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lecture33

# lecture33 - Lecture 33 6.5 Dierential Forms on Manifolds...

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Lecture 33 6.5 Differential Forms on Manifolds Let U R n be open. By definition, a k -form ω on U is a function which assigns to each point p U an element ω p Λ k ( T R n ). p We now define the notion of a k -form on a manifold. Let X R N be an n - dimensional manifold. Then, for p X , the tangent space T p X T p R N . Definition 6.14. A k -form ω on X is a function on X which assigns to each point p X an element ω p Λ k (( T p X ) ). Suppose that f : X R is a C map, and let f ( p ) = a . Then df p is of the form df p : T p X T a R R . (6.47) = We can think of df p ( T p X ) = Λ 1 (( T p X ) ). So, we get a one-form df on X which maps each p X to df p . Now, suppose µ is a k -form on X , and (6.48) ν is an -form on X . (6.49) For p X , we have µ p Λ k ( T X ) and (6.50) p ν p Λ ( T X ) . (6.51) p Taking the wedge product, µ p ν p Λ k + ( T X ) . (6.52) p The wedge product µ ν is the ( k + )-form mapping p X to µ p ν p . Now we consider the pullback operation. Let X R N and Y R be manifolds, and let f : X Y be a C map. Let p X and a = f ( p ). We have the map df p : T p X T a Y. (6.53) From this we get the pullback ( df p ) : Λ k ( T a Y ) Λ k ( T X ) . (6.54) p Let ω be a k -form on Y . Then f ω is defined by ( f ω ) p = ( df p ) ω q . (6.55) 1

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Let f : X Y and g : Y Z be C maps on manifolds X, Y, Z . Let ω be a k -form. Then ( g f ) ω = f ( g ω ) , (6.56) where g f : X Z . So far, the treatment of k -forms for manifolds has been basically the same as our earlier treatment of k -forms.
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lecture33 - Lecture 33 6.5 Dierential Forms on Manifolds...

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