1. §1.1.4 Affine space (points)
Christopher Crawford
PHY 416G

2. 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

3. Affine Space – points Points vs. vectors: coordinates vs. components
Position vector
Operations on Points / Vectors
Point + displacement
Head - tail

4. Weighted points Weighted point as a vector Affine transformation
Center of mass
Linear errors
Affine transformation

5. 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)

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

7. Cylindrical & Spherical coordinates
Differential elements

8. Example
Position vector as a field in different coordinates