Final answer:
To determine whether a vector field is solenoidal, conservative, or both, one must compute the divergence and curl of the field. The given vector field must satisfy conditions for zero divergence and zero curl to be both solenoidal and conservative.
Step-by-step explanation:
To determine if a vector field is solenoidal, conservative, or both, we need to analyze the components of the vector field. For a vector field to be conservative, the curl of the field must be zero. Specifically, the vector field A given as A = Xx² – y2xy (assuming this is expressed as A = xi + yj where x and y are variables) must satisfy the condition that the partial derivative of the x-component with respect to y is equal to the partial derivative of the y-component with respect to x.
A vector field is considered solenoidal if the divergence of the vector field is zero. In mathematical terms, for a vector field F(x, y) = M(x, y)i + N(x, y)j, the divergence is given by ∇ · F = ∂M/∂x + ∂N/∂y. To find out if the given vector field A is solenoidal, we would need to compute this divergence.
Without the full expression of the vector field, we cannot definitively determine whether it is conservative or solenoidal. However, the procedure involves computing the derivatives as indicated and checking the necessary conditions for conservativeness and solenoidality. For a vector field to be both conservative and solenoidal, it must have zero curl and zero divergence.