If you have a CAS that plots vector fields (the command is...

If you have a CAS that plots vector fields (the command is fieldplot in Maple and PlotVectorField or VectorPlot in Mathematica), use it to plot $$ \mathbf{F}(x, y)=\left(y^2-2 x y\right) \mathbf{i}+\left(3 x y-6 x^2\right) \mathbf{j} $$ Explain the appearance by finding the set of points $(x, y)$ such that $\mathbf{F}(x, y)=\mathbf{0}$.

If you have a CAS that plots vector fields (the command is fieldplot in Maple and PlotVectorField or VectorPlot in Mathematica), use it to plot

$$
\mathbf{F}(x, y)=\left(y^2-2 x y\right) \mathbf{i}+\left(3 x y-6 x^2\right) \mathbf{j}
$$

Explain the appearance by finding the set of points $(x, y)$ such that $\mathbf{F}(x, y)=\mathbf{0}$.

Key Concepts

Vector Field

A vector field is a function that assigns a vector to every point in a space, typically indicating both magnitude and direction. In the context of the plane, a vector field provides a geometrical representation of phenomena such as fluid flow or force fields by illustrating how the vectors vary from point to point.

Zeros of a Vector Field

Zeros, or critical points, of a vector field are the points where the vector field takes the zero vector value. These points are crucial as they often represent equilibrium states, where, for instance, forces balance out, and they can help explain the overall behavior or appearance of the vector field in a plot.

Solving Systems of Equations

Determining the zeros of a vector field typically involves solving a system of equations created by setting each component of the vector field equal to zero. This process relies on algebraic techniques such as factorization and substitution to find the set of points that simultaneously satisfy all the equations.

Computer Algebra Systems (CAS) for Visualization

Computer Algebra Systems (CAS) provide powerful tools to graph and analyze vector fields by automatically computing and plotting vectors over a defined grid. This assists in visualizing complex vector behaviors and verifying analytical findings, making them an essential resource in both theoretical and applied studies.

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