1 4 Affine space Curvilinear coordinates Christopher Crawford

§ 1. 4 Affine space; Curvilinear coordinates Christopher Crawford PHY 311 2014 -01 -24

Outline • Affine space – linear space of points Position vectors, displacement, differential Affine combinations, transformations Points vs. vectors – comparison and contrast • • Cylindrical and spherical coordinates Coordinate & component transformations Coordinate lines and surfaces Differential line (dl), area (da), volume (dτ) elements • Generalized curvilinear coordinates Contravariant and covariant basis and components Differentials & vector derivatives 2

Affine Space – points • Position vector • Operations POINTS VECTORS – Affine combination • Basis – N+1 vs. N • Decomposition – Coordinates vs. components • Transformations – Affine vs. linear • Fields / Differental / Integral – Parameterization vs. field 3

Cylindrical & Spherical coordinates • Coordinate transformation – Physics vs. math convention; singularities – Can you mix coordinate systems? • Component transformation 4

Cylindrical & Spherical coordinates • Differential elements 5

Example • Position vector as a field in different coordinates 6

General curvilinear coordinates 7

General Differential Elements • line element • area element • volume element 8

Example – circular coordinates 9

Unification of vector derivatives • Three rules: a) d 2=0, b) dx 2 =0, c) dx dy = - dy dx • Differential (line, area, volume) elements as transformations 10

… in generalized coordinates • Same differential d as before; hi comes from unit vectors 1 1