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Unformatted text preview: Math 285. Winter 2004/05. Differential forms A differential 0-form on R 3 is a scalar function ω : D → R 3 of class C 1 (i.e. all partial derivatives are continuous functions) on a domain D in R 3 . Let S be a set of points p 1 , . . . , p k in R 3 . We assign to each point a sign + or- and call this assignment an orientation of the set S . The integral of a differential 0-form ω over an oriented set S of points is the sum Z S ω = 1 ω ( p 1 ) + . . . + k ω ( p k ) , where i = 1 if the sign attached to p i is equal to + and i =- 1 otherwise. A differential 1-form on R 3 is an expression of the form ω = Adx + Bdy + Cdz, where F = ( A ( x, y, z ) , B ( x, y, z ) , C ( x, y, z )) : D → R 3 is a vector field on a domain D in R 3 for which the functions A, B, C belong to class C 1 . For each point P ∈ D and a vector t = P Q originated at P the differential 1-form assigns the number ω ( t ; P ) = F ( P ) · P Q. Note we put P in the notation for ω ( t ; P ) because the value depends not only on the coordinates of the vector t but also on the choice of the point P . In particular, the differential 1-form dx just reads off the first coordinate of P Q , the form dy reads the second coordinate, and dz reads the third coordinate. Now let C be a oriented simple curve in R 3 . We define the integral of a differential 1-form ω along C by Z C ω = Z b a F ( c ( t )) · c ( t ) dt = Z b a ω ( c ( t ); c ( t )) dt, where c : [ a, b ] → C is a parametrization which agrees with the orientation of C . A differential 2-form on R 3 is an expression of the form ω = Ady ∧ dz + Bdz ∧ dx + Cdx ∧ dy, where F = ( A, B, C ) : D → R 3 is a vector field of class C 1 on a domain D in R 3 . Let P ∈ R 3 and t 1 = P Q, t 2 = P R be two vectors originating at a point P ∈ D . We define ω ( t 1 , t 2 ; P ) = F ( P ) · ( t 1 × t 2 ) . If P = ( a, b, c ), Q = ( x, y, z ), R = ( x , y , z ), then t 1 = ( x- a, y- b, z- c ), t 2 = ( x- a, y- b, z- c ) and t 1 × t 2 = (( y- b )( z- c )- ( y- b )( z- c ) , ( z- c )( x- a )- ( z- c )( x- a ) , ( x- a )( y- b )- ( x- a )( y- b )) . In particular, the differential 2-form dy ∧ dz reads off the first coordinate of the cross-product, the form dz ∧ dx reads off the second coordinate, and dx ∧ dy reads off the third coordinate of the cross-product. In other words, if we considercross-product....
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