Consider the following region r and the vector field f. a. compute the two-dimensional curl of the vector field. b. evaluate both integrals in green's theorem and check for consistency. bold upper f equals left angle negative x comma negative y right angle; upper r equals startset (x comma y ): x squared plus y squared less than or equals 5 endset a. the two-dimensional curl is 0. (type an exact answer.) b. set up the integral over the region r. write the integral using polar coordinates, with r as the radius and theta as the angle. integral from 0 to nothing integral from 0 to nothing (nothing )r font size decreased by 3 dr font size decreased by 3 d theta (type exact answers.)

Looks like we're given

which in three dimensions could be expressed as

and this has curl

which confirms the two-dimensional curl is 0.

It also looks like the region is the disk . Green's theorem says the integral of along the boundary of is equal to the integral of the two-dimensional curl of over the interior of :

which we know to be 0, since the curl itself is 0. To verify this, we can parameterize the boundary of by

with . Then

black/brown

step-by-step explanation:

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no

step-by-step explanation: