M 427L - Forms Practice

M 427L Forms - M 427L Differential form Practice for the third exam 6 Explain why a 2-form on Rn is the same thing as a anti-symmetric n × n

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Unformatted text preview: M 427L Differential form Practice for the third exam 6. Explain why a 2-form on Rn is the same thing as a anti-symmetric n × n matrix. 7. Let F = (F1 , F2 , . . . , Fn ) (take the corresponding column vector) be a vector field on Rn . Let ω = Fi dxi be the corresponding 1-from. Show that d(ω ) has the same data as A, the anti-symmetric part of the matrix D = D(F ), i.e. A = 1/2(D − DT ). Explain why for n = 3, exterior derivative of 1-forms is the same thing as curl of vector fields. Explain why exterior derivative of 2-forms (on R3 ) is the same thing as divergence. 8. Let f : R2 → R2 be the map f (r, θ ) = (rcos(θ ), rsin(θ )). Compute f ∗ (dxdy ) (here the first R2 has coordinates (r, θ ) and the second has (x, y )). 9. Compute the exterior derivative of the 2 form g (x, y, z )dy ∧ dz ? 10. Is it true or false that every n-form on Rn 3 is of form d(α) for some n − 1-form α? 11. Explain why the integral of an exact 2-form (i.e. a form of type d(α) for a 1-form α) over a surface (in R3 ) without boundary, is zero. 12. Translate the previous problem into a statement about the flux of the curl of something. 13. Suppose α = d(β ) for β a 1-form on R3 . Show that α is of form d(a(x, y, z )dx + b(x, y, z )dy ). (So we can find a solution to α = d(β ) for β having no dz coefficient). ...
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This note was uploaded on 09/18/2011 for the course M 427L taught by Professor Staff during the Spring '11 term at Oklahoma State.

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