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方向导数与梯度演示

方向导数与梯度演示
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2020-03-12 14:23
向量函数微积分导数几何作图
①本范例通过动态图形演示方向导数与梯度的几何意义，直观展示梯度向量的方向与大小；②涉及核心数学概念包括方向导数、梯度向量、向量场、函数图像；③构造要点：使用 Circle, Point, Vector, Function, Plane, Rotate, Intersect 等命令构建函数图像、梯度向量及方向导数可视化；④适合高中及以上学生及微积分学习者，用于理解多元函数的微分几何性质。

GeoGebra 命令及参数

  1. Circle(P, 1) → t
  2. Point(t) → B
  3. Derivative(f(x, y), x) → fx
  4. Derivative(f(x, y), y) → fy
  5. Vector(P, P + (fx(x(P), y(P)), fy(x(P), y(P)))) → grad
  6. ImplicitCurve(f(x, y) - f(x(P), y(P))) → c
  7. ImplicitCurve(f(x, y)) → d
  8. ImplicitCurve(f(x, y) - 1) → g
  9. ImplicitCurve(f(x, y) - 2) → h
  10. ImplicitCurve(f(x, y) - 3) → i
  11. ImplicitCurve(f(x, y) - 4) → j
  12. ImplicitCurve(f(x, y) - 5) → k
  13. ImplicitCurve(f(x, y) + 1) → l
  14. ImplicitCurve(f(x, y) + 2) → m
  15. ImplicitCurve(f(x, y) + 3) → n
  16. ImplicitCurve(f(x, y) + 4) → o
  17. ImplicitCurve(f(x, y) + 5) → p
  18. Plane(P, P', P + (fx(x(P), y(P)), fy(x(P), y(P)))) → r
  19. CurveCartesian(x(P) + (fx(x(P), y(P)) * w), y(P) + (fy(x(P), y(P)) * w), w, -100, 100) → q
  20. CurveCartesian(x(P') + (fx(x(P), y(P')) * Q), y(P') + (fy(x(P), y(P)) * Q), z(P') + ((fx(x(P), y(P))^(2) + fy(x(P), y(P))^(2)) * Q), Q, -100, 100) → e
  21. Textfield(f) → InputBox1
  22. Textfield(Px) → InputBox2
  23. Textfield(Py) → InputBox3
  24. Length(grad) → normgrad
  25. Segment(P'', P') → s
  26. Function(f(x(P'), y(P')) + (fx(x(P'), y(P')) * (x - x(P'))) + (fy(x(P'), y(P')) * (y - y(P'))), x(P) - 1, x(P) + 1, y(P) - 1, y(P) + 1) → b
  27. Rotate(B, θ, P) → A
  28. Vector(P, A) → u
  29. OrthogonalLine(P, u, xOyPlane) → f_1
  30. OrthogonalPlane(P'', f_1) → v
  31. IntersectPath(v, f) → w
  32. IntersectPath(v, b) → a_1
  33. Angle(B, P, A) → β
  34. Textfield(θ) → InputBox4
  35. Point(zAxis) → C
  36. Intersect(zAxis, xAxis) → D
  37. Vector(C, D) → u_1
  38. Point(yAxis) → E
  39. Vector(E, D) → v_1
  40. Point(xAxis) → F
  41. Vector(F, D) → w_1
  42. Element(list1, faces) → Selectedvector