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§1.4 Curvilinear coordinates

§1.4 Curvilinear coordinates. Christopher Crawford PHY 416 2014-09-10. Outline. Cylindrical and spherical coordinates Coordinate & component transformations Coordinate lines and surfaces Differential line ( dl ), area ( da ), volume (d τ) elements

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§1.4 Curvilinear coordinates

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  1. §1.4Curvilinear coordinates Christopher Crawford PHY 416 2014-09-10

  2. Outline • 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

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

  4. Cylindrical & Spherical coordinates • Differential elements

  5. Example • Position vector as a field in different coordinates

  6. General curvilinear coordinates

  7. General Differential Elements • line element • area element • volume element

  8. Example – circular coordinates

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

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

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