hw3 - Physics 570 Homework No 3 due Wednesday 10 February...

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Unformatted text preview: Physics 570 Homework No. 3 due Wednesday, 10 February, 2010 1. Let { x,y,z } be local coordinates on a 3-dimensional manifold, and define the following four 1-forms: 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 1-forms. b. Calculate the 3-form ψ ∼ ∧ dψ ∼ . c. Find the 1-form ( φ ∼ ∧ ξ ∼ )( e u, · ), where the · inside the parentheses means that the 2-form is still “waiting” for an additional vector in order to determine the appropriate scalar, and therefore can be considered as a 1-form. d. Lastly, since you did show that the exterior derivative of the 1-form 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....
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This note was uploaded on 03/08/2010 for the course PHYSICS AN 570 taught by Professor Davids.king during the Spring '10 term at Caltech.

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hw3 - Physics 570 Homework No 3 due Wednesday 10 February...

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