1. The problem statement, all variables and given/known data I am only currently in multivariate calculus, so i haven't even touched differential geometry yet, but a question that i had while learning about gradients came up and it led me to the topic of geodesics and differential geometry, so here goes: Class problem: Find the equation representing the path of the particle as it moves in the direction of maximum temperature increase on a plate whose temperature at (x,y) is f(x,y) = 20 - 4x^2 - y^2. The particle starts at (2,-3). Solution: take contour map and the curves just end up being the family of orthogonal trajectories, then plug in the point for the particular solution. My thought: if you were you graph the temperature function in 3D you would get a paraboloid, and if you graphed the path in 3D it would be a parabolic cylinder (just the path stretched in the z direction). Is the curve of intersection of these two surfaces the shortest path between any point on the that curve and the "source of heat"? clarification: in my case, the particle is confined to travel on the paraboloid as opposed to just a flat plane for the class problem. 2. Relevant equations well i mean gradient i guess.... 3. The attempt at a solution I googled geodesics and such, but the rigor of calculus is a little beyond my weak high school abilities heh. thanks, I did my best.