Verify that the line intergral and the surface integral of stokes theorem are equal for the following vector field, surface s and closed curve c. assume that c has counterclockwise orientation and s has a consistent orientation.

f= < x, y,z> ; s is the paraboloid z = 13 - x^2 - y^2, for 0 less than or equal z less than or equal 13 and c is the circle x^2 + y^2 = 13 in the xy plane.

evaluate both integrals showing steps of each

See steps below

Step-by-step explanation:

We will verify Stokes' theorem for the vector field  

F(x,y,z) = (x,y,z)

and the surface bounded by the paraboloid  

and the plane z = 0

(See picture)

Specifically, we must verify that

First we compute the integral over the boundary curve, which is the circle C.

Now, we see that the force field F(x,y,z) is conservative since it is the gradient of the scalar function

That is,

Hence the curve integral along any closed path equals 0

That is to say ,

On the other hand

Hence

and Stokes' theorem holds.

i think the answer is d im not sure