§1.4 Affine space; Curvilinear coordinates Christopher Crawford PHY

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 (d τ) elements Generalized curvilinear coordinates Contravariant and covariant basis and components Differentials & vector derivatives 2

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

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; h i comes from unit vectors 11