Lecture 2; Math 636
Natural vs. ad hoc models revisited
Metatheorem: The more you match the structure of the problem, the better the model.
System – varies with application but generally coincides with the object being modeled*
Manifold = variables + equations between them (Differential Geometry)
Static vs dynamic models revisited
Group = generators + equations between them (Abstract Algebra)
Arguments from scale:
Groups acting on sets -- symmetries
Scaling: Group of similitudes (homotheties, multiples of the identity)
Alias vs. alibi in group actions – the two halves of chapter 2
Example: the rotation group on R2
Given an orthogonal matrix, is it a linear map or a change of basis?
Units and dimensional analysis
*Image from http://energy.sdsu.edu/testcenter/testhome/Test/problems/images/ch01/piston-coil.jpg