PX153 - A5 - coordinate systems
- recall [basis vectors](PX153 - A1 - notation and geometrical representation)
- in 3D, a coordinate system can be defined by specifying 3 linearly independent basis vectors at every point in space
- usually defined to be orthonormal

- to find , several ways
- eg: tangent to the curve marking the change in coordinate
cartesian coordinates
cartesian coordinates
circular polar coordinates
- 2D
- here, is the magnitude of the vector and is the angle made by the vector with the x-axis in counter-clockwise direction such that or
- transforming between polar and cartesian coordinates:
- the basis vectors for polar coordinates vary in space, but are always orthonormal

- to check: these are orthonormal
cylindrical polar coordinates
- 3D
- polar coordinates in x-y plane, extend by keeping the z-axis
- defined by

- transformations to/from cartesian:
spherical polar coordinates
- 3D
- for problems with spherical symmetry
-

- transformations to/from cartesian:
again, , , vary in space