积极 AI Logo 积极 AI

球坐标系中的体积微元

球坐标系中的体积微元
几何画板及绘图内容加载中，请稍等……
2020-02-21 11:06
三角函数解析几何立体几何微积分参数方程
①本范例演示了在球坐标系下如何表示和绘制体积微元dV；②涉及的核心数学概念包括球坐标系（ρ, θ, φ）、微积分中的体积积分、三角函数关系；③作图要点——从下方『构造摘要』提炼用到了Segment（线段）、Angle（角度）、Surface（曲面）、CircleArc（圆弧）等GeoGebra命令/对象来构建球坐标系中的微元体积；④适合大学数学、物理等相关专业的学生学习和理解多重积分在球坐标系下的应用。

GeoGebra 命令及参数

  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