 
Summary: §1.1: NEWTON'S LAW OF COOLING
Imagine some object like a cannon ball suspended in a swimming
pool by a rope. The cannon ball is at some temperature; perhaps
hotter thant the water, perhaps cooler. We can think of it as being a
`point source' of heat flow in or out. We are interested in the tem
perature T (in degrees centigrade) as a function of time t (in min
utes). It will be simplest if we measure the temperature relative to
the surrounding water, so T = 0 represents the water temperature.
We would like to understand the function T(t). Another function we
can consider is the derivative T (t) (in degrees per minute.)
One way of approaching this problem is to throw away one of the
variables, time t. We can imagine the STATE SPACE which is the plane
with coordinates (T, T ). A point in the state space is a pair of num
bers representing a temperature, and a rate of change of tempera
ture. This represents a possible state the cannon ball could be in. It is
hard to think about the rate of change without thinking about time.
T
T'
T
T'
