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Geometric Interpretation of the Jacobian

Geometric Interpretation of the Jacobian
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2016-09-10 16:04
CalculusDerivativeMatrixTransformationLocus
This GeoGebra example demonstrates the geometric interpretation of the Jacobian determinant in multivariable calculus. It visualizes how a coordinate transformation scales area. Key concepts include Jacobian matrices, derivatives, and area calculation. Construction utilizes commands like Line, Point, Sequence, Polygon, Locus, Derivative, Area, and Dilate to compare original and transformed regions. Ideal for university calculus students and educators seeking visual aids for understanding abstract transformation properties.

GeoGebra Commands & Parameters

  1. Line(a, xAxis) → h
  2. Point(h) → A
  3. Sequence(CurveCartesian[x(a + (r * t) + ((ί * r) * k)), y(a + (r * t) + ((ί * r) * k)), t, -1, 1], k, -1, 1, 2 / n) → list1
  4. Sequence(CurveCartesian[x(a - (r * k) + ((ί * r) * t)), y(a - (r * k) + ((ί * r) * t)), t, -1, 1], k, -1, 1, 2 / n) → list2
  5. Sequence(CurveCartesian[x((a + (r * t) + ((ί * r) * k))^(2)), y((a + (r * t) + ((ί * r) * k))^(2)), t, -1, 1], k, -1, 1, 2 / n) → list3
  6. Sequence(CurveCartesian[x((a - (r * k) + ((ί * r) * t))^(2)), y((a - (r * k) + ((ί * r) * t))^(2)), t, -1, 1], k, -1, 1, 2 / n) → list4
  7. Polygon(B, B', 4) → poly1, i, j, k, l, C, D
  8. Point(poly1) → E
  9. Locus(z_2, E) → loc1
  10. Derivative(f, x) → xu
  11. Derivative(f, y) → xv
  12. Derivative(g, x) → yu
  13. Derivative(g, y) → yv
  14. Sequence((x(Element[list5, i]) + (ί * y(Element[list5, i])))^(2), i, 1, 16) → list6
  15. Polygon(list6) → poly2
  16. Area(z_1, z_3, z_4, z_5, z_1) → b
  17. Centroid(poly1) → F
  18. Circle(z_6, z_4) → p
  19. Line(z_6, xAxis) → m
  20. Line(z_6, yAxis) → q
  21. Intersect(p, q, 1) → H
  22. Line(H, xAxis) → s
  23. Intersect(p, m) → I, J
  24. OrthogonalLine(J, s) → f_1
  25. OrthogonalLine(z_4, f_1) → g_1
  26. Intersect(s, f_1) → L
  27. Intersect(f_1, g_1) → N
  28. OrthogonalLine(I, s) → t
  29. Intersect(g_1, t) → K
  30. Intersect(s, t) → M
  31. Polygon(M, L, 4) → poly3, h_1, i_1, j_1, k_1, O, P
  32. Dilate(M, 1.4, z_6) → M'
  33. Dilate(L, 1.4, z_6) → L'
  34. Dilate(N, 1.4, z_6) → N'
  35. Dilate(K, 1.3, z_6) → K'