Let F= < x²e², xe, 2²e²y >. •ff, curl F S S is the hemisphere x² + y² + z² = 1, z ≥ 0, oriented upwards. Use Stokes' Theorem to evaluate curlF.ds, where

Let Q be the rectangle formed by (0, 0, 0), (1, 0, 0), (0, 1, 1) and (1, 1, 1) oriented CCW as x² dx + viewed from above (from the positive z-axis). Use Stokes' Theorem to calculate 1. y²dy - xdz.

Let F = . If curl S is the hemisphere x² + y² + ² = 4, z ≥ 0, oriented upwards Use Stokes' Theorem to evaluate curlFdS, where

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Vector F is mathematically defined as F = M x N, where M = p 2p² cos + 2p2 sind while N is a vector normal to the surface S. Determine F as well as the area of the plane perpendicular to F if surface S = 2xy + 3z.

Let u = u(x, y), and let (r, 0) be polar coordinates. Verify the re- lation |V? = u; + zuổ Hint: Compute the right-hand side by expressing ug and u, in terms of Ux and u y.

20. Let S be an oriented surface parametrized by r(u,v). Define the notation do = r, du × r, dv so that do is a vector normal to the surface. Also, the magnitude do = face area (by Equation 5 in Section 16.5). Derive the identity do is the element of sur- do = (EG – F²)'/² du dv where E = r?, F = r„ • r,, and G = |r,|?.

Convert to polar coordinates and evaluate