Example
If F(x,y)=(7+2xy)i+(x^(2)-9y^(2))j, find a function f such that F=gradf
Solution
From a previous example, we know that F is conservative, and so there exists a function f with gradf=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)=,+g(y).
(4) f_(y)(x,y)=
+g^(')(y)
Comparing (2) and (4) we see that g^(')(y)=
Integrating with respect to y, using C for the constant of integration, we have g(y)=
f(x,y)=
as the desired potential function.
+C. Putting this in (3), we have
+C. Putting this in (3), we have
Example If F(x, y) = (7 + 2xy) i + (2 9y2)j, find a function f such that F = Vf.
Solution
From a previous example, we know that F is conservative, and so there exists a function f with Vf = F, that is
1fxx,y=7+2xy 2 fy(x,y) = 2 - 9y2.
Integrating (1) with respect to x, letting g(y) be the constant of integration, we obtain
3fx,y=
+ 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.
4fyx,y=
+ g(y)
Comparing (2) and (4) we see that g(y)=
Integrating with respect to y, using C for the constant of integration, we have g(y) =
+ C.Putting this in (3, we have
f(x,y) =
as the desired potential function.