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Surface integral of a vector field

Surface integral of a vector field
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2016-11-14 19:13
VectorSolid GeometryCalculusIntegral
In collections MATH2001 - Calculus & Linear algebra II
This example demonstrates the calculation and visualization of the surface integral of a vector field over a given surface. Key mathematical concepts include vector fields, surfaces, surface integrals, derivatives, and integrals. It utilizes GeoGebra commands/objects like Derivative, Surface, Integral, Textfield, Vector, UnitVector, and Sequence. Suitable for university students and educators in mathematics, physics, and related fields for teaching and concept exploration.

GeoGebra Commands & Parameters

  1. Derivative(Phix, u) → DuPhix
  2. Derivative(Phix, v) → DvPhix
  3. Derivative(Phiz, v) → DvPhiz
  4. Derivative(Phiy, v) → DvPhiy
  5. Derivative(Phiz, u) → DuPhiz
  6. Derivative(Phiy, u) → DuPhiy
  7. Surface(Phix(u, v), Phiy(u, v), Phiz(u, v), u, ux, uy, v, vx, vy) → PHI
  8. Integral(d, v) → e
  9. Integral(f, ux, uy) → Integral
  10. Textfield(Fi) → InputBox1
  11. Textfield(Fj) → InputBox2
  12. Textfield(Fk) → InputBox3
  13. Textfield(Phix) → InputBox4
  14. Textfield(Phiy) → InputBox5
  15. Textfield(Phiz) → InputBox6
  16. Textfield(ux) → InputBox7
  17. Textfield(uy) → InputBox8
  18. Textfield(vx) → InputBox9
  19. Textfield(vy) → InputBox10
  20. Vector((0, 0, 0), (Ti(ux / 2, vy / 2), Tj(ux / 2, vy / 2), Tk(ux / 2, vy / 2))) → u
  21. UnitVector(u) → v
  22. Sequence(Sequence[(j, k), j, ux, uy, abs(uy - ux) / grid1], k, vx, vy, abs(vy - vx) / grid2) → PointsRegion
  23. Sequence(Sequence[PHI(x(Element[PointsRegion, i, j]), y(Element[PointsRegion, i, j])), i, 1, grid1 + 1], j, 1, grid2 + 1) → PointsSurface
  24. Sequence(Sequence[Vector[(x(Element[PointsSurface, i, j]), y(Element[PointsSurface, i, j]), z(Element[PointsSurface, i, j])), (x(Element[PointsSurface, i, j]) + (scale * Fi(x(Element[PointsSurface, i, j]), y(Element[PointsSurface, i, j]), z(Element[PointsSurface, i, j]))), y(Element[PointsSurface, i, j]) + (scale * Fj(x(Element[PointsSurface, i, j]), y(Element[PointsSurface, i, j]), z(Element[PointsSurface, i, j]))), z(Element[PointsSurface, i, j]) + (scale * Fk(x(Element[PointsSurface, i, j]), y(Element[PointsSurface, i, j]), z(Element[PointsSurface, i, j]))))], i, 1, grid1 + 1], j, 1, grid2 + 1) → VectorsSurface

AI Drawing Prompt Reference

Visualize the surface integral of a vector field over a given surface, demonstrating the calculation and visualization process. Construct a 3D surface using parametric equations Phix, Phiy, Phiz, then plot vectors on the surface whose magnitudes and directions are determined by the vector field components Fi, Fj, Fk. Indicate vectors on the surface to illustrate the relationship between the vector field and the surface.
The prompt above was compiled from the original drawing commands and is for reference only.