Find a unit vector normal to the surface x3 + y3 + 3xyz = 3 at the point (1, 2, −1).

Explanation

∇φ = (3x² + 3yz) i + (3y² + 3xz) j + 3xy k. At (1, 2, −1) this is −3i + 9j + 6k = 3(−i + 3j + 2k). Its magnitude is 3√14, so the unit normal is (−i + 3j + 2k)/√14.