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# Questions tagged [orientation]

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

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### Interpreting the Sign of the Jacobian Determinant

The sign of the Jacobian determinant of a two-dimensional transformation tells us if the transformation is locally orientation-preserving (if it's positive) or locally orientation-reversing (if it's ...
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### Orientation bundle and its flat connection

Let $M$ be a smooth $n$-manifold (which is not assumed to be orientable), and write $o(TM)\to M$ for its orientation bundle. Equivalently, it is the top exterior bundle $\Lambda^n(TM)\to M$. In any ...
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### Quaternion - calibration sensor [closed]

I have attached my sensor to a box. What I am interested in is the rotation of the box with respect to the world. In my case, to avoid the gimball lock, and beacause the sensor already provides it, I ...
1 vote
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### Integrating a 2-form over 2-manifold with an orientation

This excercise asks me to integrate $\omega=z dx \wedge dy$ over the 2-manifold $A = \{(x, y, z): x^2+z^2=y, y<4\}$ with orientation $o(x)$ such that $o^{31}(x)>0$ using Divergence's theorem. So ...
1 vote
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### Weak Parallelisability/Strong Orientability

An orientable manifold is defined by looking at small patches, looking at the interaction of local homology groups on these small patches, and focusing on when this gives us a coherent global story. ...
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### Oriented matroids: A map between chirotopes and the set of covectors?

I am just getting used to the topic of oriented matroids. I was surprised to read that a central aspect of oriented matroid theory is that there exists a "non-trivial equivalency" between ...
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### How to produce an N-dimensional *direct* basis?

I need a procedure to generate an N-dimensional orthonormal direct basis. The 1D, 2D, and 3D case are not a problem. For example, in the 3D case, if I have 3 vectors $v1, v2, v3$ (suppose they behave ...
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### surface integrals orientation. when is it positive or negative

For this question below, I don't understand what they're asking. what does it mean to find the net flow rate upward through the surface? How do I know if I need to put a negative before computing the ...
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### Orientation of simplicial complexes [duplicate]

I'm trying to understand simplicial homology over $\mathbb Q$ and I'm having some problems in grasping with the definition of orientation of an abstract simplicial complex, the major one being that I ...
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### orientation difference between two triangles in 3D space

Lets there be 2 sets of 3 points in 3D space, representing 2 congruent isosceles triangles. The apexes of both triangles are located at the point (0,0,0). How do I calculate the difference between the ...
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### Using Möbius band to define orientability of surfaces

Given a surface $S$, the usual definition of orientability is that there is an atlas of charts with all transition maps having non-negative Jacobian. Donaldson, in his book on Riemann surfaces, says a ...
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### Hat Knot Floer Homology with Z coefficients calculation

I would like to ask if there is a reference which carries out the calculation of the hat knot Floer homology of a knot with $\mathbb{Z}$ coefficients, i.e., $\widehat{HFK}(K;\mathbb{Z})$, where the ...
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### Orientation of a vector space from a homological point of view

I was reading Milnor and Stasheff's Characteristic Classes but I got stuck at the beginning of Chapter 9 when a homological interpretation of the orientation of a real vector space is given. More ...
I have been reading from Lee's Introduction to Smooth Manifolds and I have gotten to the section on how an orientation on a manifold is defined. First Lee defines an orientation on a vector space $V$ ...
Let $\phi: U \rightarrow O$ be a coordinate chart of an $n$-dimensional differentiable oriented manifold where $U$ is an open subset of the manifold and $O$ is an open subset of $\mathbb{R}^n$. If we ...