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向量场的积分：正方形

向量场的积分：正方形
几何画板及绘图内容加载中，请稍等……
2016-08-30 15:14
向量解析几何微积分积分几何作图
①本范例演示了在正方形区域内对向量场进行积分的计算过程；②涉及的核心数学概念包括向量场、积分、导数、曲面积分等；③作图要点使用了GeoGebra的Sequence, Line, Circle, CurveCartesian, Derivative, Sum, Segment, Distance, Textfield等命令和对象；④适合高中及以上学生学习和理解向量场积分的几何意义及计算方法。

GeoGebra 命令及参数

  1. Sequence(Sequence[Vector[(j, k), (j + (scale * F_1(j, k)), k + (scale * F_2(j, k)))], j, xmin, xmax, xstep], k, ymin, ymax, ystep) → vField
  2. Line(I, xAxis) → f
  3. Line(J, xAxis) → g
  4. Line(K, xAxis) → h
  5. Sequence(Segment[Rotate[curlPt - (0.2, 0), k + curlAngle, curlPt], Rotate[curlPt + (0.2, 0), k + curlAngle, curlPt]], k, 0, 2.0944, 1.0472) → spokes
  6. Circle(divPt, 0.2) → divGlowCircle
  7. CurveCartesian((t * x(Drag)) + ((1 - t) * x(Drag)), (t * (y(Drag) + Side)) + ((1 - t) * y(Drag)), t, 0, 1) → path_{s1}
  8. CurveCartesian((t * (x(Drag) - Side)) + ((1 - t) * x(Drag)), (t * (y(Drag) + Side)) + ((1 - t) * (y(Drag) + Side)), t, 0, 1) → path_{s2}
  9. CurveCartesian((t * (x(Drag) - Side)) + ((1 - t) * (x(Drag) - Side)), (t * y(Drag)) + ((1 - t) * (y(Drag) + Side)), t, 0, 1) → path_{s3}
  10. CurveCartesian((t * x(Drag)) + ((1 - t) * (x(Drag) - Side)), (t * y(Drag)) + ((1 - t) * y(Drag)), t, 0, 1) → path_{s4}
  11. Derivative(path_{s1}) → Dpath_{s1}
  12. Derivative(path_{s2}) → Dpath_{s2}
  13. Derivative(path_{s3}) → Dpath_{s3}
  14. Derivative(path_{s4}) → Dpath_{s4}
  15. Sum(Sequence[(((F_1(x(path_{s1}(i + dS / 2)), y(path_{s1}(i + dS / 2))), F_2(x(path_{s1}(i + dS / 2)), y(path_{s1}(i + dS / 2)))) * (y(Dpath_{s1}(i + dS / 2)), (-x(Dpath_{s1}(i + dS / 2))))) * dS), i, 0, 1 - dS, dS]) → Flux_{s1}
  16. Sum(Sequence[(((F_1(x(path_{s2}(i + dS / 2)), y(path_{s2}(i + dS / 2))), F_2(x(path_{s2}(i + dS / 2)), y(path_{s2}(i + dS / 2)))) * (y(Dpath_{s2}(i + dS / 2)), (-x(Dpath_{s2}(i + dS / 2))))) * dS), i, 0, 1 - dS, dS]) → Flux_{s2}
  17. Sum(Sequence[(((F_1(x(path_{s3}(i + dS / 2)), y(path_{s3}(i + dS / 2))), F_2(x(path_{s3}(i + dS / 2)), y(path_{s3}(i + dS / 2)))) * (y(Dpath_{s3}(i + dS / 2)), (-x(Dpath_{s3}(i + dS / 2))))) * dS), i, 0, 1 - dS, dS]) → Flux_{s3}
  18. Sum(Sequence[(((F_1(x(path_{s4}(i + dS / 2)), y(path_{s4}(i + dS / 2))), F_2(x(path_{s4}(i + dS / 2)), y(path_{s4}(i + dS / 2)))) * (y(Dpath_{s4}(i + dS / 2)), (-x(Dpath_{s4}(i + dS / 2))))) * dS), i, 0, 1 - dS, dS]) → Flux_{s4}
  19. Sum(list1, (s * 4)) → TT
  20. Sum(Sequence[((path_{s1}(i + dS) - path_{s1}(i)) * (F_1(x(path_{s1}(i + dS / 2)), y(path_{s1}(i + dS / 2))), F_2(x(path_{s1}(i + dS / 2)), y(path_{s1}(i + dS / 2))))), i, 0, s - dS, dS]) → Circ_1
  21. Sum(Sequence[((path_{s2}(i + dS) - path_{s2}(i)) * (F_1(x(path_{s2}(i + dS / 2)), y(path_{s2}(i + dS / 2))), F_2(x(path_{s2}(i + dS / 2)), y(path_{s2}(i + dS / 2))))), i, 0, s - dS, dS]) → Circ_2
  22. Sum(Sequence[((path_{s3}(i + dS) - path_{s3}(i)) * (F_1(x(path_{s3}(i + dS / 2)), y(path_{s3}(i + dS / 2))), F_2(x(path_{s3}(i + dS / 2)), y(path_{s3}(i + dS / 2))))), i, 0, s - dS, dS]) → Circ_3
  23. Sum(Sequence[((path_{s4}(i + dS) - path_{s4}(i)) * (F_1(x(path_{s4}(i + dS / 2)), y(path_{s4}(i + dS / 2))), F_2(x(path_{s4}(i + dS / 2)), y(path_{s4}(i + dS / 2))))), i, 0, s - dS, dS]) → Circ_4
  24. Sum(list2, (4 * s)) → TT_2
  25. Segment(path_{s1}(0.5), path_{s1}(0.5 + 0.01)) → arrw01
  26. Segment(path_{s2}(0.5), path_{s2}(0.5 + 0.01)) → arrw02
  27. Segment(path_{s3}(0.5), path_{s3}(0.5 + 0.01)) → arrw03
  28. Segment(path_{s4}(0.5), path_{s4}(0.5 + 0.01)) → arrw04
  29. Distance(Drag, (0, 0)) → cond
  30. Textfield(F_1) → InputBox1
  31. Textfield(F_2) → InputBox2