Conservative Vector Fields — Question 6

Question 6 – Solution

Strategy. Verify compatibility of all three components, then integrate successively while retaining correction functions.

Step 1: Compatibility Let $\mathbf F=\langle P,Q,R\rangle$. Then $$P_y=2x=Q_x, \qquad P_z=1=R_x, \qquad Q_z=0=R_y.$$ On the simply connected domain $\mathbb R^3$, these identities show the field is conservative.

Step 2: Construct $f$ Integrating $f_x=P$ gives $$f=x^2y+xz+g(y,z).$$ From $f_y=x^2+g_y=Q$, we get $g_y=2y$, so $g=y^2+h(z)$. Finally, $$f_z=x+h'(z)=R=x+2z,$$ so $h=z^2+C$. Thus $$\boxed{f=x^2y+xz+y^2+z^2+C}.$$

Step 3: Endpoint evaluation $$\int_C\mathbf F\cdot d\mathbf r=f(1,2,3)-f(0,0,0) =2+3+4+9=\boxed{18}.$$

Verification Differentiating the boxed potential returns all three original components; the additive constant cancels from the endpoint difference.