Conservative Vector Fields — Question 9

Question 9 – Solution

Strategy. Integrate the compatibility equation for the unknown component, then use the supplied trace data to fix its arbitrary function.

Step 1: Compatibility equation With $P=2xy+y$, $$P_y=2x+1.$$ Conservativeness requires $Q_x=P_y$, so integrating with respect to $x$ gives $$Q(x,y)=x^2+x+h(y).$$

Step 2: Apply the boundary data At $x=0$, $$Q(0,y)=h(y)=e^y.$$ Thus $$\boxed{Q(x,y)=x^2+x+e^y}.$$

Step 3: Find a potential Integrating $f_x=P$ yields $$f=x^2y+xy+g(y).$$ Then $$f_y=x^2+x+g'(y)=Q$$ requires $g'(y)=e^y$, so $$\boxed{f(x,y)=x^2y+xy+e^y+C}.$$

Verification Its gradient is $\langle 2xy+y,x^2+x+e^y\rangle$, and setting $x=0$ in the second component gives exactly $e^y$.