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The restriction of differential form

  1. Dec 10, 2015 #1
    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?
     
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  3. Dec 10, 2015 #2

    lavinia

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    Do the two forms have the same values on tangent vectors to z = 2?
     
  4. Dec 10, 2015 #3
    Thank you, so what should I do?
     
  5. Dec 10, 2015 #4

    lavinia

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    Just check the values.
     
  6. Dec 11, 2015 #5
    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?
     
  7. Dec 11, 2015 #6

    WWGD

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    Use ## ##'s at the beginning and end if you want to do Latex editing here.
     
  8. Dec 11, 2015 #7

    WWGD

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  9. Dec 11, 2015 #8

    WWGD

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    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
  10. Dec 11, 2015 #9
    @WWGD , thank you very much, so I think I am right, doesn't it?
     
  11. Dec 11, 2015 #10

    mathwonk

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    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?
     
  12. Dec 11, 2015 #11
    @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.
     
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