrin's "Multivariable Mathematics", and his discussion on Differential Forms and Integration on Manifolds is impossible for me to follow. Can someone recommend ...
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1answer
42 views

Differential forms should be invariant under coordinate transformations

I am wondering why, if we transform the following differential form, it does not seem to be invariant under the coordinate transformation. The $1$-form on $\mathbb R^2$ is $$ \omega = \sqrt{x^2 + y^2}...
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23 views

On the meaning of formal sums of $k$-cubes, i.e. $k$-chains (in integration on manifolds)

A singular $k$-cube in $A \subseteq \mathbb R^n$ is a continuous function $c : [0,1]^k \to A$. A singular $0$-cube in $A$ is then a function $f : \{0\}\to A$, what amounts to the same thing, a point ...
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43 views

Unnecessary assumption in exercise (from Spivak, Calculus on Manifolds)

I have a question on the following exercise (which is taken from Spivak, Calculus on Manifolds, page 105). If $\omega$ is a $1$-form $f dx$ on $[0,1]$ with $f(0) = f(1)$, show that there is a ...
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2answers
44 views

Lie derivative of a covector field

The lecturer here wants the viewer to derive the components of the Lie derivative of a (1,1) tensor-field. To this end, I want to derive the components of the Lie derivative of a covector field: let $...
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1answer
60 views

Differential forms on a projective curve: two constructions

Let $X$ be a projective curve on a perfect field $k$ ($k$-scheme integral, separated, of finite type) and let $K$ be the function field of $X$. Let's compare the following two constructions: 1. ...
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26 views

Kodaira decomposition of 1-form on a Real manifold

I recently stumbled across the Hodge decomposition theorem, which states that on any compact orientable manifold, for any form the following holds $$ \omega = \text{d}\alpha + \delta \beta + \gamma $$ ...
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20 views

Integrating a form over an image

I am trying to integrate the 2-form $$\eta=\frac{1}{\|\mathbb{x}\|^m}(x_1dx_2\wedge dx_3-x_2dx_1\wedge dx_3+x_3dx_1\wedge dx_2)$$ over $Y_\alpha$ where $$\alpha(u, v)=(u, v, (1-u^2-v^2)^{1/2})$$ and $...
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20 views

How to get a a 1-form $\eta$ from $d\eta$?

I have the form $$d\eta = x_2dx_2\wedge dx_3+x_1x_3dx_1\wedge dx_3$$ and I want to evaluate the integral of it, but I don't know how to get $\eta$ from this thing. Any hints?
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Short question about differential forms / exterior algebra

I am working towards understanding the wedge product of two vectors. Here is what I have so far: Let $V$ be an $n$-dimensional vector space over $\mathbb R$ (or $\mathbb C$, I'm not sure it makes ...
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1answer
92 views

What is the rank of a differential form

I've been searching the internet and books for a definition but none of the books on differential geometry and manifolds that I have contain the term rank in the index. While trying to find a ...
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1answer
51 views

understanding of $d(\log f(z))$ in complex analysis

In Gameline's Complex Analysis Chapter 8, the notation $d(\log f(z))$ is used: Here are my questions: In the real case, suppose for any $x\in\mathbb{R}$, one has $f(x)\neq 0$ and $f$ is ...
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1answer
48 views

Frobenius theorem for differential forms

I have to check the next version of the theorem Frobenius: Let $M$ a smooth manifold and $\{\omega^1,\ldots,\omega^k\}\subset\Omega^{1}(U)$ $l.i.$ on $U\subset M$ and $P(x)=\{ v\in T_x M\vert \omega^...
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2answers
53 views

Pullback of a complex $ 1$-form

Let $p = \operatorname{exp} : \mathbb{C} \to \mathbb{C}^*$ be a covering and $(U,z)$ a chart of $\mathbb{C}^*$ with $z = x + iy$. Let $\omega = dz/z$ be a one-form on $U$. Problem: Find the pullback $...
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1answer
101 views

Intrinsic definition of differential k-form on smooth manifold

Suppose I have a $k$-dimensional manifold embedded in $\mathbb{R}^n$. Munkres defines a $k$-form on $M$ as a a function $\omega$ that assigns an alternating tensor at each point $p \in M$ that acts on ...
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24 views

Prove: $u = \phi_1 ^* \mu_1 \wedge \phi_2 ^* \mu_2$ is a volume form on $M = M_1 \times M_2$.

Given two manifolds, $M_1 \in \mathbb{R}^{n_1}$ and $M_2 \in \mathbb{R}^{n_2}$, with respective volume forms $\mu_1,\mu_2$ prove that $\mu = \phi_1 ^* \mu_1 \wedge \phi_2 ^* \mu_2$ is a volume form on ...
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13 views

Find the volume of h(P) in terms of volume of P

Where h:$\mathbb R^n \rightarrow R^n$ is the function h(x) = $\lambda x$ and P is a k-dimensional parallelopiped in $\mathbb R^n$ Attempt at the solution : Let $x_1,....,x_k$ be vectors in $R^n$ ...
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1answer
44 views

Confusion over the definition of a volume element

I'm fairly new to differential geometry (currently self-teaching) and I'm a bit confused over the definition of a volume form. I've read that, given an $m$-dimensional manifold, a top-form $\omega\...
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1answer
76 views

Verifying Stokes' Theorem on an example

Let $M = \{(x,y,z) \in \mathbb{R}^3 | x^2+y^2+z^2=1,z>0\}$, and let $\omega = xdy$. I would like to verify that $\int_M d\omega = \int_C \omega$, where $C = \partial M$ given by $C = \{(x,y,z) \in \...
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2answers
65 views

On the definition of the volume form in general vector spaces as given in Spivak, Calculus on Manifolds

For a vector space $V$ denote by $\Lambda^k(V)$ the space of alternating $k$-tensors, or alternating $k$-fold multilinear maps on $V$. I have some difficulty following the intention of the author in ...
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1answer
78 views

Do we have $\int f dxdy = \int fdydx$ or $\int f dxdy = -\int f dydx$?

