Q:

Gradient "climbing a mountain"

For the function:
$$f(x,y)=100-x^2-y^3$$
if we take the gradient $\nabla f$ we get $(-2x,{-2y^2})$ so for every positive point in the plane we will get a vector pointing downward? so if we look at the function as a "mountain" and we want to go up in the steepest path, we just take the direction $(2x,{2y^2})$?

A:

Why do you think vector $(-x, -2y^2)$ looks downward? It's a vector on a plane, it's flat. However, it shows you the direction (still on a plane), where the function grows. There is nothing “downwardy” about the minus signs before $x$ and $y$ components

