find the following. (a). 𝑑𝑖𝑣𝐹⃑ (b). 𝑔𝑟𝑎𝑑(𝑑𝑖𝑣𝐹⃑) (c). 𝑐𝑢𝑟𝑙 (𝑔𝑟𝑎𝑑(𝑑𝑖𝑣𝐹⃑))

Question

Given 𝐹⃑ = 〈𝑥𝑦𝑧^2 , 𝑥^2𝑦𝑧^3 , 𝑦^2 〉, find the following.

(a). 𝑑𝑖𝑣𝐹⃑

(b). 𝑔𝑟𝑎𝑑(𝑑𝑖𝑣𝐹⃑)

(c). 𝑐𝑢𝑟𝑙 (𝑔𝑟𝑎𝑑(𝑑𝑖𝑣𝐹⃑))

Answer 1

a) divF = ∂/∂x (xyz²) + ∂/∂y (x²yz³) + ∂/∂z (y²) = yz² + x²z³ + 0 = yz² + x²z³

b) grad(divF) = (∂/∂x, ∂/∂y, ∂/∂z) where each partial derivative is with respect to divF.

So grad(divF) is (2xz³, z², 2yz + 3x²z²)

c) curl(grad(divF)) is hard to write out on Wyzant but involves taking a determinant of a matrix with i j k as the first row, ∂/∂x ∂/∂y ∂/∂z as the third row, and each of the components of grad(divF) as the last row.