Using math in science

In science,  we use math to model the physical world and not just as a set of numbers and symbols to be manipulated. This makes the math carry meaning in a way it typically doesn't in a math course. It makes our equations a lot "richer" — more complicated to interpret — than they are in math. Making our equations physical (as opposed to purely mathematical) gives us some very powerful tools, such as dimensional analysis, scaling, and estimation, that allow us to use math more effectively.

To get an overview of the chapter, and to understand the motivation for our approach, look at the page:

• Overview: Modeling with mathematics

Here's the full Table of Contents of the chapter. The notation "(EX)" means that there is a worked example attached to that page. The notation "{WO}" means there is an activity (a Workout) attached to that page, usually one that involves working with a simulation.

• How math in science is different from math in math
• The structure of physical modeling
• Measurement
• Dimensionalities
  • Complex dimensions and dimensional analysis (EX)
  • Changing units
  • Natural scales
    • Dimensionless equations (advanced)
  • Considering changing our arbitrary choices
  • The parameters of matter
• Estimation (WO, EX)
  • Some useful numbers
  • Scales in biology
• Approximations
• Mathematics Recap
  • Significant (and insignificant) figures
  • Scientific notation
  • The idea of algebra: Unknowns and relationships
    • Symbols in science
    • Variables, constants, and parameters
      • More complex constants and parameters
  • Functions
    
    • Units, Scaling, and functional dependence
    • Powers and exponentials
      • Log-log plots
    • The concept of field
  • Values, change, and rates of change
    • Derivatives 
      • What is a derivative, anyway? (WO, EX)
      • The chain and product rules
        • Proving the chain and product rules (technical)
      • Differential Equations (EX)
    • Integrals: The mathematical meaning
      
      • How can I actually use integrals?
  • Trigonometry
    • The small angle approximation
    • Trig functions for any value of the argument
    • The derivatives of trig functions
  • Probability
    
    • Why probability matters
    • Expected values
    • Permutations and combinations
• Building your mathematical toolbelt
  • The dimensional analysis tool
  • The scaling tool
  • The special case tool
  • Toy models
  • Reading the physics in a graph (EX)
  • Anchor equations -- Reading the physics in an equation
  • The repackaging tool: Changing physics equations to math (and back)
  • Building equations from the physics (EX)
  • Telling the story
  • The estimation tool
  • The diagram tool
  • Putting your tools together

Joe Redish 5/15/19