§1.1.4 Affine space (points)

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Presentation transcript:

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

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

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

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)

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

Cylindrical & Spherical coordinates Differential elements

Example Position vector as a field in different coordinates