What are the names of surfaces that are tangent to a vector field?

[BobbyBear]

Given a vector field, what do you call a surface such that the vector field is tangent to it at each point? I think in fluid mechanics they're called stream surfaces, but I just wondered if there's a name for them in general.

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?

 

[wofsy]

In calculus such a surface is called an integral surface of the vector field. This idea generalizes to finite dimensional distributions on differentiable manifolds. The distribution can be integrated i.e. it is tangent to a sub-manifold iff it is closed under the Lie bracket. This is the Froebeiuns Theorem.

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

 

[Sina]

Conversely,

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.

 

[wofsy]

Conversely,

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.

 

[BobbyBear]

GUYS! thanks for all your replies:) Unfortunately, I don't know what a manifold is, what a lie bracket is, or what a bundle is :P

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

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

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 Differential Equations and the Calculus of Variations, calls the curves tangent to the curl of a vector field V, 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):

let F = Curl V

Then the streamlines of F are the vortex lines of V.
The streamsurfaces of F are the vortex surfaces of V.
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 integral surface, if I mean the surface tangent to F then I'd have to say integral surface of CurlV, right?

 

[wofsy]

GUYS! thanks for all your replies:) Unfortunately, I don't know what a manifold is, what a lie bracket is, or what a bundle is :P

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

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

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 Differential Equations and the Calculus of Variations, calls the curves tangent to the curl of a vector field V, 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):

let F = Curl V

Then the streamlines of F are the vortex lines of V.
The streamsurfaces of F are the vortex surfaces of V.
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 integral surface, if I mean the surface tangent to F then I'd have to say integral surface of CurlV, 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.