Example
If F(x, y) = (7 + 2xy) i + (x² - 9y²)j, find a function f such that F = ?f.
Solution
From a previous example, we know that F is conservative, and so there exists a function f with ?f = F, that is
(1) $f_x(x, y) = 7 + 2xy$
(2) $f_y(x, y) = x^2 - 9y^2$.
Integrating (1) with respect to x, letting g(y) be the constant of integration, we obtain
(3) f(x, y) = $7x + xy^2$ + g(y).
Notice that the constant of integration is a constant with respect to x, that is, a function of y, which we have called g(y). Next we differentiate both sides of (3) with respect to y.
(4) $f_y(x, y) = 2xy$ + g'(y)
Comparing (2) and (4) we see that g'(y) = $-9y^2$
Integrating with respect to y, using C for the constant of integration, we have g(y) = $-3y^3$ + C. Putting this in (3), we have
f(x, y) = $7x + xy^2 - 3y^3 + C$
as the desired potential function.
