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可视化二元函数

可视化二元函数
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2017-01-07 02:44
向量函数解析几何
所属专题 intersection: surface-surface, curve-curve intersection: surface-surface, curve-curve intersection: surface-surface, curve-curve intersection: surface-surface, curve-curve
本范例演示了如何通过 GeoGebra 可视化二元函数的图形与性质。涉及的核心数学概念包括函数、向量、多边形、解析几何等。作图要点包括使用 Polygon、ImplicitCurve、Locus、Vector 等命令构造函数图像、向量场及轨迹；通过 Sequence、Sum 等命令生成序列与求和；结合 Textfield、RotateText 等实现动态文本显示。适合高中及以上数学教学，用于理解多元函数的几何表示与向量分析。

GeoGebra 命令及参数

  1. Polygon((x_{max}, y_{max}), (x_{min}, y_{max}), (x_{min}, y_{min}), (x_{max}, y_{min})) → VV, j, t, c, d
  2. If(b_a, Nx_a, 1) → Nx1_a
  3. Polygon((x((ro; α)), y((ro; α)), z_{min}) + PTest, (x((ro; α + 180°)), y((ro; α + 180°)), z_{min}) + PTest, (x((ro; α + 180°)), y((ro; α + 180°)), z_{max}) + PTest, (x((ro; α)), y((ro; α)), z_{max}) + PTest) → poly1, k, l, m, p
  4. ImplicitCurve(f(x, y) - f(x(C), y(C))) → ICurve_2
  5. IntersectPath(plane0, f(x, y)) → ICurve_3
  6. Point(ICurve_2) → R0
  7. Length(lr2) → llr2
  8. Sequence((x(Element[lr2, i]), y(Element[lr2, i]), f(x(Element[lr2, i]), y(Element[lr2, i]))), i, 1, llr2) → lllPmin01
  9. If(Length[lllPmin01] ≟ 0, false, true) → bwllPmin01
  10. Length(lr1) → llr1
  11. Sequence((x(Element[lr1, i]), y(Element[lr1, i]), f(x(Element[lr1, i]), y(Element[lr1, i]))), i, 1, llr1) → lllPmax01
  12. If(Length[lllPmax01] ≟ 0, false, true) → bwllPmax01
  13. If(bwllPmax01, Max[z(lllPmax01)], 1000) → Maxz0
  14. If(bwllPmax01, IndexOf[Maxz0, Sequence[z(Element[lllPmax01, i]), i, 1, Length[lllPmax01]]], 1000) → Indexmaxz0
  15. If(b, true, false) → h
  16. If(h, Nx, 1) → Nx1
  17. Length(lr3) → llr3
  18. Point(Segment[(0, 0), (6.28319, 0)]) → Af
  19. Point(Segment[Px2, Px1]) → Ax
  20. Point(Segment[Py2, Py1]) → Ay
  21. Textfield(N_{V1}) → inputBox1
  22. Textfield(x_{min}) → inputBox2
  23. Textfield(x_{max}) → inputBox3
  24. Textfield(y_{min}) → inputBox4
  25. Textfield(y_{max}) → inputBox5
  26. Textfield(f) → inputBox6
  27. Textfield(z_{min}) → inputBox7
  28. Textfield(z_{max}) → inputBox8
  29. Point(Segment[Pz2, Pz1]) → Az
  30. Textfield(ω_c) → inputBox9
  31. Textfield(vt) → inputBox10
  32. Sum(Sequence[If[f(x(PTest), y(PTest)) - f(x(Element[lr01, i]), y(Element[lr01, i])) > 0, 1, 0], i, 1, n]) → ext1
  33. RotateText("z_{min}=" + z_{min} + "", 90°) → Text2
  34. RotateText("z_{max}=" + z_{max} + "", 90°) → Text6
  35. Sequence(Segment[Element[l0maxmin2, k], Element[l0maxmin2, k + 1]], k, 1, Length[l0maxmin2]) → linelmaxmin2
  36. Sequence(Element[lr1, i], i, 1, llr1) → lPmax
  37. Sequence(Element[lr2, i], i, 1, llr2) → lPmin
  38. Sequence(Element[lr3, i], i, 1, llr3) → lPsad
  39. If(bwllPmin01, Min[z(lllPmin01)], 1000) → Minz0
  40. If(bwllPmin01, IndexOf[Minz0, Sequence[z(Element[lllPmin01, i]), i, 1, Length[lllPmin01]]], 1000) → Indexminz0
  41. Element(lr2, Indexminz0) → P_{min0}
  42. If(bwllPmin01, Element[lllPmin01, Indexminz0], (1000, 1000, 1000)) → zmin2
  43. Point(Liste1) → C111a
  44. Locus(C1110a, C111a) → loc
