# Questions tagged [orientation]

For question regarding the notion of orientation both in topology and in global analysis.

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### Jacobian of converting Euler angles to rotation vector or rotation matrix

please consider this paper: A Primer on the Differential Calculus of 3D Orientations - Bloesch 2016. Equations 27, 29 and 30, for example, give nice results about differentiating the rotation of a ...
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### What is the correct term to describe direction of travel on a curve that intersects itself exactly once? [closed]

In a non-self-intersecting curve such as a circle, we can describe direction of travel as clockwise (CW) or counter-clockwise (CCW). In a self-intersecting curve such as a figure-eight, clockwise is ...
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### Explicit description of the homology of the n-sphere

I'm currently studying the homology of manifolds (so it's mainly orientation and duality questions) and I was wondering how to explicitly compute homology of spaces and particularly $n$-spheres. ...
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### Orientation of left and right translations on a Lie Group

I read that if $G$ is a connected Lie Group, then for every $g \in G$, the left and right translations ($L_g$ and $R_g$, respectively), preserve orientation. The reason given is that $L_g$ and $R_g$ ...
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### orientability of surface and odd/even no. of punctures [closed]

Suppose that T is a surface with Euler characteristic -4. Is it orientable or non-orientable? Does it have odd or even number of punctures (disks)?
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### Quaternions to keep track of orientation: The effective quaternion does not reproduce the same results as multiple quaternions sequentially.

In short, I am trying to find the preferred orientation of a complex non-symmetric object on a surface using a Monte Carlo method. During each cycle, the object is moved with a random (limited) ...
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### Fluid dynamics - Orientation of a mathematical plane surface

The following is a problem in "Transport Phenomena" by Bird et al : "A mathematical plane surface of area $S$ has an orientation given by a unit normal vector $\bar{n}$, pointing ...
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### Is this a valid definition of an orientable manifold?

That is, an n-dimensional manifold M is said to be orientable iff the tangent spaces of its points are consistently oriented(i.e. for any 2 distinct points p and q, the matrix $B_{pq}$ which ...
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### Fundmental class of a homology manifold

Suppose $X$ is a compact, connected $n$-dimensional homology ($\Bbb Z$-)manifold (https://en.wikipedia.org/wiki/Homology_manifold). Since orientability is defined using only homology (for example, in ...
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### What is the induced orientation on a product of vector spaces in singular cohomology?

In singular cohomology, we understand an orientation of $\Bbb{R}^d$ as a choice of generator for the relative cohomology group $H^d(\Bbb{R}^d,\Bbb{R}^d\setminus \{0\})\cong \Bbb{Z}$. My question is ...
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### Intuition for orientation of a simplex (in 3 dimensions)

In trying to begin to learn basic homological algebra, i am confronted with orientation of simplices. The definition seems unmotivated and unintuitive: for $n$-simplices with $n \in \{-1,0,1,2\}$, it ...
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### Transitive action on non orientable $M$ lifts to orientable double cover

Suppose that $M$ is non orientable with transitive action by a Lie group $G$. Does that imply that some Lie group $G'$ acts transitively on the orientable double cover $M'$? This is true for ...
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### Oriented surface, oriented curve, and vector field

I have an oriented surface S and an oriented curve C. The surface and the curve intersect in points A and B. In A, the orientation of the curve is the same of the surface, in B they are in opposite ...
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### Can I compute contour orientation without using polygon area sign?

Most of the times, I determine contour orientation generating 2D points and computing the closed polygon area. Depending on the area value sign I can understand if the contour is oriented clockwise or ...
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### Intuitive understanding of oriented volume and trivectors

I get that the way a vector's arrowhead points corresponds to its orientation for a given direction (line). We can also understand vectors within $V$ as isomorphic to a set of endomorphic translations....
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### Exercise 5-15 from Spivak's Calculus on Manifolds

I came across the following question (Exercise 5-15) in Spivak's Calculus on Manifolds and am not sure how to solve it. Let $M$ be an $(n-1)$-dimensional manifold in $\mathbb{R}^n$. Let $M(\epsilon)$ ...
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### Regarding the Definition of an Orientable Foliation on a Manifold in terms of Transition Maps

Introduction to the Geometry of Foliations, Part $A$, Authors Gilbert Hector and Ulrich Hirsch, Page $15$. Let $\mathcal{F}$ be a Foliation on a Manifold $M$ defined by the atlas $\{(U_i,\phi_i)\}$. ...
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### Example of orientable surface $M\subset\mathbb{R^n}$ such that has no normal vector fields

There is the following well known result about manifolds in $\mathbb{R^n}$: Theorem: Let $M\subset\mathbb{R^n}$ be a surface (manifold) of class $C^k$ $k\geq 1$ and dimension $m$. If there are $n-m$ ...
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### Could there be a notion of orientation for domains of multi-dimensional integrals?

Other answers have explained that the reason why we take the absolute value of the Jacobian $|J|$ in multi-dimensional integrals but not in 1D integrals is because in 1D integrals a change of ...
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### Fundamental class of $M\times N$
(Hatcher 3.B.4) Show that the cross product of fundamental classes for closed $R$-orientable manifolds $M$ and $N$ is a fundamental class for $M\times N$. Assume $\dim M = m,\dim N = n$. Let $[M]$ ...