8—Multivariable Calculus 217For the examplef(x,y) =x^2 + 4y^2 ,G~= 2xˆx+ 8yˆy. At each point in thex-yplane it provides a vector
showing the steepness offat that point and the direction in whichfis changing most rapidly.
Notice that the gradient vectors are twice as long where the ellipses are closest together as they are at the
ends where the ellipses are farthest apart. The function changes more rapidly in they-direction.
The United States Coast and Geodetic Survey has available for sale a large number of maps, and of particular
interest to hikers are the contour maps. They show curves indicating the lines of constant altitude. As the highest
altitude in Florida is less than 100 meters, you may never have seen one of these maps, but they’re important
where there are mountains.
The gravitational potential energy of a massmnear the Earth’s surface ismgh. This divided by the mass is
the gravitational potential,gh. These lines of constant altitude are then lines of constant potential, equipotentials
of the gravitational field. Walk along such 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