§1.4 Affine space;Curvilinear coordinates

Christopher CrawfordPHY 311

2014-01-24

Outline• Affine space – linear space of points

Position vectors, displacement, differentialAffine combinations, transformationsPoints vs. vectors – comparison and contrast

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• Cylindrical and spherical coordinatesCoordinate & component transformationsCoordinate lines and surfacesDifferential line (dl), area (da), volume (dτ) elements

• Generalized curvilinear coordinatesContravariant and covariant basis and componentsDifferentials & vector derivatives

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Affine Space – points• Position vector

• Operations– Affine combination

• Basis– N+1 vs. N

• Decomposition– Coordinates vs. components

• Transformations– Affine vs. linear

• Fields / Differental / Integral– Parameterization vs. field

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POINTS VECTORS

Cylindrical & Spherical coordinates• Coordinate transformation

– Physics vs. math convention; singularities– Can you mix coordinate systems?

• Component transformation

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Cylindrical & Spherical coordinates• Differential elements

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Example

• Position vector as a field in different coordinates

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General curvilinear coordinates

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General Differential Elements• line element

• area element

• volume element

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Example – circular coordinates

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Unification of vector derivatives• Three rules: a) d2=0, b) dx2 =0, c) dx dy = - dy dx• Differential (line, area, volume) elements as transformations

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… in generalized coordinates• Same differential d as before; hi comes from unit vectors

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