Consider \(\vec{F}(x,y,z)=\langle P(x,y,z),Q(x,y,z),R(x,y,z) \rangle.\) Note this has 3 inputs and three outputs. For convenience \(\frac{\partial\vec{F}_x}{\partial x} = \frac{\partial P(x,y,z)}{\partial x}.\)
Activity74.
The purpose of this activity is to illustrate the concept of divergence in a vector field.
(a)
If people arrive at the gas station at a rate of 6 every 15 minutes and leave at a rate of 3 every 15 minutes what will be the result?
(b)
If people are arriving at lover’s leap at a rate of 1 person per hour and leaving at a rate of 1 per minute, what will be the result?
(c)
In terms of rates what information do \(\frac{\partial \vec{F}_x}{\partial x},\)\(\frac{\partial \vec{F}_y}{\partial y}\text{,}\) and \(\frac{\partial \vec{F}_z}{\partial z}\) provide?
(d)
As a result how can \(\frac{\partial \vec{F}_x}{\partial x}+\frac{\partial \vec{F}_y}{\partial y}+\frac{\partial \vec{F}_z}{\partial z}\) be described?
Subsection5.4.2Definition
Definition5.4.1.Div.
For a vector field \(F\text{,}\) the divergence is
