Simple Nature - Light and Matter

write it using the symbol∂in place of the differential d’s. Putting
all this notation together, we have

∇U=

∂U

∂x

ˆx+

∂U

∂y

yˆ+

∂U

∂z

ˆz [definition of the gradient].

The gradient looks scary, but it has a very simple physical inter-
pretation. It’s a vector that points in the direction in whichU is
increasing most rapidly, and it tells you how rapidlyUis increasing
in that direction. For instance, sperm cells in plants and animals
find the egg cells by traveling in the direction of the gradient of the
concentration of certain hormones. When they reach the location
of the strongest hormone concentration, they find their destiny. In
terms of the gradient, the force corresponding to a given interaction
energy isF=−∇U.
Force exerted by a spring example 78
In one dimension, Hooke’s law isU = (1/2)k x^2. Suppose we
tether one end of a spring to a post, but it’s free to stretch and
swing around in a plane. Let’s say its equilibrium length is zero,
and let’s choose the origin of our coordinate system to be at the
post. Rotational invariance requires that its energy only depend
on the magnitude of thervector, not its direction, so in two di-
mensions we haveU = (1/2)k|r|^2 = (1/2)k

(

x^2 +y^2

)

. The force
exerted by the spring is then

F=−∇U

=−

∂U

∂x

xˆ−

∂U

∂y

yˆ

=−k xˆx−k yyˆ.

The magnitude of this force vector isk|r|, and its direction is to-

ward the origin.

This chapter is summarized on page 1073. Notation and terminology
are tabulated on pages 1066-1067.

Section 3.4 Motion in three dimensions 221