If $f : \mathbb R^2 \to \mathbb R$ is an integrable function, then do we have $$ \int f dxdy = \int f dydx $$ or $$ \int f dxdy = -\int f dydx? $$ (I am leaving the domain of integration as it does ...
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0answers
21 views

Computing $\alpha^*w$ in general

Let $A$ be open in $R^k$, let $\alpha : A \rightarrow R^n$ be of class $C^\infty$. Let $x$ denote the general point of $R^k$, let $y$ denote the general point of $R^n$ If $I = (i_i, ...,i_l)$ is an ...
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15 views

Show that $\alpha^*(dy_i) = d(\alpha^*y_i) $

where $y_i:\mathbb R^n \rightarrow R$ is the $i^{th}$ projection function in $\mathbb R^n$ Given: Let A be open in $R^k$; let $\alpha: A \rightarrow R^n$ be a $C^{\infty}$ map. Let x denote the ...
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1answer
70 views

De Rahm cohomology of a sphere, help with proof

I am working through Guillemin and Pollack's proof that the de Rahm cohomology of the sphere is $H^p(\mathbf{S}^k) = \mathbf{R}$ for $p = 0$ and $p = k$ and $H^p(\mathbf{S}^k) = 0$ otherwise. Here, $...
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2answers
72 views

Relation between exterior derivative and Lie bracket

There is a formula connecting the exterior derivative and the Lie bracket $$d\omega (X,Y) = X \omega(Y) - Y \omega(X) - \omega([X,Y]).$$ What is a good way to remember this? By which I mean, what ...
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22 views

clarification on what order of differential form means

I'm confused about what the order of a differential form is, from my understanding it is the k-tuple of vectors that the function takes in, but this doesn't seem right. For instance, according to my ...
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1answer
48 views

Proof that $x \ dy \wedge dz$ is NOT exact on any open subset of $\mathbb{R}^{3}$

Consider the smooth differential 2-form $\omega = x \ dy \wedge dz$ defined on $\mathbb{R}^{3}$ (using the coordinates $(x,y,z)$ of course). I'm trying to show that $\omega$ is NOT exact on any open ...
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31 views

If a k-form vanishes in the neighborhood of p then it vanishes at p

Let w be a k-form defined in an open set A of $\mathbb R^n$. We say that w vanishes on x if w(x) is the zero tensor. Show that if w vanishes at each x in a neighborhood of $x_0$ then dw vanishes at $...
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1answer
44 views

Finding Integrating factor

I have been solving this ODE $ydx+(y+\tan(x+y))dy=0$ My approach was like this; As this equation is not exact; $\frac{\partial}{\partial y}y = 1$ $\frac{\partial}{\partial x}(y+\tan(x+...
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24 views

Writing a linear transformation in terms of elementary alternating k-tensors

Let T: $\mathbb R^m \rightarrow R^n $ be the linear transformation T(x) = Bx If $\alpha_I$ is an elementary alternating k-tensor on $\mathbb R^n$, the $T^*\alpha_I$ has the form $T^*\alpha_I = \sum_{[...
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1answer
47 views

Lie derivative of a two-form

Let $M$ be a manifold, $\alpha$ a two form and $x,y,z$ vector fields. How can we show that $$L_x(\alpha(y,z))=(L_x \alpha)(y,z)+\alpha(L_x y,z)+\alpha(y,L_x z)$$ (where $L_x$ is the Lie derivative ...
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21 views

The determinant of $X_I$ = wedge product of elementary tensors

Let $(x_1,\dots,x_k)$ be vectors in $\mathbb{R}^n$; let $X$ be the matrix $X = [x_1 \cdots x_k]$. If $I = (i_1, \dots, i_k)$ is an arbitrary $k$-tuple from the set $\{1,\dots, n\}$, show that $\phi_{...
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1answer
33 views

Checking alternating tensors on $\mathbb R^4$

I'm trying to solve the following question : Which of the following are alternating tensors in $\mathbb R^4$ and express those that are in terms of the elementary tensors on $\mathbb R^4$: \begin{...
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1answer
130 views

Does there exist a smooth curve satisfying certain conditions?

Let $\gamma := dc - a\,db$ be a $1$-form on $\mathbb{R}^3$ with coordinates $a$, $b$, $c$. Have $\omega$ be the $2$-dimensional subbundle of $T\mathbb{R}^3$ consisting of vectors in the kernel of $\...
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0answers
50 views

Why don't I get the same answer when I calculate the pullback vs integral over a manifold?

Let's take the differential form $\omega = xy \, dx \wedge dy$. We say that $M$ is the surface $z = x^2 + y^2 \leq 1$ with the standard orientation. I can calculate $\int_M \omega$ via pullback and ...
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1answer
69 views

What's the exterior derivative of the two form: $x \,dy \wedge dz + y \,dx \wedge dz + z\,dx \wedge dy$?

What's the exterior derivative of the two form: $x \,dy \wedge dz + y\, dx \wedge dz + z\,dx \wedge dy$? I saw some calculations on this site and I'm pretty sure that the answer is $3\, dx\wedge dy\...