Evaluate F. dr. where F(x, y, z) = -2y + 2xj + zk and C is...

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Evaluate F. dr. where F(x, y, z) = -2y + 2xj + zk and C is the curve of the intersection of the plane y + z = 3 and the cylinder x + y = 4. (Orient C to be counterclockwise when viewed from above.) SOLUTION The curve C (an ellipse) is shown in the figure. Although JF. dr could be evaluated directly, it's easier to use Stokes Theorem. We first compute i |# dx curl F = - // curl F. = 1h 1 + [ (2+ 2 = [ht IF. dr = Although there are many surfaces with boundary C, the most convenient choice is the elliptical region S in the plane y + z 3 that is bounded by C. If we orient S upwards, then C has the induced positive orientation. The projection D of S on the xy-plane is the disk x + y s 4 and so with z = g(x, y) = 3-y, we have curl F. ds = 2 x8 s = // [ (-+) j 8 dy 8 = (2T) + 0 = 8T 2 k a z Jr dr de de )d8 )k dA Evaluate F. dr. where F(x, y, z) = -2y + 2xj + zk and C is the curve of the intersection of the plane y + z = 3 and the cylinder x + y = 4. (Orient C to be counterclockwise when viewed from above.) SOLUTION The curve C (an ellipse) is shown in the figure. Although JF. dr could be evaluated directly, it's easier to use Stokes Theorem. We first compute i |# dx curl F = - // curl F. = 1h 1 + [ (2+ 2 = [ht IF. dr = Although there are many surfaces with boundary C, the most convenient choice is the elliptical region S in the plane y + z 3 that is bounded by C. If we orient S upwards, then C has the induced positive orientation. The projection D of S on the xy-plane is the disk x + y s 4 and so with z = g(x, y) = 3-y, we have curl F. ds = 2 x8 s = // [ (-+) j 8 dy 8 = (2T) + 0 = 8T 2 k a z Jr dr de de )d8 )k dA

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