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Level Curves with Gradient Field

Level Curves with Gradient Field
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2018-01-29 13:59
FunctionCoordinate GeometryCalculusDerivativeGraphing
In collections intersection: surface-surface, curve-curve intersection: surface-surface, curve-curve intersection: surface-surface, curve-curve intersection: surface-surface, curve-curve
This example demonstrates the visualization of level curves and gradient fields for a function. It involves core mathematical concepts like level curves, gradient fields, function graphs, and analytical geometry. Construction highlights include using commands such as Line, Point, Midpoint, Mirror, Polygon, Function, Textfield, IntersectPath, ImplicitCurve, Max, Min, Sequence, and Derivative to create interactive visualizations. Suitable for high school students and math enthusiasts to understand the geometric properties of multivariable functions and gradient fields.

GeoGebra Commands & Parameters

  1. Line(A, yAxis) → r_2
  2. Point(r_2) → D
  3. Line(A, xAxis) → r_1
  4. Point(r_1) → B
  5. Midpoint(D, B) → M
  6. Mirror(A, M) → C
  7. Polygon(A, B, C, D) → Polígon1, a, b, c, d
  8. Function(f(s, t), s, x(A), x(B), t, y(A), y(D)) → g
  9. Textfield(f) → CampEntradaSortida1
  10. Point(zAxis) → P
  11. IntersectPath(h, f) → i
  12. ImplicitCurve(f(x, y) - f_{(x,y)}) → j
  13. Max(l1) → mx
  14. Min(l1) → mn
  15. Sequence(ImplicitCurve[f(x, y) - s], s, -5, 5, k) → llista1
  16. Point(j) → E
  17. Derivative(f, x) → f_x
  18. Derivative(f, y) → f_y
  19. Sequence(Sequence[(x(A) + (k * (x(B) - x(A)) / dens), y(A) + (l * (y(D) - y(A)) / dens)), l, 0, dens], k, 0, dens) → lli_1
  20. Sequence(Sequence[Vector[Element[lli_1, k, l], Element[lli_1, k, l] + (f_x(Element[lli_1, k, l]), f_y(Element[lli_1, k, l]))], l, 0, dens], k, 0, dens) → lli_2

AI Drawing Prompt Reference

Draw level curves and gradient field for a function. First, draw the x-axis and y-axis, then find their midpoint to construct a rectangle. Plot the function graph and input the function expression. Take a point on the z-axis, find its intersection with the function graph, and draw the level curves. Calculate the partial derivatives of the function to generate the sequence of gradient field vectors.
The prompt above was compiled from the original drawing commands and is for reference only.