Differentials

13 Differentials

So far, we have learned that

velocity is the derivative of position.
displacement is the integral of velocity.

In equation form, these two statements are:

$\vec v = \frac{d\vec r}{dt}; \hspace{20pt} \vec r(t) - \vec r(t_i) = \int_{t_i}^t \vec v(t')dt'$ .

Of course, in reality, these two equations are really the same equation: when you integrate the equation on the left, you get the equation on the right. If you take the derivative of the equation on the right, we get the equation on the left. We can make this relation especially clear using differentials.

For our purposes, the differential $dt$ refers to “an infinitesimally small amount of time.” The differential $d\vec r$ refers to the corresponding infinitesimally small displacement (i.e. change in position) that takes place during the time interval $dt$ . Velocity is the quantity that relates these two differentials, i.e.

$d\vec r = \vec v dt$ .

I find that this equation is by far the simplest way of thinking about the relation between position and velocity. To get back that “velocity is the derivative of position”, we just divide by $dt$ . To get “displacement is the integral of velocity”, we just integrate, i.e.

$\vec r(t) - \vec r(t_i) = \int_{t_i}^t d\vec r = \int_{t_i}^t \vec v(t')dt'$ .

Moreover, the equation $d\vec r =\vec v dt$ reads exactly like (distance) = (velocity) $\times$ (time).

For these reasons, I view the differential form of this equation as the most useful way of thinking about velocity. In fact, differentials are a fantastic tool for thinking about physical problems, and I will use them liberally throughout this book.

Key Takeaways

13 Differentials

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