1. (10 分) Evaluate ∬S(F⋅N)dS, where F = (xy, y², x³y³), N is the outer normal and S is the surface bounded by the parabolic cylinder z = y² and the planes z = z, z = 0 and x = 1.

2. (10 分) Let C be a positively oriented, piecewise-smooth, simple closed curve in the plane and let D be the region bounded by C. If f(x, y) is a scalar function with continuous first and second partial derivatives in D, derive the following formula: ∬D ( fxx + fyy )dxdy =where = ∇f⋅N is the normal derivative, N is an outer unit normal, s is the arc-length.

1. (10 分) Evaluate ∬_S(F⋅N)dS, where F = (xy, y², x³y³), N is the outer normal and S is the surface bounded by the parabolic cylinder z = y² and the planes z = z, z = 0 and x = 1.

相關申論題

相關試卷