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Directional Derivative and the Gradient

Directional Derivative and the Gradient
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2016-12-12 11:16
VectorFunctionCoordinate GeometryCalculusGeometric Construction
This example demonstrates the geometric meaning of directional derivatives and gradients in 3D space, visualizing the direction and magnitude of gradient vectors. Key math concepts include directional derivatives, gradients, vector fields, and 3D surfaces. Construction highlights involve commands like PointIn, Vector, Derivative, CurveCartesian, CornerThreeD, Polygon, ImplicitCurve, Segment from GeoGebra to build 3D graphics, vector fields, and gradient visualizations. Suitable for university students in mathematics or physics, or learners interested in calculus and vector analysis.

GeoGebra Commands & Parameters

  1. PointIn(f) → P
  2. Vector(B, B + (cos(c), sin(c))) → v
  3. Derivative(f, x) → fx
  4. Derivative(f, y) → fy
  5. Vector(B, B + G / abs(G)) → u
  6. Vector(B, B + w / abs(w)) → m
  7. Vector(P, P + m) → q
  8. CurveCartesian((t * cos(c)), (t * sin(c)), f((t * cos(c)), (t * sin(c))), t, -5, 5) → d
  9. CornerThreeD(-1, 1) → L
  10. CornerThreeD(-1, 7) → U
  11. Polygon((x(L), y(L), 0), (x(L), y(U), 0), (x(U), y(U), 0), (x(U), y(L), 0)) → poly1, g, h, i, j
  12. ImplicitCurve(((3 * x^(2)) * y) - y^(3) - z(P)) → e
  13. Segment(P, B) → k
  14. Segment((x(L), 0, 0), (x(U), 0, 0)) → l
  15. Segment(B + w, B + (1 / 4 * (fx(x(B), y(B)), fy(x(B), y(B))))) → p
  16. Segment(P, D) → r
  17. Segment(D, C) → s
  18. Vector(P, D) → t
  19. Vector(B, B + G / ((abs(w) * abs(G)))) → u_1
  20. Segment((0, y(L), 0), (0, y(U), 0)) → f_1
  21. Vector((0, 0, z(L)), (0, 0, z(U))) → g_1
  22. CurveCartesian(sqrt((b_1 + t^(3)) / ((3 * t))), t, b_1, t, -10, 10) → c_1
  23. CurveCartesian((-sqrt((b_1 + t^(3)) / ((3 * t)))), t, b_1, t, -10, 10) → c_2
  24. ImplicitCurve(((3 * x^(2)) * y) - y^(3)) → e_1
  25. CurveCartesian(sqrt((z(P) + t^(3)) / ((3 * t))), t, z(P), t, -10, 10) → c_3
  26. CurveCartesian((-sqrt((z(P) + t^(3)) / ((3 * t)))), t, z(P), t, -10, 10) → c_4