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.

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step-by-step explanation:

i think something is missing in your question because y-1=(x-12) would not pass through the point (-4,-3). because if you replace x by -4 your are supposed to find y=-3 which is not the case her. if you replace x by -4 you find =-15.

i assume the right equation is y-1=1/4*(x-12) because this one is passing through both your points (if you are interested to know how i found that out i can tell you).

in this equation if i replace x by -4 i find y=-3 and if i replace x by 12, i find y=1 so the line representing this equation is passing through those 2 points.

then the standard form of the equation is finding an expression of y in function of x

y-1=1/4*(x-12)

y=x/4-3+1

y=x/4-2

hope that .

do you confirm there was a mistake in the question?