D integraldisplay γ f d s where f x y z x y y 2 z xy

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(d) integraldisplay γ F · d s , where F ( x, y, z ) = ( x + y, y 2 z, xy ) and γ ( t ) = ( t 2 , 2 t, - t 3 ), 0 t 1. 9. [10 points] For each of the following differential forms ω determine if ω is exact. If ω is exact, use the algorithm given in class to find its potential function g . (a) ω = ( yze xyz + 3 x 2 y 2 ) dx + ( xze xyz + 2 x 3 y + 2 yz ) dy + ( xye xyz + y 2 z ) dz (b) ω = ( y 3 + 2 xy sin z ) dx + (2 yz + 3 xy 2 + x 2 sin z ) dy + ( y 2 + 2 z + x 2 y cos z ) dz . 10. [14 points] (a) Give the definition of each of the following: i. A trigonometric polynomial of degree N . ii. A (parametrized) path in R n . iii. A subset S R n is simply connected . iv. g is a potential function for F : R n R n v. A 2– dimensional (parametrized) surface in R n . (b) State Green’s Theorem .
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