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Understanding Affine Space: Points and Transformations in Physics

Explore affine space, comparing points and vectors, learning about position vectors, displacement, affine combinations, and transformations. Discover cylindrical and spherical coordinates, coordinate transformations, and differential elements in physics. Dive into the differences between coordinates and components, operations on points and vectors, and affine vs. linear transformations.

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Understanding Affine Space: Points and Transformations in Physics

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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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