I have to find the path of quickest decent for the surface $f(x,y)=50-x^2-2y^2$ so I find $\nabla f$ the gradient vector and at any point ill head in the opposite direction of $\nabla f$. Now to find the curve. From the above I find $r' =2x \mathbf{i} + 4y\mathbf{j}$.
Now to find $r$ I have said assume x and y to be functions of $t$ then ye can get $\frac{\frac{dy}{dt}}{\frac{dx}{dt}}=\frac{dy}{dx}=\frac{2y}{x}$ which I can solve given some initial point.
Two questions. Is this correct? I seen something similar and its the only way I can see how to do this. If it is right can someone explain how this works or what is going on at and after the point I assume both to be a function of t