The material of this chapter is usually not covered in a book on mathematics. The field of mathematics deals with numbers and numerical relationships. It does not matter what these numbers are; they may account for physical properties of a system, but they may equally well be numbers that are not related to anything physical. Consider the expression g = df/dt. From a mathematical point of view these functions can be anything, as long as g is the derivative of f, with respect to t. The situation is different in physics. When f(t) defines the position of a particle, and t denotes time, then g(t) is a velocity. This relation fixes the physical dimension of g(t). In mathematical physics, the physical dimension of variables imposes constraints on the relation between these variables. In this chapter we explore these constraints. In Section 2.2 we show that this provides a powerful technique for spotting errors in equations. In the remainder of this chapter we show how the physical dimensions of the variables that govern a problem can be used to find physical laws. Surprisingly, while most engineers learn about dimensional analysis, this topic is not covered explicitly in many science curricula.

2.1 Two rules for physical dimensions

In physics every physical parameter is associated with a physical dimension. The value of each parameter is measured with a certain physical unit. For example, when we measure how long a table is, the result of this measurement has dimension “length.” This length is measured in a certain unit, which may be meters, inches, furlongs, or whatever length unit we prefer to use. The result of this measurement can be written as

l = 3 m.

The variable l has the physical dimension of length. In this chapter we write this as

l ∼ [L].

The square brackets are used in this chapter to indicate a physical dimension.