Suppose that a mountain climber is at the point $(500, 300, 3390)$ on the surface of a mountain described by the function $h(x,y)=4000-.001x^2-.004y^2$. In what direction would the climber have the steepest up slope? What is that slope?
$\nabla h(x,y) = h_x(x,y)i+h_y(x,y)j$
$h_x = -.002xi$
$h_y = -.008yj$
$\nabla h(500,300) = \langle -.002(500), -.008(300)\rangle = \langle -1,-2.4\rangle$
Since this is a negative vector, I assume this would be the steepest down slope, so the steepest up slope would be the opposite?
$\langle 1,2.4 \rangle$
Does this solution answer both what the steepest up slope is and the the what the slope is?