This concept extends the single-variable differential to functions of two independent variables, defining the total differential DW = F_x·DX + F_y·DY as the linear approximation to a continuously differentiable function W = f(x,y), and generalizes any expression M(x,y)DX + N(x,y)DY as a differential. It introduces the concept of an exact differential — one for which a potential function f exists such that f_x = M and f_y = N — and establishes the necessary and sufficient condition (given continuity of the partials) that M_y = N_x, connecting multivariable calculus to the theory of differential equations. The domain is multivariable calculus, specifically the theory of partial derivatives and exactness conditions that underlie first-order exact differential equations.
This concept extends the single-variable differential to functions of two independent variables, defining the total differential DW = F_x·DX + F_y·DY as the linear approximation to a continuously dif…