  45. PolyLine(l0maxmin1) → linelmaxmin11
  46. If(bwllPmax01, Element[lllPmax01, Indexmaxz0], (1000, 1000, 1000)) → zmax2
  47. RemoveUndefined(Sequence[If[f(x(PTest), y(PTest)) - f(x(Element[lr01, i]), y(Element[lr01, i])) > 0, i, NaN], i, 1, n]) → lr04
  48. Zip(Vector[b, a], a, Sequence[Element[lr01, Element[lr04, i]], i, 1, Length[lr04]], b, Sequence[Element[lr02, Element[lr04, i]], i, 1, Length[lr04]]) → lr06
  49. Sequence((x(Element[lr01, i]), y(Element[lr01, i]), f(x(Element[lr01, i]), y(Element[lr01, i]))), i, 1, n) → lr013
  50. Sequence((x(Element[lr02, i]), y(Element[lr02, i]), f(x(Element[lr02, i]), y(Element[lr02, i]))), i, 1, n) → lr023
  51. Zip(Vector[b, a], a, Sequence[Element[lr013, Element[lr04, i]], i, 1, Length[lr04]], b, Sequence[Element[lr023, Element[lr04, i]], i, 1, Length[lr04]]) → lr063
  52. Zip(Vector[a, b], a, Sequence[Element[lr01, Element[lr05, i]], i, 1, Length[lr05]], b, Sequence[Element[lr02, Element[lr05, i]], i, 1, Length[lr05]]) → lr07
  53. Zip(Vector[a, b], a, Sequence[Element[lr013, Element[lr05, i]], i, 1, Length[lr05]], b, Sequence[Element[lr023, Element[lr05, i]], i, 1, Length[lr05]]) → lr073
  54. Textfield(zoom_{3D}) → InputBox1
  55. Element(lr1, rmax - 1) → Pzmax
  56. Element(lr2, rmin - 1) → Pzmin
  57. Element(lr3, rsad - 1) → Pzsad
  58. Text("( x , y , z )") → TextR01
  59. Join({{TextR01}, lllPmax01}) → llPmax01
  60. Element(lr1, Indexmaxz0) → P_{max0}
  61. Sequence((x(Element[lr3, i]), y(Element[lr3, i]), f(x(Element[lr3, i]), y(Element[lr3, i]))), i, 1, llr3) → lllPsp
  62. Join({{TextR01}, lllPmin01}) → llPmin01
  63. Join({{TextR01}, lllPsp}) → llPsad01
  64. Element(lllPsp, rsad - 1) → Pzsad_{3D}
  65. Element(lllPmax01, rmax - 1) → Pzmax_{3D}
  66. Element(lllPmin01, rmin - 1) → Pzmin_{3D}
  67. Textfield(A) → InputBox2
  68. RemoveUndefined(Sequence[If[f(x(PTest), y(PTest)) - f(x(Element[lr01, i]), y(Element[lr01, i])) > 0, i, (-i)], i, 1, n]) → lr04c
  69. Append(lr04c, Element[lr04c, 1]) → lr04cc
  70. Sum(Sequence[If[(Element[lr04cc, i] * Element[lr04cc, i + 1]) < 0, 1, 0], i, 1, n + 1]) → lr04ccc
  71. Line(A, RoR) → i
  72. ImplicitCurve(f(x, y) - zi) → list3
  73. ImplicitCurve(f(x, y) - zii) → list4
  74. Vector(R0, R41) → w_11
  75. Vector(R0, R0 + R11) → u_11
  76. Textfield(ω0_c) → InputBox3
  77. Textfield(r_a) → InputBox5
  78. Point(c) → G
  79. Point(j) → H
  80. Point(w_11, 1) → I
  81. Vector(F, I_{3D}) → v
  82. If(ext1 ≟ n, 1, If[ext1 ≟ 0, 2, If[(lr04ccc > 2) ∧ (lr04ccc ≤ 4), 3, 4]]) → roo
  83. Textfield(n_c) → InputBox6
  84. Textfield(a_c) → InputBox7
  85. Textfield(r_{lc}) → InputBox8
  86. Textfield(vt) → InputBox9
  87. PolyLine(Bl) → Bly
  88. PolyLine(Gr) → Gry
  89. PolyLine(Ro) → Roy
  90. Textfield(Nx) → Resolution
  91. Textfield(r) → InputBox11
  92. Textfield(v_{m4}) → InputBox4
  93. Textfield(Nz) → InputBox10
  94. Textfield(n) → InputBox12
  95. Textfield(rr) → InputBox13
  96. Textfield(r_{ext}) → InputBox14
  97. Textfield(v_{zi}) → InputBox15
  98. Textfield(d_1) → InputBox16
  99. Textfield(f_1) → InputBox17