Math help needed

Question

Given that F = x³zi - 2yz²j + x y³z⁴k. Find the following at (1,1,3):

i)div F.

ii)curl F.

iii)∇ · (∇ x F).

Answer 1

(i) div(F) = ∂_x(x³z) + ∂_y(-2yz²) + ∂_z(xy³z⁴) = 3x²z - 2z² + 4xy³z³

At (1, 1, 3), you just plug in x = 1, y = 1, z = 3 in the above.

div(F(1, 1, 3)) = 3•1²•3 - 2•3² + 4•1•1³•3³ = 99

(ii) curl(F) = (∂_y(xy³z⁴) - ∂_z(-2yz²))i - (∂_x(xy³z⁴) - ∂_z(x³z))j + (∂_x(-2yz²) - ∂_y(x³z))k

= (3xy²z⁴ + 2z²)i - (y³z⁴ - x³)j + (0 - 0)k

At (1, 1, 3), you just plug in x = 1, y = 1, z = 3 in the above, you'll get what you want curlF(1, 1, 3).

(iii) It is basically asking for the divergence of the curl of F, which is zero (for any vector field with continuous partial derivatives) div(curl(F)) = 0.