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