7 to 2 1 1 9 10 points for each of the following

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7) to (2 , 1 , 1). 9. [10 points] For each of the following differential forms ω determine if there exists a function g such that ω = dg . Whenever g exists, use the algorithm given in class to find the g . (a) ω = ( yz cos( xyz ) + 2 xy 3 z 2 ) dx + ( xz cos( xyz ) + 3 x 2 y 2 z 2 ) dy + ( xy cos( xyz ) + 2 x 2 y 3 z 2 ) dz . (b) ω = parenleftbigg z 1 + x 2 z 2 + yz 2 parenrightbigg dx + ( xz 2 - z cos y ) dy + parenleftbigg x 1 + x 2 z 2 + 2 xyz - sin y + 3 z 2 parenrightbigg dz . 10. [10 points] (a) Give a curve γ and a differential form ω such that integraldisplay γ ω is equivalent by Green’s Theorem to integraldisplay 1 - 1 integraldisplay 1 - x 2 - 1 - x 2 e x 2 dy dx . EVALUATION OF THE INTEGRAL IS NOT REQUIRED. (b) State the corollary to Green’s Theorem which expresses the area of a region as a line integral.
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