Question

You are standing above the point (5,5)(5,5) on the surface z=15−(2x2+2y2)z=15−(2x2+2y2). (a) in which direction should you walk to descend fastest? (give your answer as a unit 2-vector.)

Answer 1

Since the surface is the top part of a sphere centred on the origin, the directions of steepest descent are any vector originating from the origin (0,0).

At (5,5), the direction would be dictated by the line joining (5,5) and the origin (0,0), i.e. along <5,5> or equivalently along <1,1>.

You can also go through vector calculus to arrive at the same result.