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Volume Element in Spherical Coordinates

Volume Element in Spherical Coordinates
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2020-02-21 11:06
TrigonometryCoordinate GeometrySolid GeometryCalculusParametric Equation
This example demonstrates how to represent and draw the volume element dV in spherical coordinates. The core mathematical concepts involved include spherical coordinates (ρ, θ, φ), volume integration in calculus, and trigonometric relationships. The construction uses GeoGebra commands/objects like Segment, Angle, Surface, and CircleArc to build the differential volume element in spherical coordinates. It is suitable for university students in mathematics, physics, and related fields to understand the application of multiple integrals in spherical coordinates.

GeoGebra Commands & Parameters

  1. Segment((0, 0, 0), (((ρ * sin(φ)) * cos(θ)), ((ρ * sin(φ)) * sin(θ)), (ρ * cos(φ)))) → g
  2. Angle((0, 0, ρ), (0, 0, 0), (((ρ * sin(φ)) * cos(θ)), ((ρ * sin(φ)) * sin(θ)), (ρ * cos(φ)))) → α
  3. Segment((0, 0, 0), (((ρ * sin(φ)) * cos(θ)), ((ρ * sin(φ)) * sin(θ)), 0)) → h
  4. Angle((((ρ * sin(φ)) * cos(θ)), 0, 0), (0, 0, 0), (((ρ * sin(φ)) * cos(θ)), ((ρ * sin(φ)) * sin(θ)), 0)) → β
  5. Surface(((rl * sin(f)) * cos(t)), ((rl * sin(f)) * sin(t)), (rl * cos(f)), f, fl, fh, t, tl, th) → a
  6. Surface(((rh * sin(f)) * cos(t)), ((rh * sin(f)) * sin(t)), (rh * cos(f)), f, fl, fh, t, tl, th) → b
  7. CircleArc((0, 0, 0), (((ρ * sin(fl)) * cos(θ)), ((ρ * sin(fl)) * sin(θ)), (ρ * cos(fl))), (((ρ * sin(fh)) * cos(θ)), ((ρ * sin(fh)) * sin(θ)), (ρ * cos(fh)))) → ρdφ
  8. Segment((((rl * sin(φ)) * cos(θ)), ((rl * sin(φ)) * sin(θ)), (rl * cos(φ))), (((rh * sin(φ)) * cos(θ)), ((rh * sin(φ)) * sin(θ)), (rh * cos(φ)))) → dρ
  9. CircleArc((0, 0, 0), (((ρ * sin(φ)) * cos(tl)), ((ρ * sin(φ)) * sin(tl)), (ρ * cos(φ))), (((ρ * sin(φ)) * cos(th)), ((ρ * sin(φ)) * sin(th)), (ρ * cos(φ)))) → ρsinφdθ
  10. Segment((xc, yc, 0), (xc, yc, zc)) → i
  11. Segment((((rl * sin(sweepφ)) * cos(tl)), ((rl * sin(sweepφ)) * sin(tl)), (rl * cos(sweepφ))), (((rh * sin(sweepφ)) * cos(tl)), ((rh * sin(sweepφ)) * sin(tl)), (rh * cos(sweepφ)))) → j
  12. Segment((((rl * sin(fh)) * cos(tl)), ((rl * sin(fh)) * sin(tl)), (rl * cos(fh))), (((rh * sin(fh)) * cos(tl)), ((rh * sin(fh)) * sin(tl)), (rh * cos(fh)))) → m
  13. Segment((((rl * sin(sweepφ)) * cos(th)), ((rl * sin(sweepφ)) * sin(th)), (rl * cos(sweepφ))), (((rh * sin(sweepφ)) * cos(th)), ((rh * sin(sweepφ)) * sin(th)), (rh * cos(sweepφ)))) → k
  14. Segment((((rl * sin(fl)) * cos(sweepθ)), ((rl * sin(fl)) * sin(sweepθ)), (rl * cos(fl))), (((rh * sin(fl)) * cos(sweepθ)), ((rh * sin(fl)) * sin(sweepθ)), (rh * cos(fl)))) → n
  15. Segment((((rl * sin(fh)) * cos(sweepθ)), ((rl * sin(fh)) * sin(sweepθ)), (rl * cos(fh))), (((rh * sin(fh)) * cos(sweepθ)), ((rh * sin(fh)) * sin(sweepθ)), (rh * cos(fh)))) → l