Part 2: Gradient Vector Fields

If U( x,y,z) is a function of 3 variables, then

F( x,y,z) = ÑU

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

F( x,y) = á 2x,2y ñ

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

F( x,y,z) = ÑU

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

U( x,y,z) +C

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

U( x,y) = x3y2 + z.

Solution: The gradient of U( x,y,z) = x³y²+z is

ÑU = á 3x2y2,2x3y,1 ñ

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

U( x,y,z) = x3y2+z+C

LiveGraphics3d Applet

       

In physics, a potential U( x,y,z) is often used to model potential energy, in which case the vector field

F( x,y,z) = -ÑU

is known as the force vector field for that given potential.