14. (a) Verify Stokes' theorem for F = −2yi + rzj - yz²k and S is the surface of the paraboloid 2z+y2 0≤=≤2 (b) Verify Stokes' theorem for F as in (a) and the surface being the disc r² + y² ≤4, z = 2.

Does the sphere x2+y2+z2=100 have symmetry with respect to the a x-axis? b xy-plane?

Write parametric equations for a cycloid traced by a point P on a circle of radius a as the circle rolls along the x -axis given that P is at a maximum when x=0.

Let S be the portion of the plane 2x + 3y + z = 2 lying between the points (−1, 1, 1), (2, 1, −5), (2, 3, −11), and (-1, 3, -5). Find parameterizations for both the surface S and its boundary S. Be sure that their respective orientations are compatible with Stokes' theorem. from (-1, 1, 1) to (2, 1,-5) from (2, 1, 5) to (2, 3, -11) from (2, 3, -11) to (-1, 3, -5) from (-1, 3, 5) to (-1, 1, 1) boundary S₁ (t) = S₂(t) = S3(t) = S4(t) = Φ(u, v) = te [0, 1) te [1, 2) te [2, 3) te [3, 4) UE [-1, 2], VE [1, 3]