Part 2: Gradient Vector Fields
If U( x,y,z) is a function of 3 variables, then
is called the gradient vector field of U( x,y,z) . A
gradient vector field assigns to each point the direction in which the
levels of U are increasing most quickly. Indeed, it was shown in section
2-6 that the gradient of a function f(x,y) points in the direction
that f has the greatest rate of change.
EXAMPLE 2 Find the gradient vector field of U(x,y) = x2+y2. Sketch it along with the level curves of U(x,y) .
Solution: Since ÑU =
á 2x,2y
ñ , the
gradient vector field is
Consequently, the gradients are normal to the level curves of U(x,y) and point in the direction that the levels are increasing most
quickly in.
Conversely, if F( x,y,z) is the gradient of some
function U( x,y,z), which is to say that
then F( x,y,z) is called a conservative vector
field, or equivalently, F( x,y,z) is said to be exact. The function U( x,y,z) is said to be a potential
for F( x,y,z). It follows that all the potentials
of a conservative vector field F( x,y) are of the
form
where C is an arbitrary constant.
EXAMPLE 3 Show that F( x,y,z) =
á 3x2y2, 2x3y,
1
ñ is exact (that is,
conservative) by showing that it is the gradient of
Solution: The gradient of U( x,y,z) = x3y2+z is
which is the same as F( x,y,z) . Thus, F( x,y,z) =
á 3x2y2,2x3y,1
ñ is
conservative, and all of its potential functions are of the form
In physics, a potential U( x,y,z) is often used to
model potential energy, in which case the vector field
is known as the force vector field for that given potential.
Check your Reading: Why are potential functions unique
only up to a constant?