Conservative Vector Fields — Question 3

Question 3 – Solution

Strategy. Make the cross-partial identity hold for every point, not just at a selected point.

Step 1: Determine the parameter Write $P=ay+2xy$ and $Q=3x+x^2$. Then $$P_y=a+2x, \qquad Q_x=3+2x.$$ Equality for all $x$ requires $$\boxed{a=3}.$$ No other constant can make the two affine expressions identical.

Step 2: Construct the potential For $a=3$, $$f_x=3y+2xy.$$ Integrating with respect to $x$ gives $$f=3xy+x^2y+g(y).$$ Now $f_y=3x+x^2+g'(y)$ must equal $Q=3x+x^2$, so $g'(y)=0$. $$\boxed{f(x,y)=3xy+x^2y+C}.$$

Verification The gradient is $\langle 3y+2xy,3x+x^2\rangle$, exactly $\mathbf F_3$. The coefficient comparison also proves the value $a=3$ is unique.