1 1 4 Affine space points Christopher Crawford
- Slides: 8
§ 1. 1. 4 Affine space (points) Christopher Crawford PHY 416 G 2014 -09 -05
Outline • Affine space – linear space of points Points vs. vectors – comparison and contrast Position vectors, displacement, differential Affine combinations, transformations • Cylindrical and spherical coordinates Coordinate & component transformations Coordinate lines and surfaces Differential line (dl), area (da), volume (dτ) elements 2
Affine Space – points • Points vs. vectors: coordinates vs. components • Position vector • Operations on Points / Vectors – Point + displacement – Head - tail 3
Weighted points • Weighted point as a vector – Center of mass – Linear errors • Affine transformation 4
Summary: Points vs. Vectors • Operations – Affine combination • Basis – N+1 vs. N • Decomposition – Coordinates vs. components • Transformations – Affine vs. linear • Fields / Differential / Integral – Parameterization vs. field (coordinates) 5
Cylindrical & Spherical coordinates • Coordinate transformation – Physics vs. math convention; singularities – Can you mix coordinate systems? • Component transformation 6
Cylindrical & Spherical coordinates • Differential elements 7
Example • Position vector as a field in different coordinates 8
