### 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 \(s=a\cdot t^2\) between the distance \(s\) travelled in a time \(t\) by an object that is at time \(t=0\) is at rest and thereafter undergoes a uniformly accelerated rectilinear motion with an acceleration \(a\), which is constant in this example. The distance \({\vartriangle}s(t_1)\) between time \(t=t_1\) and \(t=t_1+{\vartriangle}t\) can be calculate in an exact way:

\[{\vartriangle}s(t_1)=a(t_1+{\vartriangle}t)^2 -at_1^2=2at_1{\vartriangle}t+a({\vartriangle}t)^2\tiny.\]

If the time duration \({\vartriangle}t\) is short, then\(({\vartriangle}t)^2\) is much smaller than \({\vartriangle}t\) and the last term is negligibly small in the above equation. so:

\[{\vartriangle}s(t_1)\approx 2at_1{\vartriangle}t,\quad \text{for small }{\vartriangle}t\tiny.\]

The quantity \(s\) 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 \({\vartriangle}s(t_1)\) on a time interval \([t_1,t_1+{\vartriangle}t]\) and this approximation is better when the change of time \({\vartriangle}t\) 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 \(ds\) and \(dt\), and then the approximation becomes an equality \[ds(t_1)=2a\cdot t_1\cdot dt\] Because both \(dt\) as \(ds\) are understood as changes, that is, as difference between final and initial value, they are called **differentials.** The ratio of these differentials, the **differential quotient** \(\displaystyle\frac{ds}{dt}\), is the slope of the tangent line at a point \((t,s(t))\) , that is, is equal to the derivative of \(s\) in \(t.\)