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Also in particular, if the vector field in question is the curl of another vector field, is there any special name for surfaces such that the curl field is tangent to them at each point?

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- Thread starter BobbyBear
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- #1

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Also in particular, if the vector field in question is the curl of another vector field, is there any special name for surfaces such that the curl field is tangent to them at each point?

- #2

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I think a surface tangent to the curl of a vector field is called a vortex tube.

- #3

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Given any n-manifold, the union of the tangent spaces at each point gives you the tangent bundle which is of 2n dimension (since its elements are couples of n-dimensional points and n-dimensional vectors). If you take a cross section from your tangent bundle, at each point this gives you a single vector tangent to the manifold hence this cross section is a vector field.

- #4

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Given any n-manifold, the union of the tangent spaces at each point gives you the tangent bundle which is of 2n dimension (since its elements are couples of n-dimensional points and n-dimensional vectors). If you take a cross section from your tangent bundle, at each point this gives you a single vector tangent to the manifold hence this cross section is a vector field.

Right. But I think the idea is used when applied to vector fields whose integral manifolds are proper submanifolds of a larger one. The spirit of the idea is to solve an ordinary differential equation and then find the submanifold that contains the flow lines. Vortex tubes are a classic example.

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Okay I think I'll just go with

[tex]

P \frac{\partial\z}{\partial x} + Q \frac{\partial\z}{\partial y} = R

[/tex]

are surfaces that are tangent at each point to the vector field (P,Q,R). These surfaces are made up of characteristic curves that are tangent at each point to the vector field. I'm not sure what you'd call these characteristic curves, but by analogy to fluid mechanics, I suppose one could call them streamlines. Streamlines passing through any closed curve (that is not a streamline itself) form a tubular surface called a stream-tube. If the curve is not closed, I suppose you could just call it a stream-surface.

Lev Elsgolts, in his book

So I think in summary one can use the following nomenclature (even though it sounds like one is referring to fluid mechanics concepts rather than just in general):

let

Then the streamlines of

The streamsurfaces of

Streamlines passing through any closed curve form a tubular surface called a streamtube.

And likewise,

Vortex lines passing through any closed curve form a tubular surface called a vortex tube.

Is this okay then?

And in the same manner, if I just say

- #6

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Okay I think I'll just go withintegral surfaceto refer to a surface that is tangent to a vector field (eg the solution the PDE

[tex]

P \frac{\partial\z}{\partial x} + Q \frac{\partial\z}{\partial y} = R

[/tex]

are surfaces that are tangent at each point to the vector field (P,Q,R). These surfaces are made up of characteristic curves that are tangent at each point to the vector field. I'm not sure what you'd call these characteristic curves, but by analogy to fluid mechanics, I suppose one could call them streamlines. Streamlines passing through any closed curve (that is not a streamline itself) form a tubular surface called a stream-tube. If the curve is not closed, I suppose you could just call it a stream-surface.

Lev Elsgolts, in his bookDifferential Equations and the Calculus of Variations, calls the curves tangent to the curl of a vector fieldV, Vortex lines. And the surfaces generated by such curves, Vortex surfaces.

So I think in summary one can use the following nomenclature (even though it sounds like one is referring to fluid mechanics concepts rather than just in general):

letF=CurlV

Then the streamlines ofFare the vortex lines ofV.

The streamsurfaces ofFare the vortex surfaces ofV.

Streamlines passing through any closed curve form a tubular surface called a streamtube.

And likewise,

Vortex lines passing through any closed curve form a tubular surface called a vortex tube.

Is this okay then?

And in the same manner, if I just sayintegral surface, if I mean the surface tangent toFthen I'd have to say integral surface ofCurlV, right?

Yes you are right. You should keep in mind thought that vortex lines and their surfaces are a special example - not every vector field is the curl of another yet all vector fields have integral surfaces.

In the books I learned from vortex surfaces were called vortex tubes.

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