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Unformatted text preview: Physics 570 Homework No. 3 Solutions: due Wednesday, 10 February, 2010 1. Let { x,y,z } be local coordinates on a 3dimensional manifold, and define the following four 1forms: φ ∼ ≡ yz dx + dz , ψ ∼ ≡ (sin z ) dx + (cos z ) dy , ξ ∼ ≡ dy + z dz , α ∼ ≡ y dx xdy x 2 + y 2 . and also define the following tangent vector: e u = ∂ x + ∂ y . a. Calculate the exterior differential of each of the four 1forms. b. Calculate the 3form ψ ∼ ∧ dψ ∼ . c. Find the 1form ( φ ∼ ∧ ξ ∼ )( e u, · ), where the · inside the parentheses means that the 2form is still “waiting” for an additional vector in order to determine the appropriate scalar, and therefore can be considered as a 1form. d. Lastly, since you did show that the exterior derivative of the 1form α ∼ above was zero, there must be some function f such that α ∼ = df . Please present such a function, but be assured that the function f is not unique. ........................................................................................... a. The calculations are straightforward, I hope: dφ ∼ = y dz ∧ dx + z dy ∧ dx , dψ ∼ = cos z dz ∧ dx sin z dz ∧ dy ,dξ ∼ = 0 , d α ∼ = 2 dy ∧ dx x 2 + y 2 2 ( xdx + y dy ) ∧ ( y dx xdy ) ( x 2 + y 2 ) 2 = 2 dy ∧ dx x 2 + y 2 2 ( x 2 + y 2 ) dy ∧ dx ( x 2 + y 2 ) 2 = 0 . b. Already having dψ ∼ again we can simply follow through the manipulations: ψ ∼ ∧ dψ ∼ = (sin z dx + cos z dy ) ∧ (cos z dz ∧ dx sin z dz ∧ dy ) = sin 2 z dx ∧ dy ∧ dz + cos 2 z dx ∧ dy ∧ dz = dx ∧ dy ∧ dz . c. We first need to determine the desired 2form: φ ∼ ∧ ξ ∼ = yz dx ∧ dy + dz ∧ dy + yz 2 dx ∧ dz . Then when we are letting a 2form act on a vector, we need to remember that it is skew symmetric. A simple example is that ( dx ∧ dy )( ∂ x + ∂ y , · ) = ( dx ⊗ dy dy ⊗ dx )( ∂ x + ∂ y , · ) = dy dx . Therefore, for our current case we have ( yz dx ∧ dy + dz ∧ dy + yz 2 dx ∧ dz )( ∂ x + ∂ y , · ) ≡ ( ∂ x + ∂ y ) c ( yz dx ∧ dy + dz ∧ dy + yz 2 dx ∧ dz ) = yz dy + yz 2 dz yz dx dz = yz ( dx dy ) + ( yz 2 1) dz ....
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 Spring '10
 DavidS.King
 Derivative, Vector Space, ........., Tensor field

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