# The elevation on a portion of a hill is given by $f(x, y) = 100 − 4x^2 − 2y$. From the location above $(2, 1)$, in which direction will water run?

So, I know that the gradient will be $$<-8(2), -2>$$ since we are measuring at the point $$(2,1)$$. However, I'm not really sure where to go from there. I'm looking for the direction in which the water will travel, but how will I go about finding that? Should I try to somehow relate this to finding steepest ascent or descent?

• Maybe $-\|\vec\nabla f(x,y)\|$? – manooooh Oct 23 '18 at 4:16
• I mean, evaluated at $(2,1)\in\Bbb R^2$, i.e. $-\|\vec\nabla f(2,1)\|$. – manooooh Oct 23 '18 at 4:48

Things acted on by gravity move to minimize their potential energy. For example a bar in midair, will travel downward as to minimize its gravitational potential energy. Minimizing gravitational potential energy, just amounts to minimizing the elevation from the ground. You need to find in which direction from $$(2,1)$$ the water will run to minimize its height. In other words, you need to find the direction of steepest decent.