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Lagrange Multipliers Example 6: Visualizing Constrained Optimization on a 3D Sur

Lagrange Multipliers Example 6: Visualizing Constrained Optimization on a 3D Sur
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2020-07-15 20:33
VectorSolid GeometryCalculusOptimizationParametric Equation
① This example demonstrates the method of Lagrange multipliers for solving constrained optimization problems, visually showing how the objective function and the constraint surface (a cylinder) are tangent at the extremum. ② Core mathematical concepts include multivariable extrema, Lagrange multipliers, gradient vectors, constraint surfaces, and the geometric relationship between tangent planes and normals. ③ Construction highlights: using GeoGebra commands such as CurveCartesian for parametric curves, Surface for the constraint surface, Vector for gradient vectors (Vf, Vg), Point for tangent points, Text and LaTeX for formulas, Slider (a, α) for interactive parameters, and Zip/Sequence for batch object generation, enabling dynamic exploration. ④ Suitable for university-level calculus or optimization courses, as well as advanced high school students interested in the geometric intuition behind Lagrange multipliers.

GeoGebra Commands & Parameters

  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

AI Drawing Prompt Reference

Create a 3D visualization of constrained optimization: a cylinder as the constraint surface, with a parametric curve on it representing feasible points. The objective function surface is tangent to the cylinder at extremum points. At each tangency point, draw two gradient vectors—one perpendicular to the objective level curve and one perpendicular to the constraint surface—and mark the angle between them. Show the projection of the objective function along the constraint curve, and place multiple sample points on the curve with short segments indicating gradient directions. Add text labels for gradients and function expressions, and include a table listing coordinates, function values, and gradient components for each sample point.
The prompt above was compiled from the original drawing commands and is for reference only.