eIndependentPath

Example Find the line integral $\displaystyle{\int_{(0,0,0)}^{(4,1,2)}(3ydx + 3xdy + 2zdz)}$ ．

Answer We show that F has a potential $f$ ．

$\displaystyle f_x = F_1 = 3y, f_y = F_2 = 3x, f_z = F_3 = 2z$

Then, we integrate $f_x$

with respect to $x$

$\displaystyle f = \int f_x dx = 3xy + g(y,z), f_y = 3x + g_{y}(y,z) = 3x$

From this，we have $g_{y}(y,z) = 0$ Next， $f_z = g_{z}(y,z) = 2z$ implies that $g(y,z) = z^2$

. Thus,

$\displaystyle f = 3xy + z^2$

${\bf F}$ has a potential. This line integral is independent of path. Therefore,

$\displaystyle \int_{(0,0,0)}^{(4,1,2)}(3ydx + 3xdy + 2zdz) = f(4,1,2) - f(0,0,0) = 16$