§1.4 affine space; curvilinear coordinates christopher crawford phy 311 2014-01-24
DESCRIPTION
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 3 POINTSVECTORSTRANSCRIPT
§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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