Find the values of the constants a, b, and c so that the directional derivative of ϕ = axy² + byz + cz²x³ at (1, 2, -1) has a maximum of magnitude 64 in a direction parallel to the z axis. Find the acute angle between the surfaces xy²z = 3x + z² and 3x² - y² + 2z = 1 at the point (1, -2, 1). Find the constants a and b so that the surface ax² - byz = (a + 2)x will be orthogonal to the surface 4x²y + z³ = 4 at the point (1, -1, 2). 4.68. (a) Let u and v be differentiable functions of x, y, and z. Show that a necessary and sufficient condition that u and v are functionally related by the equation F(u, v) = 0 is that ∇u × ∇v = 0. (b) Determine whether u = arc tan x + arc tan y and v = (x + y)/(1 - xy) are functionally related. 4.69. (a) Show that ∇u · ∇v × ∇w = 0 a necessary and sufficient condition that u(x, y, z), v(x, y, z), and w(x, y, z) be functionally related through the equation F(u, v, w) = 0. (b) Express ∇u · ∇v × ∇w in determinant form. This determinant is called the Jacobian of u, v, and w with respect to x, y, and z, and is written ∂(u, v, w)/∂(x, y, z) or J(u, v, w / x, y, z). (c) Determine whether u = x + y + z, v = x² + y² + z² and w = xy + yz + zx are functionally related. A = 3xyz²i + 2xy³j - x²yzk and ϕ = 3x² - yz. Find, at the point (1, -1, 1), (a) ∇ · A, (b) A · ∇ϕ, (c) ∇(ϕA), (d) ∇ · (∇ϕ). Evaluate div(2x²zi - xy²zj + 3yz²k).