Show transcribed image text Problem . Conservative field If a differential function f(x, y) is given, we can calculate the partial derivatives af af and We can also ask the ax ay opposite question: . f(ar, Y) = ? Question:Suppose two functions P(x, y), Q(x,y) are given, can you find a functions = f(x,y) such that Q(, Y) TX – af af P(t, y) = P(x, y), and Q(x,y) ax ay A theorem answering this question is Theorem. Such function f exists when the following relation holds: a P aQ ay ax – C The proof of it relies on a famous theorem called Green's theoremwhich we will learn in section 16.4. If the relation aboves holds, we say the vector field (P(X,Y,Q(x, y)) is conservative, and the function f is called the potential function. Your task is to find the potential function f when the relation in the above thereom holds (1) Now lets consider functions: P(x,y) = 3+2xy and Qxy) = x23у2. (1.1) Check that (P, Q) is a conservative field. af af (1.2) Now try to find the form of the potential function f with =P and =Q. ax ay (2) Determine if the following fields are conservative. If so, find the corresponding potential function f. (2.1) (P, Q) = (In yy/x, Inx +x/y); (2.2) (P,Q) = (ybe , (1+xy)ey);