final expression above) is also called the total derivative operator (with respect to ). For example, the total derivative of is Here there is no term since itself does not depend on the independent variable directly. Total differential equation A total differential equation is a differential equation expressed in terms of total derivatives. Since the exterior derivative is coordinate-free, in a sense that can be given a technical meaning, such equations are intrinsic and geometric. Application to equation systems In economics, it is common for the total derivative to arise in the context of a system of equations. For example, a simple supply-demand system might specify the quantity q of a product demanded as a function D of its price p and consumers' income I, the latter being an exogenous variable, and might specify the quantity supplied by producers as a function S of its price and two exogenous resource cost variables r and w. The resulting system of equations determines the market equilibrium values of the variables p and q. The total derivative of p with respect to r, for example, gives the sign and magnitude of the reaction of the market price to the exogenous variable r. In the indicated system, there are a total of six possible total derivatives, also known in this context as comparative static derivatives: , , , , , and . The total derivatives are found by totally differentiating the system of equations, dividing through by, say , treating and as the unknowns, setting , and solving the two totally differentiated equations simultaneously, typically by using Cramer's rule. Application to continuity equation See also - generalization of the total derivative References A. D. Polyanin and V. F. Zaitsev, Handbook of Exact Solutions for Ordinary Differential Equations (2nd edition), Chapman & Hall/CRC Press, Boca Raton, 2003. From thesaurus.maths.org total derivative External links Ronald D. Kriz (2007) Envisioning total derivatives of scalar functions of two dimensions using raised surfaces and tangent planes from Virginia Tech Differential calculus Differential operators Lagrangian mechanics Mathematical analysis Multivariable calculus