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

the answer is y < -3 or   y > 3

-3y > 7 + 2

-3 y > 9

y < -3

option a is correct.

step-by-step explanation:

scale factor: it is used to find proportional measurements.

proportions means having the same ratio.

scale factor is the ratio of the model measurement to the actual measurement in a simplest form.

or we can say that the ratio of any two corresponding lengths in two similar geometric figures

if the scale factor is more than 1, then it is an enlargement,

if the scale factor less than 1, then it is a reduction.

given the two geometric figure:

by definition of proportions, we have