Divergence and curl example

For F : R3 R3, the formulas for the divergence and curl are

div F = ∂F1-
∂x + ∂F2-
∂y + ∂F3-
 ∂z
curl F = (                                    )
  ∂F3    ∂F2  ∂F1    ∂F3  ∂F2    ∂F1
  -----  ----,-----  ----,-----  ----
  ∂y     ∂z   ∂z     ∂x   ∂x     ∂y.
(The formula for curl was somewhat motivated in an earlier reading.)

Given these formulas, there isn’t a whole lot to computing the divergence and curl. Just “plug and chug,” as they say.

Example

Calculate the divergence and curl of F = (-y,xy,z).

Solution: Since

∂F1
----
∂x = 0, ∂F2
----
 ∂y = x, ∂F3
----
 ∂z = 1
we calculate that
div(F) = 0 + x + 1 = x + 1.
Since
∂F1
----
 ∂y = -1,∂F2
----
∂x = y,
∂F1-
 ∂z = ∂F2-
∂z = ∂F3-
∂x = ∂F3-
∂y = 0,
we calculate that
curl(F) = (0 - 0, 0 - 0,y + 1) = (0, 0,y + 1).

Good things we can do this with math. If you can figure out the divergence or curl from the picture of the vector field (below), you doing better than I can.