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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
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
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.
- Spring '11