Differentials and integrals: Differentials
What is a differential?
The concept of differential can be introduced in different ways: in a somewhat messy but intuitive style or mathematically rigorous, and in the context of mathematical functions (as change of linearisations of functions) or in mathematical disciplines such as calculus, differential geometry or non-standard analysis. In this module, e have chosen for the first intuitive approach.
Example of a quadratic function First a simple example in a physics context: the quadratic relation between the distance travelled in a time by an object that is at time is at rest and thereafter undergoes a uniformly accelerated rectilinear motion with an acceleration , which is constant in this example. The distance between time and can be calculate in an exact way:
If the time duration is short, then is much smaller than and the last term is negligibly small in the above equation. so:
The quantity is not only the distance traveled, but represent at the same time the position of the object at a certain time. Thus, the last equation can also be considered as an approximation for the change of position on a time interval and this approximation is better when the change of time is smaller. Note, we are discussing here a relationship between changes in two variables (position and time). In order to indicate that the changes are negligibly small (infinitesimal'), one uses the symbols and , and then the approximation becomes an equality