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One Forms

  1. Jul 19, 2014 #1
    Hi

    I am a new(ish) student of general relativity. I am currently reading 'Relativitiy DeMystified'

    However this guys explanation of one forms is completely mystifying to me.

    He says that basis vectors

    [tex] e_a = ∂_a = {\frac{∂a}{∂X^a}} [/tex]


    And then says that this type of basis is called a coordinate basis, and that is allows us to 'think of a vector as an operator, one that maps a function into a new function

    Then

    [tex] Vf = (V^a e_a) = V^a ∂_a f [/tex]

    The vector V can be represented by covariant components V_a and this vector is called a 'one form'

    I just did not get that. Can anyone explain in really simple terms what a one form is?

    Thanks

    Peter
     
  2. jcsd
  3. Jul 19, 2014 #2

    Nugatory

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    Staff: Mentor

    Stab an onion with an icepick, and the icepick will pierce some number of layers of the onion. The one-form tells you how many layers of the onion will be pierced by a stab of unit length in a particular direction. Higher numerical values of the components of the one-form correspond to more closely spaced and thinner layers of the onion.

    The traditional gradient of a function can be treated as a one-form. It tells you how the value of the function changes when you move an infinitesimal distance in a given direction. The surfaces of constant value of the function are analogous to the layers of the onion.
     
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