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.