线积分:功

线积分:功
几何画板及绘图内容加载中,请稍等……
2021-10-14 09:53
所属专题 MATH1052-edX
①本范例演示了线积分在计算变力沿曲线做功时的应用;②涉及的核心数学概念包括向量场、曲线积分、微积分中的定积分;③作图要点:使用了SelectedIndex, Midpoint, Element, Derivative, ParametricDerivative, Distance, CurveCartesian, Integral, Textfield, Segment, If, Sequence, Translate等GeoGebra命令/对象来构建向量场、参数化曲线及计算积分;④适合大学理工科学生学习和理解向量微积分应用。

GeoGebra 命令及参数

  1. SelectedIndex(pathNames) → pathIndex
  2. Midpoint(N, P) → Q
  3. Element(pathList, pathIndex) → path
  4. Derivative(path) → DDpath
  5. ParametricDerivative(path) → Dpath
  6. Distance(N, P) → NP
  7. Element(CompX, pathIndex) → RTX
  8. Derivative(RTX, x) → DRTX
  9. Element(CompY, pathIndex) → RTY
  10. Derivative(RTY, x) → DRTY
  11. CurveCartesian(x(path(t)), y(path(t)), t, 0, s) → partPath
  12. Integral(MUL, 0, 1) → Work
  13. Textfield(F_1) → InputBox1
  14. Textfield(F_2) → InputBox2
  15. Segment(path(0.3), path(0.3 + 0.01)) → arrow1
  16. Segment(path(0.6), path(0.6 + 0.01)) → arrow2
  17. If(Work ≟ Work, Work, Sum[Sequence[((path(i + dS) - path(i)) * (F_1(x(path(i + dS / 2)), y(path(i + dS / 2))), F_2(x(path(i + dS / 2)), y(path(i + dS / 2))))), i, 0, s - dS, dS]]) → WOORK
  18. Sequence(Vector[path(i), path(i) + ((scale * F_1(x(path(i)), y(path(i)))), (scale * F_2(x(path(i)), y(path(i)))))], i, 0, s, 0.15) → PathVectors, {(0.10000000000000009, -0.10000000000000009), (0.06999999999999995, -0.06999999999999995), (0.03999999999999998, -0.03999999999999998), (0.010000000000000009, -0.010000000000000009), (-0.01999999999999999, 0.01999999999999999), (-0.050000000000000044, 0.050000000000000044), (-0.08000000000000007, 0.08000000000000007)}
  19. Sequence(path(i), i, 0.25, 0.75, 0.25) → PtPathList, {(0.5, -0.5), (0, 0), (-0.5, 0.5)}
  20. Sequence(Vector[(0, 0), (1 / sqrt((y(DDpath(i)))^(2) + (x(DDpath(i)))^(2)) * (x(DDpath(i)), y(DDpath(i))))], i, 0.25, 0.75, 0.25) → tangvectorsList, {(-0.7071067811865475, 0.7071067811865475), (-0.7071067811865475, 0.7071067811865475), (-0.7071067811865475, 0.7071067811865475)}
  21. Translate(Vector[Element[tangvectorsList, 1]], Element[PtPathList, 1]) → uu
  22. Translate(Vector[Element[tangvectorsList, 2]], Element[PtPathList, 2]) → vv
  23. Translate(Vector[Element[tangvectorsList, 3]], Element[PtPathList, 3]) → ww
  24. Textfield(xA) → InputBox3
  25. Textfield(yA) → InputBox4
  26. Textfield(xB) → InputBox5
  27. Textfield(yB) → InputBox6

AI 绘图提示语参考

绘制一个二维向量场,其中包含一条由参数方程定义的曲线路径,路径上分布着表示力场方向的箭头。在路径上选取三个等距点,分别画出该点处的单位切向量,并沿路径绘制若干小段位移向量,用于近似计算力沿路径所做的功。同时显示两个输入框,用于输入力场的两个分量函数,以及四个输入框用于设置路径的起点和终点坐标。最终在图形中标注出计算得到的功的数值。
以上提示语为根据原始绘图命令整理而来,仅供参考。