Question

Let D be an open, connected but not simply-connected subset of R²

. Give an example of vector field F(x,y)=(P(x,y),Q(x,y)) satisfying each of the following cases. Explain your answer in detail. (a) [ 1pt]∀(x,y)∈D,Py (x,y)=Qx (x,y), and F is a gradient (conservative) vector field. (b) ∀(x,y)∈D,Py (x,y)=Qx (x,y), and F is not a gradient (conservative) vector field.

Answer 1

To satisfy case (a), F(x, y) = (x^2, x^2 + y) is an example of a conservative vector field. To satisfy case (b), F(x, y) = (-y, x) is an example of a non-conservative vector field.

Step-by-step explanation:

To satisfy case (a), we need to find a vector field F(x, y) that is a gradient (conservative) vector field, where Py(x, y) = Qx(x, y) for all (x, y) in D. One example of such a vector field is F(x, y) = (x^2, x^2 + y), where D could be any open, connected but not simply-connected subset of R^2. This vector field is conservative because its curl is zero, and it satisfies the condition Py(x, y) = Qx(x, y).

To satisfy case (b), we need to find a vector field F(x, y) that is not a gradient (conservative) vector field, even though Py(x, y) = Qx(x, y) for all (x, y) in D. One example of such a vector field is F(x, y) = (-y, x), where D could be any open, connected but not simply-connected subset of R^2. This vector field is not conservative because its curl is nonzero, even though it satisfies the condition Py(x, y) = Qx(x, y).