Let F= r²i+ (2xy + x) j+zk. Let C be the circle x² + y² = 1 and S the disk a² + y² ≤ 1 within the plane = 0. (a) Determine the flux of F out of S. (b) Determine the circulation of F around C. (c) Find the flux of VX F. Verify Stokes' theorem directly in this case.

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Show that the acceleration of a point particle in polar coordinates (d'r d²0 dr de +2 de 0 is equivalent to the acceleration in Cartesian a = r r dt? dt dt? dt dt d?x coordinates ả : dt? d'y -ŷ using r = dt? x² + y² , î = cos Oâ + sin0ŷ, 2 Ö = - sin Ox + cos Oŷ , x =r cos0, and y = r sin©.

(d) Use Stokes' theorem to evaluate f. F.dr where F = z?î + y²j + xk and C is the triangle with vertices (1, 0, 0), (0, 1, 0), and (0, 0, 1). The unit vector normal is upward.