Describing the path of a heat seeking particle

[azure kitsune]

Homework Statement

Find the path of a heat-seeking particle placed at (4,3,10) with a temperature field

T(x,y,z) = 400 - 2x^2 - y^2 -4z^2

Homework Equations

Formulas for directional derivative and gradient.

The Attempt at a Solution

At any point in space (x,y,z), the particle must be moving in the direction which causes the greatest increase in T. This direction is given by the direction of

\nabla T = < -4x, -2y, -8z >.

I do not know how to continue from here.

 

[HallsofIvy]

The velocity vector points in the direction of \nabla T and so must be some multiple, say "k", of it.

That means you must have dx/dt= -4kx, dy/dt= -2ky, and dz/dt= -8kz for some number k.
You also are given that x(0)= 4(0), y(0)= 3, z(0)= 10. You can solve each of those for x, y, and z in terms of kt. That gives parametric equations for the path with parameter s= kt.