Imagine you have a terrain which height is determined by the function $$H(x,y)=1500e^{-x^2-2y^2}$$
If we let a ball down the top of the highest mountain, and we know it will always follow the maximum slope, prove the function it will follow will be $y=x^2$.
The highest mountain will be the point $(0,0)$, with a height of $1500$.
The parcial derivatives are :
$$\frac{\partial H}{\partial x}=-1500·2x·e^{-x^2-2y^2}$$
$$\frac{\partial H}{\partial y}=-1500·4y·e^{-x^2-2y^2}$$
And so, the gradient is $$∇(x,y)=(-3000xe^{-x^2-2y^2},-6000ye^{-x^2-2y^2})$$
This is where I get stuck. Am I on the good way? What would be the next step?