拉格朗日乘数法示例6：三维约束曲面上的最值可视化

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. LaTeX(Cu0_y, true) → txtCu0_y
13. Textfield(f) → InputBox3
14. Textfield(Cu_g) → InputBox4
15. Zip((x(a), y(a), f(a)), a, Sequence[Point[Cu_g, t / n], t, 0, n]) → lg_1
16. Textfield(n) → InputBox5
17. CurveCartesian(t, (0.8 - t^(2)) / (t + 1), f(t, (0.8 - t^(2)) / (t + 1)), t, (-π), π) → q1
18. Surface(t, (0.8 - t^(2)) / (t + 1), u, t, -3, 3, u, -0.1, 0.00001) → g1
19. IndexOf(Min[z(l2f)], z(l2f)) → Indexmin2
20. Element(l1, Indexmax1) → Amax1
21. Element(l1, Indexmin1) → Amin1
22. Element(l1f, Indexmax1) → Amax1_{3D}
23. Element(l1f, Indexmin1) → Amin1_{3D}
24. Element(l2f, Indexmin2) → Amin2_{3D}
25. Element(l2, Indexmin2) → Amin2
26. Element(l2, Indexmax2) → Amax2
27. Element(l2f, Indexmax2) → Amax2_{3D}
28. RotateText(text14, 90°) → text15
29. 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
30. 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
31. Segment(B, C) → f_1
32. Text("∇f", Point[u_f, 1]) → text25f
33. Text("∇g", Point[u_g, 1]) → text25g
34. Simplify(fx'(x, y) - (gx'(x, y) * fy'(x, y) / gy'(x, y))) → Cu_xf
35. Simplify(fy'(x, y) - (gy'(x, y) * fx'(x, y) / gx'(x, y))) → Cu_yf
36. Textfield(txtCu01_x) → InputBox1
37. Textfield(txtCu01_y) → InputBox2
38. RemoveUndefined(Zip[Vector[c, c + UnitVector[a] / 2], a, l1xy'_f, c, l1]) → lvec1_f
39. RemoveUndefined(Zip[Vector[c, c + UnitVector[a] / 2], a, l2xy'_f, c, l2]) → lvec2_f
40. RemoveUndefined(Zip[Vector[c, c + UnitVector[a] / 2], a, l1xy'_g, c, l1]) → lvec1_g
41. RemoveUndefined(Zip[Vector[c, c + UnitVector[a] / 2], a, l2xy'_g, c, l2]) → lvec2_g
42. Zip(Segment[a, b], a, l1, b, l1f) → l1seg
43. Sequence(Text[i, l1(i) - (0.2, 0.1)], i, 1, lä1) → l1txt
44. Sequence(Text[i, l2(i) - (0.2, 0.1)], i, 1, lä2) → l2txt
45. Zip(Segment[a, b], a, l2, b, l2f) → l2seg