拉格朗日乘数法示例5：三维约束优化可视化

GeoGebra 命令及参数

1. Simplify(Derivative[f, y, 1]) → fy'
2. Simplify(Derivative[f, x, 1]) → fx'
3. LeftSide(Cu_g) → g
4. Simplify(Derivative[g, y, 1]) → gy'
5. Simplify(Derivative[g, x, 1]) → gx'
6. Point(Cu_g) → B
7. Vector(B, B + UnitVector[Vector[B, Dg]] / 2) → u_g
8. Vector(B, B + UnitVector[Vector[B, Df]] / 2) → u_f
9. Angle(u_g, u_f) → α
10. LaTeX(f, true) → text8a
11. LaTeX(Cu_g, true) → text8b
12. Textfield(f) → InputBox3
13. Textfield(Cu_g) → InputBox4
14. Zip((x(a), y(a), f(a)), a, Sequence[Point[Cu_g, t / n], t, 0, n]) → lg_1
15. Textfield(n) → InputBox5
16. Element(l1, Indexmax1) → Amax1
17. Element(l1, Indexmin1) → Amin1
18. Element(l1f, Indexmax1) → Amax1_{3D}
19. Element(l1f, Indexmin1) → Amin1_{3D}
20. Element(l2f, Indexmin2) → Amin2_{3D}
21. LaTeX(Amin2_{3D}, true, true) → text17
22. Surface((-4.4 * cos(t)), sqrt(1 - cos(t)^(2)), u, t, (-π), π, u, -5, 60) → g1a
23. Surface((-4.4 * cos(t)), (-sqrt(1 - cos(t)^(2))), u, t, (-π), π, u, -5, 60) → g2a
24. CurveCartesian((-sqrt(t)), sqrt((1 - t^(3)) / 3), f((-sqrt(t)), sqrt((1 - t^(3)) / 3)), t, 0, 1) → q2
25. Surface((-(1 - (3 * t^(2)))^(1 / 6)), t, u, t, -1 / sqrt(3), 1 / sqrt(3), u, -1.2, 1.2) → g2
26. CurveCartesian((-sqrt(t)), (-sqrt((1 - t^(3)) / 3)), f((-sqrt(t)), (-sqrt((1 - t^(3)) / 3))), t, 0, 1) → q1a
27. CurveCartesian(sqrt(t), (-sqrt((1 - t^(3)) / 3)), f(sqrt(t), (-sqrt((1 - t^(3)) / 3))), t, 0, 1) → q2a
28. CurveCartesian((rr(t) * cos(t / 2)), (rr(t) * sin(t / 2)), f((rr(t) * cos(t / 2)), (rr(t) * sin(t / 2))), t, 0, 12) → q1
29. Surface((rr(t) * cos(t / 2)), (rr(t) * sin(t / 2)), v, t, 0, (4 * π), v, 0, 4) → g1
30. RotateText(text1, 90°) → text3
31. TableText(Append["N", 1…Length[l1]], Append["\fgcolor{red}{Cp}", l1], Append["\fgcolor{blue}{f}(\fgcolor{red}{C_p })", l1fa], Append["(\dot{f_x},\dot{f_y})", l1xy'_fa], Append["(\dot{g_x},\dot{g_y})", l1xy'_ga], Append["λ_x", lλ1ax], Append["λ_y", lλ1ay], Append["\widehat{\dot{g}\dot{f}} ", lα1], "c|_") → tTab1
32. TableText(Append["N", 1…Length[l2]], Append["\fgcolor{red}{Cp}", l2], Append["\fgcolor{blue}{f}(\fgcolor{red}{C_p })", l2fa], Append["(\dot{f_x},\dot{f_y})", l2xy'_fa], Append["(\dot{g_x},\dot{g_y})", l2xy'_ga], Append["λ_x", lλ2ax], Append["λ_y", lλ2ay], Append["\widehat{\dot{g}\dot{f}} ", lα2], "c|_") → tTab2
33. Segment(B, C) → f_1
34. Text("∇g", Point[u_g, 1]) → text25g
35. Text("∇f", Point[u_f, 1]) → text25f
36. Element(l2, Indexmax2) → Amax2
37. Element(l2f, Indexmax2) → Amax2_{3D}
38. Element(l2, Indexmin2) → Amin2
39. Simplify(fx'(x, y) - (gx'(x, y) * fy'(x, y) / gy'(x, y))) → Cu_xf
40. Simplify(fy'(x, y) - (gy'(x, y) * fx'(x, y) / gx'(x, y))) → Cu_yf
41. RemoveUndefined(Zip[Vector[c, c + UnitVector[a] / 2], a, l1xy'_f, c, l1]) → lvec1_f
42. RemoveUndefined(Zip[Vector[c, c + UnitVector[a] / 2], a, l1xy'_g, c, l1]) → lvec1_g
43. RemoveUndefined(Zip[Vector[c, c + UnitVector[a] / 2], a, l2xy'_f, c, l2]) → lvec2_f
44. RemoveUndefined(Zip[Vector[c, c + UnitVector[a] / 2], a, l2xy'_g, c, l2]) → lvec2_g
45. Textfield(txtCu01_x) → InputBox1
46. Textfield(txtCu01_y) → InputBox2
47. Zip(Segment[a, b], a, l1, b, l1f) → l1seg
48. Sequence(Text[i, l1(i) - (0.3, 0.1)], i, 1, lä1) → l1txt
49. Sequence(Text[i, l2(i) - (0.3, 0.1)], i, 1, lä2) → l2txt
50. Zip(Segment[a, b], a, l2, b, l2f) → l2seg