8—Multivariable Calculus 187The U.S.Coast and Geodetic Survey makes a large number of maps, and hikers are particularly
interested in the contour maps. They show curves indicating the lines of constant altitude. When
Apollo 16 went to the Moon in 1972, NASA prepared a similar map for the astronauts, and this is a
small segment of that map. The contour lines represent 10 meter increments in altitude.*
The gravitational potential energy of a massmnear the Earth’s (or Moon’s) surface ismgh.
This divided bymis the gravitational potential,gh. These lines of constant altitude are then lines of
constant potential, equipotentials of the gravitational field. Walk along an equipotential and you are
doing no work against gravity, just walking on the level.
8.6 Electrostatics
The electric field can be described in terms of a gradient. For a single point charge at the origin the
electric field is
E~(x,y,z) =kq
r^2
ˆr
whererˆis the unit vector pointing away from the origin andris the distance to the origin. This
vector can be written as a gradient. Because thisE~is everywhere pointing away from the origin, it’s
everywhere perpendicular to the sphere centered at the origin.
E~=−gradkq
r
You can verify this a several ways. The first is to go straight to the definition of a gradient. (There’s
a blizzard of minus signs in this approach, so have a little patience. It will get better.) This function is
increasing most rapidly in the direction moving toward the origin. ( 1 /r) The derivative with respect to
distance in this direction is−d/dr, so
−d/dr(1/r) = +1/r^2. The direction of greatest increase is along−ˆr, so grad(1/r) =−rˆ(1/r^2 ). But
the relation to the electric field has another− 1 in it, so
−gradkq
r
= +ˆr
kq
r^2
There’s got to be a better way.
Yes, instead of insisting that you move in the direction in which the function isincreasingmost
rapidly, simply move in the direction in which it is changing most rapidly. The derivative with respect
to distance in that direction is the component in that direction and the plus or minus signs take care of
themselves. The derivative with respect torof ( 1 /r) is− 1 /r^2. That is the component in the direction
ˆr, the direction in which you took the derivative. This says grad(1/r) =−ˆr(1/r^2 ). You get the same
result as before but without so much fussing. This also makes it look more like the familiar ordinary
derivative in one dimension.