The Question
Suppose that a mountain has the shape of an elliptic paraboloid given by $z = c - ax^2 - by^2, a,b,c \in (0,\infty)$ and $x$ and $y$ are the east-west and north-south map coordinates, and $z$ is the altitude above sea level. At point $(1,1)$ , in what direction is the altitude increasing most rapidly? If a marble were released at $(1,1)$ in what direction would it begin to roll?
My answer:
To start I turned the paraboloid into a level set given by $ax^2 + by^2 + z = c$ and then found the gradient function $\nabla f = (2ax, 2by, 1)$ which at $(1,1)$ is $(2a, 2b, 1)$. But I know that this is the direction of the normal to the 'mountain' so it can't be the direction that the altitude is increasing most rapidly. Can someone please shed some light?
For the second part I assume that the marble would roll in the opposite direction to the direction the altitude is increasing most rapidly. So just the negative of the answer to the last part!