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Vector Fields
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
...
Definition
A vector field on two (or three) dimensional space is a function F that assigns to each point
( x, y ) (or ( x, y, z ) ) a two (or three dimensional) vector given by F ( x, y )
(or F ( x, y, z ) )
...
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 function P, Q, R (if it is
present) are sometimes called scalar functions
...
Example 1 Sketch each of the following direction fields
...
This means
plugging in some points into the function
...
1
1
⎛1 1⎞
F⎜ , ⎟=− i + j
2
2
⎝2 2⎠
1
1
1
⎛1 1⎞
⎛ 1⎞
F ⎜ ,− ⎟ = −⎜− ⎟i + j = i + j
2
2
2
⎝2 2⎠
⎝ 2⎠
1
3
⎛3 1⎞
F⎜ , ⎟=− i + j
4
2
⎝2 4⎠
( 12 , 12 ) we
Likewise, the third evaluation tells us that at the point ( 32 , 14 ) we
So, just what do these evaluations tell us? Well the first one tells us that at the point
will plot the vector − 12 i + 12 j
...
We can continue in this fashion plotting vectors for several points and we’ll get the following
sketch of the vector field
...
Here is a sketch with many more vectors
included that was generated with Mathematica
...
Despite that let’s go ahead and do a couple of evaluations
anyway
...
Sometimes this will happen so don’t get excited about it when it
does
...
The sketch on the left is from the “front”
and the sketch on the right is from “above”
...
In the second chapter we looked at the gradient vector
...
In these cases the function f ( x, y, z ) is often called a scalar function to differentiate it from the
vector field
...
(a) f ( x, y ) = x 2 sin ( 5y )
(b) f ( x, y, z ) = ze − xy
Solution
(a) f ( x, y ) = x 2 sin ( 5y )
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
...
Here is the gradient vector field for this function
...
∇f = − yze − xy , −xze − xy , e − xy
Let’s do another example that will illustrate the relationship between the gradient vector field of a
function and its contours
...
Solution
Recall that the contours for a function are nothing more than curves defined by,
f ( x, y ) = k
for various values of k
...
Here is the gradient vector field for this function
...
Notice that the vectors of the vector field are all perpendicular (or orthogonal) to the contours
...
The k’s we used for the graph above were 1
...
5, 6, 7
...
5, 12, and 13
...
Now notice
that as we increased k by 1
...
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
...
Therefore, it should make sense that the two ideas should match up as they do here
...
A vector field F is called a
conservative vector field if there exists a function f such that F = ∇f
...
All this definition is saying is that a vector field is conservative if it is also a gradient vector field
for some function
...
On the other hand, F = − y i + x j is not a conservative vector field since there is no function f
such that F = ∇f
...
In that section we will also show how to find the potential
function for a conservative vector field