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jford1906

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- Thread starter jford1906
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In summary, a flow being transverse to the boundary means that the flow is never tangent to the boundary and must either point inward or outward on each component of the boundary. Since the solid torus only has one boundary component, a flow transverse to its boundary must be everywhere outward or inward. Additionally, any flow transverse to the boundary of a manifold cannot be periodic, as it would carry all points back to where they started in a certain time.

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jford1906

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WWGD

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Transverse I think means here not lying in the tangent space to the boundary points.

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jford1906

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WWGD

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That sounds right.

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jford1906

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Rock. Thanks!

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lavinia

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jford1906 said:

Transverse to the boundary means that the part of the flow that is on boundary does not lie in the tangent space to the boundary. So the flow must point either outwards or inwards.

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zinq

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jford1906 said:

First of all, there are a lot of different periodic flows on a solid torus, so no one flow is "the" periodic flow.*

Secondly (as Lavinia has said), a flow being transverse to a manifold's boundary means that the flow is never tangent to the boundary. It follows that

Clearly, any flow transverse to the boundary of any manifold cannot be periodic, since a periodic flow means that at a certain time T > 0, the flow carries all points back to where they started.

_____________________________

* For example: If θ is the angular coordinate in the solid torus D

A vector field is a mathematical concept that assigns a vector to each point in a given space, such as a plane or a three-dimensional space.

A manifold is a mathematical object that is locally similar to a Euclidean space. It can be thought of as a surface that can be smoothly bent or curved without tearing or stretching.

A vector field is transverse to the boundary of a manifold if at each point on the boundary, the vector is perpendicular to the surface of the boundary. This means that the vector field is not tangent to the boundary, but instead crosses it at a right angle.

Having a vector field that is transverse to the boundary of a manifold allows for a better understanding of the behavior of the vector field near the boundary. It also helps in solving certain differential equations and in studying the dynamics of a system.

Some applications include fluid dynamics, electromagnetism, and general relativity. In fluid dynamics, the velocity of a fluid can be represented as a vector field that is transverse to the boundary of a manifold. In electromagnetism, the electric and magnetic fields can be represented as vector fields that are transverse to the boundary of a manifold. In general relativity, the curvature of spacetime can be represented as a vector field that is transverse to the boundary of a manifold.

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