Question

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)

          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)

let two vector fields and a scalar field be defined by fx y z x2 y2 i 2zk 10 gx y z 3xi 5yj ck c r 11 fx 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 07406

Added by Beatriz J.

Question

Please give Ace some feedback