Coordinate transformations and the metric of space

Classical Mechanics and Relativity: Lecture 6
0:00 Introduction
1:09 Cartesian coordinates
9:14 Translations and Rotations
28:38 Transforming basis vectors: Jacobian
36:16 Example: 2d rotations
41:06 Time-dependent transformations
47:25 Example: cylindrical coordinates
52:37 Metric tensor
1:05:41 Inner Product
1:09:45 Example: metric for polar coordinates
1:13:54 Volume element

Theoretical physicist Dr Andrew Mitchell presents an undergraduate lecture course on Classical Mechanics and Relativity at University College Dublin. This is a complete and self-contained course in which everything is derived from scratch.

In this lecture we discuss different coordinate systems (Cartesian and curvilinear, inertial and accelerated) and the transformations between them. The Jacobian is introduced, the inner product is discussed, and the metric of space for different coordinate systems is explained from first principles. This sets the ground work for generalized coordinates in Lagrangian mechanics, and the Minkowski metric in special relativity.

Full lecture course playlist:    • Classical Mechanics and Relativity lecture...  

Course textbooks:
"Classical Mechanics" by Goldstein, Safko, and Poole
"Classical Mechanics" by Morin
"Relativity" by Rindler