Let two vector fields and a scalar field be
defined by
F(x, y, z) = (x² + y²) i + 2zk
(10)
G(x, y, z) = 3xi - 5yj + ck, c ∈ ℝ
(11)
f(x, y, z) = 2x² + 2y² + 7z²
(12)
and let the surface S be defined by
S: y² + z² = 4, -2 ≤ x ≤ 2, z < 0. (13)
Do the following:
(a) Determine the divergence and the curl of F.
(b) Determine the value of the constant c for which G is incompressible.
(c) Determine the gradient of f.
(d) Find a parametrization r(u, v) of S and calculate r_u and r_v for this parametrization.
(e) Determine the flux integral of F out through S, i.e.
∬_S F ⋅ n dA,
(14)
where n is pointing away from the origin.
(f) Let the directed curve C be parameterized as
C: r_c(t) = (4t - 2)i + 3 cos (13π/2 t) j + 3 sin (13π/2 t) k, t ∈ [0, 1]
(15)
and define the vector field
H(x, y, z) = ∇f(x, y, z).
(16)
Determine the value of the line integral
∫_C H ⋅ dr.
(17)
