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Illustration of Directional Derivative

Illustration of Directional Derivative
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2022-03-04 01:49
VectorCalculusDerivative
This example illustrates the concept of directional derivative. Using geometric methods and objects like vectors, curves, and surfaces in GeoGebra, it demonstrates the process of calculating directional derivatives. Key commands used include Vector, CurveCartesian, Surface, etc. Suitable for university calculus courses.

GeoGebra Commands & Parameters

  1. Vector(P, P + (cos(alfa), sin(alfa))) → v
  2. CurveCartesian(x0 + (vx * t), y0 + (vy * t), f(x0 + (vx * t), y0 + (vy * t)), t, -5, 5) → krzywa
  3. CurveCartesian(x0 + (vx * t), y0 + (vy * t), z0 + (pochkier * t), t, -5, 5) → styczna
  4. Surface(x0 + (x(v) * t), y0 + (y(v) * t), s, t, -7, 7, s, 0, 7) → plaszcz
  5. CurveCartesian(x0 + (vx * t), y0 + (vy * t), z0, t, 0, 5) → prostapozioma
  6. Point(styczna) → A
  7. Point(prostapozioma) → B
  8. Angle(B, Pf, A) → β
  9. Vector(Pf, Pf + v) → vNaWykresie
  10. Segment(Pf, P) → b
  11. Vector(P, P + (fx, fy)) → gradient
  12. Segment(P, (x(P), 0)) → d
  13. Segment(P, (0, y(P))) → e

AI Drawing Prompt Reference

Illustrate the directional derivative concept in a geometric way. Start from a point P and define a direction vector v based on an angle α; use v to build a parameterized curve in the plane through P. Construct a reference curve that passes through a point Pf and is parameterized along the same direction, then mark the tangency point A on that curve. Form another corresponding parameterized straight curve (a reference profile) and take its point B, then label the angle β using points B, Pf, and A. Use the same directional parameter t together with an additional parameter s to generate a surface that represents how the function varies along v. Finally draw the segment b from Pf to P, show the gradient vector at P using the shift by (fx, fy), and include the horizontal and vertical projection segments d and e from P to the coordinate axes.
The prompt above was compiled from the original drawing commands and is for reference only.