Chapter 1, Problem 1.6CYU

Check Your Understanding Is the equation $v=at$dimensionally consistent?

One further point thin needs to be mentioned is the effect of the operations of calculus on dimensions. We have seen that dimensions obey the rules of algebra, just like units, but what happens when we take the derivative of one physical quantity with respect to another or integrate a physical quantity over another? The derivative of a function is just the slope of the line tangent to its graph and slopes are ratios, so for physical quantities $v$and $t$ , we hive that the dimension of the derivative of $v$with respect to us just the ratio of the dimension of v over that of $t$ : $\left[\frac{dv}{dt}\right]=\left[\frac{v}{t}\right]$ .

Similarly, since integrals are just sums of products, the dimension of the integral of $v$with respect to $t$ is simply the dimension of $v$times the dimension of $t$ : $\left[{\displaystyle \int vdt}\right]=\left[v\right]\cdot \left[t\right]$ .

By the same reasoning, analogous rules hold for the units of physical quantities derived from other quantities by integration or differentiation.