§1.4 Curvilinear coordinates Christopher Crawford PHY

Outline 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

Cylindrical & Spherical coordinates Coordinate transformation – Physics vs. math convention; singularities – Can you mix coordinate systems? Component transformation 3

Cylindrical & Spherical coordinates Differential elements 4

Example Position vector as a field in different coordinates 5

General curvilinear coordinates 6

General Differential Elements line element area element volume element 7

Example – circular coordinates 8

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 9

… in generalized coordinates Same differential d as before; h i comes from unit vectors 10