Connect and share knowledge within a single location that is structured and easy to search. Let's try to understand the geometry carried by the orientation. Note that changing the ordering of the basis elements generally changes the orientation. For example, swapping the labels of two basis elements changes the orientation.
Generally, a re-ordering defines a consistently-oriented basis if and only if the permutation of indices is an even permutation, i.
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A simplex with a fixed order of vertices up to an even permutation is said to be oriented. The atlas is said to be orienting if the coordinate transformations between charts are all positive. In the case of a differentiable manifold this means that the Jacobians of the coordinate transformations between any two charts are positive at every point.
In this case, all orienting atlases divide into two classes such that the transition from the charts of one atlas to the charts of another is positive if and only if both atlases belong to the same class. A choice of this class is called an orientation of the manifold.
Through this homomorphism a covering is created, which is a two-sheeted covering in the case of a non-orientable manifold. It is said to be orienting since the covering space will be orientable. The homological interpretation of orientation enables this concept to be applied to generalized homology manifolds cf. Homology manifold. The concept of orientation also allows a natural generalization for the case of an infinite-dimensional manifold modelled on an infinite-dimensional Banach or topological vector space.
This requires restrictions on the linear operators which are differentials of transitions from one chart to another: They must not simply belong to the general linear group of all isomorphisms of the structure space, which is homotopically trivial in the uniform topology for the majority of classical vector spaces, but must also be contained in a disconnected subgroup of the general linear group.
The connected component of the given subgroup will then also provide the "sign" of the orientation. The subgroup usually chosen is the Fredholm group, consisting of those isomorphisms of the structure space for which the difference from the identity isomorphism is a completely-continuous operator. In the problem of describing the class of bundles which are orientable in a given theory, the following general result holds.
The opposite problem consists of describing a theory in which a given bundle or class of bundles is orientable. Constructing for a given class of vector bundles the universal theory, which maps onto any other theory in which the class of bundles is orientable, has yet to be carried out Log in.
The magnitude of the line segment is the 'length' of the vector. The 'orientation' of the line segment we can define as the angle that the line segment makes with the horizontal axis. To be clear this angle is measured counterclockwise from the positive x axis and is an angle between 0 and OK so far we just have a line segment that is situated on the plane somewhere and we know how it is oriented, but there is no arrowhead yet.
Now the last thing you need is the 'sense', that basically tells you where to put the arrowhead and implies an order. We can define this vector as AB with an arrow over it, where you read it as the vector starting from A and ending at B.
More generally you can define a vector by defining its magnitude length , its orientation, and then its sense. But keeping in mind, technically a vector is an equivalence class. So there are an infinite number of vectors which are parallel to each other have the same orientation and have the same sense or same choice of where to put the arrowhead there are only two possible senses, since the arrowhead can be placed on A or on B.
But I don't want to confuse you. Usually we discuss vectors that are situated at the origin so we don't concern ourselves over other equivalent vectors. The magnitude of the vector corresponds to the length of the arrow, and the direction of the vector corresponds to the angle between the arrow and a coordinate axis.
The sense of the direction is indicated by the arrowhead. This pretty much sums up what I just wrote above. You see the 'direction' your article uses the word orientation just gives you an angle that the vector makes with the horizontal axis, but that creates an ambiguity since an arrow can point in two opposite directions and still make the same angle.
The sense clears this ambiguity and indicates where the arrowhead actually goes. So the sense tells us the order so to speak in which to read the vector. It indicates where to start and end on the vector. Also technically you can indicate the angle of the vector however you choose, doesn't have to be a horizontal axis, could be vertical. Now you are probably more familiar with a vector description that combines orientation and sense by saying for example, give me the vector that is 2 units length and is rotated 30 degrees counterclockwise starting from the point 2,0.
But I am combining direction and sense here. In math you just have to be flexible. Different authors mean the same thing but use different words. In an ideal world every author would agree with each other's notation and terminology.
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