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Line Integrals of Vector Fields: Work & Circulation

Line Integrals of Vector Fields: Work & Circulation
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2016-11-07 13:24
VectorFunctionCalculusIntegralParametric Equation
This example illustrates the concept of line integrals in vector fields, including work and circulation. By constructing vector fields, path integrals, and geometric shapes, students can gain a visual understanding of these concepts. GeoGebra commands such as SelectedIndex, Midpoint, and Derivative are used. Suitable for university mathematics and physics courses.

GeoGebra Commands & Parameters

  1. SelectedIndex(pathNames) → pathIndex
  2. Midpoint(N, P) → Q
  3. Element(pathList, pathIndex) → path
  4. Derivative(path) → DDpath
  5. ParametricDerivative(path) → Dpath
  6. Distance(N, P) → NP
  7. Element(CompX, pathIndex) → RTX
  8. Derivative(RTX, x) → DRTX
  9. Element(CompY, pathIndex) → RTY
  10. Derivative(RTY, x) → DRTY
  11. CurveCartesian(x(path(t)), y(path(t)), t, 0, s) → partPath
  12. Integral(MUL, 0, 1) → Work
  13. Textfield(F_1) → InputBox1
  14. Textfield(F_2) → InputBox2
  15. Segment(path(0.3), path(0.3 + 0.01)) → arrow1
  16. Segment(path(0.6), path(0.6 + 0.01)) → arrow2
  17. If(Work ≟ Work, Work, Sum[Sequence[((path(i + dS) - path(i)) * (F_1(x(path(i + dS / 2)), y(path(i + dS / 2))), F_2(x(path(i + dS / 2)), y(path(i + dS / 2))))), i, 0, s - dS, dS]]) → WOORK
  18. Sequence(Vector[path(i), path(i) + ((scale * F_1(x(path(i)), y(path(i)))), (scale * F_2(x(path(i)), y(path(i)))))], i, 0, s, 0.15) → PathVectors
  19. Sequence(path(i), i, 0.25, 0.75, 0.25) → PtPathList
  20. Sequence(Vector[(0, 0), (1 / sqrt((y(DDpath(i)))^(2) + (x(DDpath(i)))^(2)) * (x(DDpath(i)), y(DDpath(i))))], i, 0.25, 0.75, 0.25) → tangvectorsList
  21. Translate(Vector[Element[tangvectorsList, 1]], Element[PtPathList, 1]) → uu
  22. Translate(Vector[Element[tangvectorsList, 2]], Element[PtPathList, 2]) → vv
  23. Translate(Vector[Element[tangvectorsList, 3]], Element[PtPathList, 3]) → ww
  24. Textfield(xA) → InputBox3
  25. Textfield(yA) → InputBox4
  26. Textfield(xB) → InputBox5
  27. Textfield(yB) → InputBox6

AI Drawing Prompt Reference

Visualize line integrals in vector fields, showing work and circulation. Construct vector fields, path integrals, and geometric shapes. Mark midpoints and tangent vectors on the path, and represent vector field directions with arrows. Finally, calculate and display the results of the line integral, distinguishing work and circulation with different colors.
The prompt above was compiled from the original drawing commands and is for reference only.