Use stokes' theorem to evaluate c f · dr where c is oriented counterclockwise as viewed from above. f(x, y, z) = yzi + 3xzj + exyk, c is the circle x2 + y2 = 16, z = 7.

This question is based on the stokes' theorem.Therefore,  the  by using stokes theorem.

Given:

f(x, y, z) = yzi + 3xzj + yk, c is the circle z = 7.

According to the question,

Stokes' theorem states that, the line integral of f(x, y, z) along C is equivalent to the surface integral of the curl f(x, y, z) of  over the disk D bounded by :

Now, the disk can be parameterized by,

s(u,v) = u cos v i + u sin v j + 7 k

Where, u lies between 0 to 4 and v lies between 0 to 2.

The curl of the given vector field is,

Hence, the surface integral is,

Therefore,  the by using stokes theorem.

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