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Directional Derivatives and Gradient Demo

Directional Derivatives and Gradient Demo
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2020-03-12 14:23
VectorFunctionCalculusDerivativeGeometric Construction
This example visually demonstrates the concept of directional derivatives and gradients through dynamic geometry. It illustrates how the gradient vector indicates the direction of steepest ascent and its magnitude corresponds to the rate of change. Key mathematical concepts include directional derivatives, gradient vectors, vector fields, and function graphs. Constructed using commands like Circle, Point, Vector, Function, Plane, Rotate, and Intersect, it provides an interactive way for high school and university students to explore multivariable calculus and differential geometry.

GeoGebra Commands & Parameters

  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