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§1.1.4 Affine space (points)

§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

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§1.1.4 Affine space (points)

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  1. §1.1.4Affine space (points) Christopher Crawford PHY 416G 2014-09-05

  2. Outline • Affine space – linear space of pointsPoints vs. vectors – comparison and contrastPosition vectors, displacement, differentialAffine combinations, transformations • Cylindrical and spherical coordinatesCoordinate & component transformationsCoordinate lines and surfacesDifferential 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 • 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

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