Multivariate Calculus - Bruce E. Shapiro

Lecture 21
Vector Fields
Definition 21.1 A vector field on R 3 is a function F(x, y, z) : D ⊂ R 3 ↦→ R 3
that assigns a vector v = (v x , v y , v z ) to every point (x, y, z) in D.
Of course, it is also possible to define a vector field on R 2 , as a function that assigns
a vector v = (v x , v y ) to each point (x, y) in some domain D ⊂ R 2 . We will limit
ourselves to a discussion 2D fields at first because they are easier to visualize on
paper. One way to visualize a vector field is the following:
1. Pick an arbitrary set of points in the domain at which you want to know the
vector field.
2. For each point (x, y) in your set, calculate the vector v = F(x, y) at (x, y).
3. For each vector you have calculated, draw an arrow starting at (x, y) in the
direction of v and whose length is proportional to ‖v‖.
Example 21.1 Plot the vector field
F(x, y) =
(
− y 2 , x )
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on the domain [−2, 2] × [−2, 2].
Solution. First, we construct a table of values for the function. We will pick the
points at integer coordinates, e.g., (−2, −2), (−2, −1), ..., (2, 2).
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