The restriction of differential form

1. Dec 10, 2015

1591238460

1. Assume M=xdy -ydx+dz ∈ Ω1(R^3). What's the restriction of M to the plane {z=2}? I think it's xdy-ydx. Is that right?

2. Dec 10, 2015

lavinia

Do the two forms have the same values on tangent vectors to z = 2?

3. Dec 10, 2015

1591238460

Thank you, so what should I do?

4. Dec 10, 2015

lavinia

Just check the values.

5. Dec 11, 2015

1591238460

I think the tangent vectors of the plane z=2 are in the form of a$\frac{\partial}{\partial x}$+b$\frac{\partial}{\partial y}$, to a$\frac{\partial}{\partial x}$+b$\frac{\partial}{\partial y}$, both of the two forms of the value, am I right?

6. Dec 11, 2015

WWGD

Use 's at the beginning and end if you want to do Latex editing here.

7. Dec 11, 2015

WWGD

8. Dec 11, 2015

WWGD

Notice that, as a plane $z=2$ is 2-dimensional. Equivalently, points in the plane are of the form $(x,y,2)$

Last edited: Dec 11, 2015
9. Dec 11, 2015

1591238460

@WWGD , thank you very much, so I think I am right, doesn't it?

10. Dec 11, 2015

mathwonk

what about just saying z is constant along z=2 so dz is zero on vectors tangent to z=2? is that what you were thinking?

11. Dec 11, 2015

1591238460

@mathwonk yes, and in the embedded submanifold {z=2} the the tangent vectors are just in the form a∂∂x+b∂∂y, so the two values of two forms are the same.