# Curves of Steepest Descent for 3D Functions

Curves of Steepest Descent for 3D Functions

The curve of steepest descent is determined by the negative of the gradient vector, which gives the direction of maximum increase. A numerical solution of the differential equations

x'(t)=-∇f(x(t),y(t))

and

y'(t)=-∇f(x(t),y(t))

with variable initial values gives the direction of steepest descent. As you increase the length slider, the value of the parameter is increased and a curve for is traced out on the surface that follows the direction of the negative gradient at each point. Neglecting the mass of the object and friction, this plot gives the path a marble would follow as it rolled down a hill. Note that the trace of the curve of steepest descent is perpendicular to each of the contour curves, as shown in the contour plot.

t

(x(s),y(s))

s∈[0,t]