Given the system defined by the scalar fields F and G, find the unique function ϕ(x, y, z) = (u, v) defined by the implicit function theorem. Also,...

Given the system defined by the scalar fields F and G, find the unique function ϕ(x, y, z) = (u, v) defined by the implicit function theorem. Also, determine a unit vector ~v in the first quadrant that forms an angle of π/4 with the X-axis and calculate ∂f/∂~v(0,-1) for the function f(x,y) = y^2 + 2xy + 1.
Find the unique function ϕ(x, y, z) = (u, v) defined by the implicit function theorem.
Determine a unit vector ~v in the first quadrant that forms an angle of π/4 with the X-axis.
Calculate ∂f/∂~v(0,-1) for the function f(x,y) = y^2 + 2xy + 1.