Some people find it useful to try and draw pictures to illustrate a vector valued function. This can be a
very useful idea in the case where the function takes points in D ⊆ ℝ2 and delivers a vector in ℝ2. For
many points
(x,y)
∈ D, you draw an arrow of the appropriate length and direction with its tail at
(x,y)
.
The picture of all these arrows can give you an understanding of what is happening. For example if the
vector valued function gives the velocity of a fluid at the point
(x,y)
, the picture of these arrows can
give an idea of the motion of the fluid. When they are long the fluid is moving fast, when
they are short, the fluid is moving slowly. The direction of these arrows is an indication of
the direction of motion. The only sensible way to produce such a picture is with a computer.
Otherwise, it becomes a worthless exercise in busy work. Furthermore, it is of limited usefulness in
three dimensions because in three dimensions such pictures are too cluttered to convey much
insight.
Example 8.2.1Draw a picture of the vector field
(− x,y)
which gives the velocity of a fluid flowingin two dimensions.
PICT
You can see how the arrows indicate the motion of this fluid.
Example 8.2.2Draw a picture of the vector field
(y,x)
for the velocity of a fluid flowing in twodimensions.
PICT
Here is another such example.
Example 8.2.3Draw a picture of the vector field
(y cos(x) + 1,x sin(y)− 1)
for the velocity of afluid flowing in two dimensions.
PICT
These pictures were drawn by maple. Note how they reveal both the direction and the
magnitude of the vectors. However, if you try to draw these by hand, you will mainly waste
time.