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Curves of Steepest Descent for 3D Functions

(x, y)

	0
0

length

0.001

f(x, y) =

5 cos(x - 2) + 5 sin(x - 2y)

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

(x(s),y(s))

for

s∈[0,t]

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.

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