Dimensions and Dimensional Analysis

As introduced in section 1.2.2, a derived physical quantity can always be expressed as some combination of the seven fundamental (basic) quantities. For convenience in writing such combinations compactly, each basic quantity is represented by a single symbol: $L$ for length, $M$ for mass, $T$ for time, $K$ for temperature, $I$ for electric current, $C$ for luminous intensity, and $\text{mol}$ for amount of substance.

The dimensions of a physical quantity are the powers to which each of these fundamental-quantity symbols must be raised in order to reproduce the unit of that quantity. When a derived quantity is written as a product of the fundamental-quantity symbols each raised to the appropriate power, the resulting expression is called its dimensional formula; by convention, a dimensional formula is always enclosed in square brackets, with no commas between the symbols.

Three illustrations:

1. Velocity: $\text{velocity} = \dfrac{\text{displacement}}{\text{time}}$, so its dimensions are $\dfrac{[L]}{[T]} = [L^1M^0T^{-1}]$.
2. Velocity gradient (the rate of change of velocity with distance): $\text{velocity gradient} = \dfrac{\text{velocity}}{\text{distance}}$, so its dimensions are $\dfrac{[L^1T^{-1}]}{[L]} = [L^0M^0T^{-1}]$.
3. Charge: $\text{charge} = \text{current} \times \text{time}$, so its dimensions are $[I][T] = [L^0M^0T^1I^1]$. …