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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!

1 Answers 1

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You do not want to consider a level set here. But rather define $f(x,y)=c-ax^2-by^2$. Then the altitude of the mountain directly above the point $(x,y)$ is $f(x,y)$. You wish to find the direction in which $f$ is increasing most rapidly at the point $\bigl(1,1, f(1,1)\bigr)$. This of course would be the gradient of $f$ evaluated at $(1,1)$.

As for the marble, your intuition is correct: it would roll in the direction in which $f$ is decreasing most rapidly at the point $\bigl(1,1,f(1,1)\bigr)$. This would be the direction opposite the gradient of $f$ evaluated at $(1,1)$. So, here, just multiply your answer to the first part by $-1$.