Definition of divergence
Rather than introducing the divergence of a vector field w as an inner product between w and the "vector" del, here we define the divergence in a way that physically makes more sense.
The divergence of w at point P is defined as
And it turns out that this is equal to
The divergence of a vector is a measure of how much the vector spreads out from the point in question.
Definition of curl
To define the curl of a vector field v at point P, we consider a circle with area a whose center is at P.
The n component of the curl of v at point P is defined as
It turns out that this is equal to
The curl of a vector is a measure of how much the vector curls around the point in question.