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# Stroke's Theorem and Direct Evaluation

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(a) Let F(x,y,z)=(x^2+y-4)i + (3xy)j + (2xy+z^2)k. Evaluate the double integral over S of (curl(F). dS) where S is the surface x^2 + y^2 + z^2 = 16, z >=0

(I) Using Stroke's theorem
(II)By direct evaluation

(b) Find the flux of the vector field
F(x,y,z) = (y-x)i + (x+y)j + y k across the side of the triangle with vertices at (1,0,0), (0,1,0) and (0,0,1)

https://brainmass.com/physics/flux-flux-density/strokes-theorem-direct-evaluation-10565

#### Solution Preview

(a) Let F(x,y,z)=(x^2+y-4)i + (3xy)j + (2xy+z^2)k. Evaluate the double integral over S of (curl(F). dS) where S is the surface x^2 + y^2 + z^2 = 16, z >=0

(I) Using Stroke's theorem
(II)By direct evaluation

Solution. (I) I don't know how to use Stroke's theorem to calculate the surface integral with respect to coordinate elements. I am ...

#### Solution Summary

This does several things with a function: evaluates a double integral using Streok's theorem and direct evaluation, and then finds the flux of a given vector field

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