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Directional Derivatives and Gradient

Directional Derivatives and Gradient
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2016-12-08 18:10
VectorCalculusDerivative
In collections ₂ΛM 1: Lagrange Multipliers with One Constraint. Examples
This example illustrates the concepts of directional derivatives and gradients. It constructs the gradient vector at a point on a 3D surface and the directional derivative along any direction. Key mathematical concepts include derivatives, vectors, and 3D surfaces. GeoGebra commands used include Circle, Point, Derivative, Vector, ImplicitCurve, Plane, CurveCartesian, Textfield, Length, Segment, Function, Rotate, OrthogonalLine, IntersectPath, Angle, Intersect, Element, etc. Suitable for university calculus courses.

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

AI Drawing Prompt Reference

Draw a 3D surface and its gradient vector and directional derivative at a point on the surface. First, draw the surface, then select a point on the surface, compute the gradient vector at that point, and draw the gradient vector. Next, draw the curve of the directional derivative at that point. Finally, draw the normal plane and the normal vector in the plane.
The prompt above was compiled from the original drawing commands and is for reference only.