We need to start this chapter off with the definition of a
vector field as they will be a major component of both this chapter and the
next. Let’s start off with the formal
definition of a vector field.
Definition
That may not make a lot of sense, but most people do know
what a vector field is, or at least they’ve seen a sketch of a vector
field. If you’ve seen a current sketch
giving the direction and magnitude of a flow of a fluid or the direction and
magnitude of the winds then you’ve seen a sketch of a vector field.
The standard notation for the function 
is,
depending on whether or not we’re in two or three
dimensions. The function P, Q,
R (if it is present) are sometimes
called scalar functions.
Let’s take a quick look at a couple of examples.
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Example 1 Sketch
each of the following direction fields.
(a) [Solution]
(b) [Solution]
Solution
(a)
Okay, to graph the vector field we need to get some
“values” of the function. This means
plugging in some points into the function.
Here are a couple of evaluations.
So, just what do these evaluations tell us? Well the first one tells us that at the
point we will plot the vector . Likewise, the third evaluation tells us
that at the point we will plot the vector .
We can continue in this fashion plotting vectors for
several points and we’ll get the following sketch of the vector field.
If we want significantly more points plotted then it is
usually best to use a computer aided graphing system such as Maple or
Mathematica. Here is a sketch with
many more vectors included that was generated with Mathematica.
[Return to Problems]
(b)
In the case of three dimensional vector fields it is
almost always better to use Maple, Mathematica, or some other such tool. Despite that let’s go ahead and do a couple
of evaluations anyway.
Notice that z
only affect the placement of the vector in this case and does not affect the
direction or the magnitude of the vector.
Sometimes this will happen so don’t get excited about it when it does.
Here is a couple of sketches generated by
Mathematica. The sketch on the left is
from the “front” and the sketch on the right is from “above”.
 
[Return to Problems]
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Now that we’ve seen a couple of vector fields let’s notice
that we’ve already seen a vector field function. In the second chapter we looked at the gradient vector. Recall that given a function the gradient vector is defined by,
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This is a vector field and is often called a gradient vector field.
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In these cases the function is often called a scalar function to
differentiate it from the vector field.
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Example 2 Find
the gradient vector field of the following functions.
(a)
(b)
Solution
(a)
Note that we only gave the gradient vector definition for
a three dimensional function, but don’t forget that there is also a two
dimension definition. All that we need
to drop off the third component of the vector.
Here is the gradient vector field for this function.
(b)
There isn’t much to do here other than take the gradient.
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Let’s do another example that will illustrate the
relationship between the gradient vector field of a function and its contours.
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Example 3 Sketch
the gradient vector field for as well as several contours for this
function.
Solution
Recall that
the contours for a function are nothing more than curves defined by,
for various values of k. So, for our function the contours are
defined by the equation,
and so they are circles centered at the origin with radius
.
Here is the gradient vector field for this function.
Here is a sketch of several of the contours as well as the
gradient vector field.
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Notice that the vectors of the vector field are all
perpendicular (or orthogonal) to the contours.
This will always be the case when we are dealing with the contours of a
function as well as its gradient vector field.
The k’s we used for the graph above were 1.5, 3, 4.5, 6, 7.5, 9, 10.5, 12, and 13.5. Now notice that as we increased k by 1.5 the contour curves get closer together and that as the contour curves get closer together the larger vectors become.
In other words, the closer the contour curves are (as k is increased by a fixed amount) the faster the function is changing at that point.
Also recall that the
direction of fastest change for a function is given by the gradient vector at
that point. Therefore, it should make
sense that the two ideas should match up as they do here.
The final topic of this section is that of conservative
vector fields. A vector field is called a conservative vector field if there exists a function such that . If is a conservative vector field then the
function, f, is called a potential function for .
All this definition is saying is that a vector field is
conservative if it is also a gradient vector field for some function.
For instance the vector field is a conservative vector field with a
potential function of because .
On the other hand, is not a conservative vector field since there
is no function f such that . If you’re not sure that you believe this at
this point be patient, we will be able to prove this in a couple of sections. In that section we will also show how to find
the potential function for a conservative vector field.