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M 427L - Forms Practice

M 427L - Forms Practice - M 427L Dierential form Practice...

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M 427L Differential form Practice for the third exam 6. Explain why a 2-form on R n is the same thing as a anti-symmetric n × n matrix. 7. Let F = ( F 1 , F 2 , . . ., F n ) (take the corresponding column vector) be a vector field on R n . Let ω = F i dx i 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 - D T ). 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 R 3 ) is the same thing as divergence. 8. Let f : R 2 R 2 be the map f ( r, θ ) = ( rcos ( θ ) , rsin ( θ )). Compute f * ( dxdy ) (here the first R 2 has coordinates ( r, θ ) and the second has ( x, y )). 9. Compute the exterior derivative of the 2 form
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