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.

def of div_03_sphere.png

The divergence of w at point P is defined as

Def of div_01.png

And it turns out that this is equal to

Def of div_02.png

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 whose center is at P.

def of curl_01_circle.png

The n component of the curl of v at point P is defined as

def of curl_02.png

It turns out that this is equal to

def of curl_03.png

Specifically,

def of curl_04.png
def of curl_05.png
def of curl_06.png

The curl of a vector is a measure of how much the vector curls around the point in question.