Show that the vector field f x y cos x e y sin x e y

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Show that the vector field F ( x , y ) < cos( x ) e y , sin( x ) e y 10 y > is actually a gradient field by producing a function φ ( x , y ) such that ∇φ ( x , y ) = F ( x , y ) for all ( x , y ) in the plane. (b) Using the Fundamental Theorem of Line Integrals, evaluate the path integral below, where C is any smooth path from the origin to the point ( π /2,ln(2)). [ WARNING: You must use the theorem to get any credit here.] C (cos( x ) e y ) dx (sin( x ) e y 10 y ) dy
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TEST3/MAC2313 Page 6 of 6 ______________________________________________________________________ 10. (10 pts.) Use the substitution u = (1/2)( x + y ) and v = (1/2)( x - y ) to evaluate the integral below, where R is the bound region enclosed by the triangular region with vertices at (0,0), (2,0), and (1,1). R cos 1 2 ( x y ) dA R cos 1 2 ( x y ) dA x , y ______________________________________________________________________ Silly 10 Point Bonus: Become a polar explorer. Reveal the details of how one can obtain the exact value of the definite integral 0 e x 2 dx even though there is no elementary anti-derivative for the function f( x ) e x 2 . Say where your work is, for it won’t fit here! [You may gloss some of the technical details related to the matter of convergence.]
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Show that the vector field F x y cos x e y sin x e y 10 y...

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