To find the directional derivative of ∇ · (∇ϕ) at the point (1, -2, 1) in the direction of the normal to the surface xy²z = 3x + z² where ϕ = 2x⁴yz³, we first need to find the gradient of ϕ and the normal to the surface.
The gradient of ϕ is given by:
∇ϕ = (∂ϕ/∂x, ∂ϕ/∂y, ∂ϕ/∂z)
= (8x³yz³, 6x⁴z², 6x⁴y²z²)
The normal to the surface is given by the gradient of the surface function xy²z - 3x - z²:
∇(xy²z - 3x - z²) = (y²z - 3 - 2z, 2xyz, xy² - 2x)
Now, to find the directional derivative of ∇ · (∇ϕ) in the direction of the normal to the surface at the given point, we compute the dot product of the gradient of ϕ and the normal to the surface at that point:
∇ · (∇ϕ) = ∇ϕ · ∇
= (8x³yz³, 6x⁴z², 6x⁴y²z²) · (y²z - 3 - 2z, 2xyz, xy² - 2x)
We evaluate this expression at the point (1, -2, 1) to find the directional derivative.
