If z f (x, y) is a function with continuous first partial derivatives in a region R

Chapter 11, Problem 31

(choose chapter or problem)

If z = f(x, y) is a function with continuous first partial derivatives in a region R, then a flow V(x, y) = (P(x, y), Q(x, y)) in R may be defined by letting \(P(x, y)=-\frac{\partial f}{\partial y}\)(x, y) and \(Q(x, y)=\frac{\partial f}{\partial x}\)(x, y). Show that if X(t) = (x(t), y(t)) is a solution of the plane autonomous system

\(x^{\prime}=P(x, y)\)

\(y^{\prime}=Q(x, y)\)

then f(x(t), y(t)) = c for some constant c. Thus a solution curve lies on the level curves of f. [Hint: Use the Chain Rule to compute \(\frac{d}{d t} f(x(t), y(t))\).]