Suppose the function
represents some physical quantity, such as temperature, in a region of the
-plane. Then the level curves of F, where
, could be interpreted as isotherms on a weather map (i.e curves on a weather map representing constant temperatures). Along one of these curves,
, of constant temperature we have, by Chain rule and the fact that the temperature, F, is constant on these curves:
Multiplying through by
we obtain
Therefore, if we were not given the original function F but only an equation of the form:
we could set
and then by integrating figure out the original
.
(1) First ensure that there is such an
, by checking the exactness-condition:
This is because if there was such an F, then
where
and
simply denote the partial derivatives with respect to the variables
and
respectively (where we hold the other variable constant while taking the derivative).
(2)Second, integrate
with respect to
respectively:
for some unknown functions
(these play the role of constant of integration when you integrate with respect to a single variable). So to obtain
it remains to determine either
or
.
(3)Equate the above two formulas for
:
(4) Since to find
it suffices to determine
or
, pick the integral that is easier to evaluate. Suppose that
is easier to evaluate. To obtain
we differentiate both expression for
in
(for fixed
):
and then integrate in
:
(5)Observe that
is only a function of
since if we differentiate the expression we found for
and use step
we find that