# Tagged Questions

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### Why is the integral of any orientation form over $\mathbb{S}^1$ non zero?

I am trying to understand the proof of Theorem 17.21 in Lee's Introduction to smooth manifolds; however I am finding myself stuck right at the beginning. The statement I am having trouble with is: ...
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### Integration of differential forms

I have just started to learn differential forms. Now, there is a concept of pulling integral back. I somewhat understood the procedure to do it. But, I don't understand why we do it and when to use ...
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### Integration of bundle-valued differential forms

The literature, at least textbooks, seems to be very scarce on the topic of integrating bundle-valued differential forms. So I wonder where can I read on the topic? I want to see usual theorems, like ...
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### Area of spherical cap with integrals

Given a sphere $S$ of fixed diameter $D$ (or radius $R=D/2$, it will be convenient to have both, I suppose), and a point $P$ on its surface, let's create a ball $B$ of variable radius $r$ with its ...
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### Integration by substitution and separation of variables.

Let's say I want to integrate over a sphere $S^2$. Take $f \in (L^1(S^2),dS)$, then we have that $$\int_{S^2} |f| dS = \int_{S^2} |f| \sin^2 (\theta) d \theta d \phi < \infty,$$ right? Now, ...
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### Integration on $\mathbb{R}^n$ in terms of differential forms

One defines integration on a smooth manifold as follows: First define $\int_M \omega$ when $\omega$ is supported on a single coordinate chart by pulling back to $\mathbb{R}^n$ an integrating there, ...
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### About integration on manifold and partition of unity (and finiteness of open covers)

Please see the definition below of integration over a boundary of a Lipschitz domain. My question is, the summation in (C.36) for example is over $n$. But when is this a finite sum? If ...
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### Shell method for calculating volume of solid of revolution - general

Let us have an injective continuous function $f : [a,b] \to [0,c]$ (such that $f(a)=0$ and $f(b)=c$). I want to calculate the volume of solid revolution of $f$ around the $y$ axis. The first method ...
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### Two definitions of integral on boundary $\int_{\partial\Omega}f$?

I have seen two definitions of an integral of a function $f:\partial\Omega \to \mathbb{R}$ from the boundary of an open set $\Omega \subset \mathbb{R}^n$ where the domain is Lipschitz. 1) ...
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### Surface integral for a scalar function defined on a discrete surface

Imagine a polyhedral, discrete surface embedded in $\mathbb{R}^3$. Its faces are all triangles. For each vertex, one can compute the discrete mean and Gaussian curvatures and evaluate the sum of ...
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### Problem 4-25 from Spivak's Calculus on Manifolds

I am reading through Spivak's Calculus on Manifolds and have come across a technicality in one of the problems that is annoying me. It is Problem 4-25, the statement of which is Let $c$ be a ...
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### Integration by parts on the $N$-dimensional torus

My problem. Consider the $N$-dimensional torus $$\mathbb{T}^N = \mathbb{R}^N /(2\pi n \mathbb{Z})^N \simeq [-\pi n, \pi n)^N$$ and consider two function $v,w \in H^2(\mathbb{T}^N)$. I want to ...
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### Definition clarification on orientation on a manifold.

I have been trying to self-learn differential geometry. I think I may have misunderstood/missed out on something along the way. It is said that for $X$ an $n$-form, $M$ a differentiable manifold, ...
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### Relation between de Rham cohomology and integration

This question is a follow-up to When does a null integral implies that a form is exact? . As mentionned in the selected answer, given certain conditions it is possible to find an isomorphism between ...
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### Is there a 0-form $\tau$ with $d\tau=\omega$?

Consider the 1-form $$\omega=(x^2-yz)dx+(y^2-xz)dy-xydz.$$ Does a 0-Form $\tau$ on $\mathbb{R}^3$ exist which fullfils $d\tau=\omega$? Hello, my simple answer is: YES, because ...
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### When does a null integral implies that a form is exact?

It is trivial to prove that the integration of a $(n-1)$ exact form on the boundary of a $n$-manifold is 0. What about the contraposative ? If the integration of a $(n-1)$-form on the boundary of a ...
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### The notion of a curve in the context of line integrals

For brevity I'm making the following assumption: I'm only talking about regular curves on $\left[a,b\right]$ with values in $\mathbb{R}^{n}$, and line integrals of scalar fields. [Since there are a ...
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### How to conclude $\frac{d}{dt}E(f)=-\int _M (\Delta f)\dot{f} d\mu$?

Can anyone explain to me how I can conclude $\frac{d}{dt}E(f)=-\int _M (\Delta f)\dot{f} d\mu$ by using integration by parts and $\langle f_1 ,f_2 \rangle_\mu:=\int_M f_1 f_2 d\mu$? Where $M$ is a ...
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### Homeomorphism vs diffeomorphism in the definition of k-chain

In "Analysis and Algebra on Differentiable Manifolds", 1st Ed., by Gadea and Masqué, in Problem 3.2.4, the student is asked to prove that circles can not be boundaries of any 2-chain in ...
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### 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 ...
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### derivative of the integral over a sphere of variable radius [Reference needed]

On a Riemannian manifold $(M,g)$, let $F(s)=\int_{\partial B_s(x_0)}udS$ where $u$ is a smooth function, ${\partial B_s(x_0)}$ is the (geodesical) sphere of center $x_0$ and radius $r$ dS is the ...
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### why $\iint \sqrt{1+\left(\frac{\partial f}{\partial x}\right)^2+\left(\frac{\partial f}{\partial y}\right)^2}\:dx\:dy$ means surface area?

Why the following integral means the area of surface $f(x,y)=z$? $$\iint \sqrt{1+\left(\frac{\partial f}{\partial x}\right)^2+\left(\frac{\partial f}{\partial y}\right)^2}\:dx\:dy$$
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### Integration on manifold, pullback

Define the maps $F_\pm:B^2\rightarrow S^2$, $(x,y)\rightarrow(x,y,\pm\sqrt{1-x^2-y^2})$. Prove that for any $\omega\in\Omega^2(S^2)$: $$\int_{S^2}\omega=\int_{B^2}F_+^*\omega-\int_{B_2}F_-^*\omega$$ ...
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### Closed not exact form on $\mathbb{R}^n\setminus\{0\}$

I'd like to construct a closed but not exact $n-1$-form $\omega$ on $\mathbb{R}^n\setminus\{0\}$ in analogy to the winding form: $$\frac{x~dy-y~dx}{x^2+y^2}$$ I think something like ...
I have a bit of a problem with the following identity: Suppose that $U, V \subset \mathbb{R}^n$, are two open sets. Let $x^1,...,x^n$ be a system of coordinates of $U$ and $y^1,...,y^n$ one on $V$. ...
Let $(N,g_N)$ be a Riemannian manifold and let $\psi: M \rightarrow N$ be a a diffeomorphism. Now I know how the Riemannian metric on $M$ defined by the pull-back of the metric on $N$ looks like (this ...