Basics of Dimensional Analysis

The goal of this module is to provide a philosophical perspective on why we measure things and how we think dimensionally about those measurements. Dimensional analysis is a common vocabulary that allows us to agree on what a given measurement means, based on the philosophy that measurements are epistemological representations of an objective reality.

The following video from Sixty Symbols (5:51 min) gives an excellent summary of the elegance and power of dimensional thinking, a skill that I think should be central to training in quantitative environmental science. Sixty Symbols is generally an excellent math and science education YouTube channel.

Understanding why scientists measure things first requires understanding why scientists do science. Let's establish some basic understanding of the philosophy of science (from an inexpert perspective, 4:22 min).

If data represent an objective perspective on reality, then how do we use them to gain understanding or awareness of causality?

Extracting objective understanding of cause and effect from patterns in measurements is the domain of the scientific method. Let's review the scientific method and the philosophy of logic associated with its practice (12:01 min).

If scientists are to agree about the meaning of patterns among measurements, we must share a common perspective on the meaning of any one measurement. Dimensional analysis provides a structured language for mapping the meaning of a given measured variable to some component of our objective reality. I like to call dimensional analysis an epistemological mapping that allows us to use physics to think about cause and effect in our shared reality. The combination of structured vocabulary and mathematical abstraction make dimensional analysis both a powerful generalized communication tool, as well as a powerful tool for checking the integrity of mathematical calculations and derivations.

Let's start with an introduction of dimensions in space, where we already have some intuition for the three dimensions in which we exist. We can start with a single dimension and compare and contrast the abstraction of dimensions with the specific numerical context of units (4:16 min).

Derived dimensions arise from thinking about multidimensional references frames. Spatial dimensions like areas and volumes provide simple examples of derived dimensions and an example of how to carry dimensions through algebraic calculations (7:51 min).

Dimensions are used to define many more basic components of reality than properties of space. Further fundamental dimensions help to describe additional components of space, time, mass, and energy (10:27 min).

Fundamental dimensions can be combined into derived dimensions that describe increasingly complex phenomena important to the study of environmental systems. A dimensional analysis of basic Newtonian physics provides an example of how to derive the dimensions of energy from the fundamental dimensions of length, time, and mass (6:49min).

Dimensions are useful for describing the fundamental meaning of variables, but they are abstract and further context is often necessary. The physical properties of density and concentration are examples of why knowing mass of what or volume of what are important to the interpretation of dimensions (3:31 min).

Finally, we can explore the dimensional relationships between heat and temperature, as well as a few more potentially tricky examples (7:24 min).

With these basics, you already have an elegant and powerful epistemological tool for analyzing the context of variables and derivations performed with those variables.

Let's start working through some examples of mathematical derivations with dimensions. The conventions used for the formatting of dimensions are primarily designed to clearly differentiate them from other symbols used for units or mathematical variables. Surrounding dimensions with square brackets and use of nonitalicized capital letters are the typical practices for providing this differentiation. Using positive and negative exponents rather than mathematical fraction notation for simplified dimensions helps avoid potential ambiguities in the order of operation (3:12 min).

Let's review some examples of how to carry out dimensional analysis on mathematics to evaluate whether dimensional consistency suggests the equation is derived from independent theory (mechanistic) or lack of dimensional consistency suggests the equation was derived using arbitrary mathematics representing patterns in data (empirical). In the process, let's cover some nuts and bolts of dimensional notation and creating professional mathematical notation in MS Word. We'll start with dimensional analysis of the Darcy Weisbach equation for hydraulics (18:32 min).

We can then contrast the dimensional analysis of the Darcy Weisbach equation with dimensional analysis of another equation used for hydraulics, Manning's equation (6:48 min).

In scientific training, many have been introduced to a form of dimensional analysis though unit conversion (based on the unit factor method) or carrying units through equations to check the integrity of mathematics. Let's take a moment to clarify why the unit factor method of unit conversion works based on its foundations in dimensional analysis (3:03 min).

This introduction to how measurements fit in the practice of science and how we define the meaning of measurements with dimensional analysis should give you the basic ability to examine the meaning of mathematics with a more systematic and careful eye. Using this basic technique, I cannot tell you how many times I have quickly spotted mistakes in my own work, my students' work, or even in work submitted to peer-reviewed journals. Furthermore, I would suggest that being "good" at applied mathematics is less about remembering all the rules of analytical calculus, and more about simply being  highly practiced at algebra and highly practiced at dimensional analysis. Beefing up those two basic skills will take you a long way.

Slides from videos

These are the slides used in the videos

Click this link to download the MS PowerPoint file

The embedded Google viewer below sometimes provides poor renderings of Microsoft files. Use the link above to download the original file with proper formatting.

Slides from class

In preparing for class, I sometimes make minor adjustments or improvements to the visual aids. These do not always justify remaking the videos, but I do provide the adjusted version of the sides here.

Click this link to download the MS PowerPoint file

The embedded Google viewer below sometimes provides poor renderings of Microsoft files. Use the link above to download the original file with proper formatting.

Dimensional analysis examples

These are the examples worked in the videos using the Microsoft equation editor.

Click this link to download the MS Word document

The embedded Google viewer below sometimes provides poor renderings of Microsoft files. Use the link above to download the original file with proper formatting.

In particular, the Google viewer may show the dimensions in an italicized font, which is not proper form.

A paper discussing common distortions created by imprecise understanding of the scientific method

Hutto (2012) in BioScience

A table of examples of dimensions commonly used in hydrology

Click this link to download the postscript PDF document

More videos with MS Word Equation Editor keyboard shortcuts

Here is a video I made for another class that works through other features that might be useful.

A Google search will find many videos like the following.

A quick reference for keyboard shortcuts in Microsoft equation editor

Click this link to download the postscript PDF document

Brochure (USA version) describing the International System of Units

Link to the NIST online version of this document

Table 3 on page 13 (PDF page 29) summarizes fundamental dimensions and was described in the video.

A paper on the topic of the philosophy of science applied to hydrology

Baker (2017) in Water Resources Research