Choose a path \(C\) from \((2,1)\) to \((3,5)\) and compute \[ \int_C(2x+y)\,dx+y\,dy. \]
Give and apply a criterion that shows whether this integral is path independent.
Solution
Parametrize the line segment by \[ r(t)=(1-t)(2,1)+t(3,5)=(2+t,1+4t),\qquad 0\leq t\leq1. \] Then \[ \int_C(2x+y)\,dx+y\,dy =\int_0^1(9+22t)\,dt =20. \]
For \(F=(P,Q)=(2x+y,y)\), path independence on \(\mathbb R^2\) would require \(P_y=Q_x\). Here \(P_y=1\) and \(Q_x=0\), so the integral is path dependent.