Higher Concepts of Mathematics
The ideas of differentials and derivatives are fundamental to the application of mathematics to
dynamic systems. They are used not only to express relationships among distance traveled,
elapsed time and velocity, but also to express relationships among many different physical
quantities. One of the most important parts of understanding these ideas is having a physical
interpretation of their meaning. For example, when a relationship is written using a differential
or a derivative, the physical meaning in terms of incremental changes or rates of change should
be readily understood.
When expressions are written using deltas, they can be understood in terms of changes. Thus,
the expression DT, where T is the symbol for temperature, represents a change in temperature.
As previously discussed, a lower case delta, d, is used to represent very small changes. Thus,
dT represents a very small change in temperature. The fractional change in a physical quantity
is the change divided by the value of the quantity. Thus, dT is an incremental change in
temperature, and dT/T is a fractional change in temperature. When expressions are written as
derivatives, they can be understood in terms of rates of change. Thus, dT/dt is the rate of
change of temperature with respect to time.
Interpret the expression DV /V , and write it in terms of a
differential. DV /V expresses the fractional change of velocity.
It is the change in velocity divided by the velocity. It can be
written as a differential when DV is taken as an incremental
may be written as
Give the physical interpretation of the following equation relating
the work W done when a force F moves a body through a
dW = Fdx
This equation includes the differentials dW and dx which can be
interpreted in terms of incremental changes. The incremental
amount of work done equals the force applied multiplied by the
incremental distance moved.