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Vector Fields and Line Integrals Explorer

Vector Fields and Line Integrals Explorer
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2022-06-28 03:45
VectorCalculusDerivativeIntegralGeometric Construction
In collections ₂ΛM 1: Lagrange Multipliers with One Constraint. Examples
This example demonstrates the exploration of vector fields, line integrals, divergence, and curl. It involves core mathematical concepts such as vector fields, line integrals, divergence (div), curl (curl), and vector operations. The construction utilizes GeoGebra commands and objects like SelectedIndex, Midpoint, Sequence, Element, Circle, CurveCartesian, Sum, Segment for dynamic path generation, divergence/curl point calculation, and visualization. Suitable for university calculus, linear algebra courses, and advanced high school students studying vector calculus applications.

GeoGebra Commands & Parameters

  1. SelectedIndex(fieldNames) → fieldIndex
  2. Midpoint(A, B) → M
  3. Sequence(Segment[Rotate[curlPt - (0.25, 0), k + curlAngle, curlPt], Rotate[curlPt + (0.25, 0), k + curlAngle, curlPt]], k, 0, 2.0943951024, 1.0471975512) → spokes
  4. Element(curlList, fieldIndex) → curl
  5. Element(divList, fieldIndex) → div
  6. Circle(divPt, 0.2) → divGlowCircle
  7. Element(gList, fieldIndex) → g
  8. Element(fList, fieldIndex) → f
  9. Sequence(Sequence[Vector[(j, k), (j + (scale * f(j, k)), k + (scale * g(j, k)))], j, xmin, xmax, xstep], k, ymin, ymax, ystep) → vField
  10. SelectedIndex(pathNames) → pathIndex
  11. Element(pathList, pathIndex) → path
  12. CurveCartesian(x(path(t)), y(path(t)), t, 0, s) → partPath
  13. Sum(Sequence[((path(i + dS) - path(i)) * (f(x(path(i + dS / 2)), y(path(i + dS / 2))), g(x(path(i + dS / 2)), y(path(i + dS / 2))))), i, 0, s - dS, dS]) → work
  14. Sequence(Vector[path(s), path(s) + ((scale * f(x(path(s)), y(path(s)))), (scale * g(x(path(s)), y(path(s)))))], s, 0, 1, 0.1) → pathVectors
  15. Segment(path(0.2), path(0.21)) → arrow1
  16. Segment(path(0.5), path(0.51)) → arrow2
  17. Segment(path(0.8), path(0.81)) → arrow3

AI Drawing Prompt Reference

Create a dynamic vector field and line integral explorer, showing the relationships between vector fields, line integrals, divergence, and curl. Begin by generating a vector field with arrows indicating direction and magnitude at each point. Select a path and draw vectors along it, marking the start, midpoint, and end of the path with arrows. Calculate and visualize divergence and curl, using circles for divergence and rotating lines for curl. Finally, plot the path as a curve and add vectors along the path to demonstrate the interaction between the vector field and the path.
The prompt above was compiled from the original drawing commands and is for reference only.