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Directional Derivative and Gradient Demo

Directional Derivative and Gradient Demo
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2017-12-17 16:01
VectorFunctionCoordinate GeometryCalculusDerivative
This example demonstrates the geometric meaning of directional derivatives and gradients, showing the gradient vector and directional derivatives along any direction at a point on a function. Key math concepts include gradient, directional derivative, vector, function graph, and coordinate geometry. GeoGebra commands and objects used in construction are CurveCartesian (curve), Vector (vector), Derivative (derivative), Angle (angle), etc. Suitable for high school and above students learning vector calculus.

GeoGebra Commands & Parameters

  1. CurveCartesian(t, y0, a(t, y0), t, -5, 5) → b
  2. CurveCartesian(x0, t, a(x0, t), t, -5, 5) → c
  3. CurveCartesian((t * cos(α)) + x0, (t * sin(α)) + y0, a((t * cos(α)) + x0, (t * sin(α)) + y0), t, -2, 2) → cut
  4. Curvature(A, cut) → k
  5. Segment(D, A_1) → d
  6. Surface((u * cos(α)) + x0, (u * sin(α)) + y0, v, u, -2, 2, v, -2, 2) → e
  7. Vector(A, A + (cos(α), sin(α))) → v1
  8. Derivative(cut) → cut'
  9. Vector(A, A + B) → v
  10. Angle(v1, v) → β
  11. Vector(A_1, A_1 + ((k1 * x0), (k2 * y0))) → grad

AI Drawing Prompt Reference

Draw a function graph and show the gradient vector at a point and the directional derivative along any direction. First, draw the function curve, then draw the gradient vector at a specific point, followed by the directional derivative vector along any direction, and finally display the curvature and angle.
The prompt above was compiled from the original drawing commands and is for reference only.