Question

How to find general equation of path taken by a particle?

Answer 1

First of all, I used GeoGebra to draw the function f(x,y)=x2y2f(x,y)=x2y2 and the point P(1,2,4)P(1,2,4). I know that the particle will move in the direction of the gradient, hence I calculated the gradient

∇⃗ f(x,y)=(2xy2,2x2y)⇒x=1,y=2∇⃗ f=(8,4)∇→f(x,y)=(2xy2,2x2y)⇒⏟x=1,y=2∇→f=(8,4)Now, I think that the particle will stop when it reaches the origin, but I'm not sure it is true, and if it is true how to explain it.

Also, I don't understand how the calculation of ∇⃗ f(x,y)∇→f(x,y) could help me in finding the explicit form of the path.