15.9 Vector Fields
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
, you draw an arrow of the
appropriate length and direction with its tail at
. 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
, 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
Example 15.9.1 Draw a picture of the vector field
which gives the velocity
of a fluid flowing in two dimensions.
You can see how the arrows indicate the motion of this fluid.
Here is another such example. This one is much more complicated.
Example 15.9.2 Draw a picture of the vector field
the velocity of a fluid flowing in two dimensions.
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.