梯度与方向导数：几何与微积分的结合

GeoGebra 命令及参数

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