Total derivative

In mathematics, the total derivative of a function at a point is the best linear approximation near this point of the function with respect to its arguments. Unlike partial derivatives, the total derivative approximates the function with respect to all of its arguments, not just a single one. In many situations, this is the same as considering all partial derivatives simultaneously. The term "total derivative" is primarily used when is a function of several variables, because when is a function of a single variable, the total derivative is the same as the derivative of the function.
"Total derivative" is sometimes also used as a synonym for the material derivative in fluid mechanics.
The total derivative as a linear map.
Let formula_1 be an open subset. Then a function formula_2 is said to be (totally) differentiable at a point formula_3 if there exists a linear transformation formula_4 such that
The linear map formula_6 is called the (total) derivative or (total) differential of formula_7 at formula_8. Other notations for the total derivative include formula_9 and formula_10. A function is (totally) differentiable if its total derivative exists at every point in its domain.
Conceptually, the definition of the total derivative expresses the idea that formula_6 is the best linear approximation to formula_7 at the point formula_8. This can be made precise by quantifying the error in the linear approximation determined by formula_6. To do so, write
where formula_16 equals the error in the approximation. To say that the derivative of formula_7 at formula_8 is formula_6 is equivalent to the statement
where formula_21 is little-o notation and indicates that formula_16 is much smaller than formula_23 as formula_24. The total derivative formula_6 is the "unique" linear transformation for which the error term is this small, and this is the sense in which it is the best linear approximation to formula_7.
The function formula_7 is differentiable if and only if each of its components formula_28 is differentiable, so when studying total derivatives, it is often possible to work one coordinate at a time in the codomain. However, the same is not true of the coordinates in the domain. It is true that if formula_7 is differentiable at formula_8, then each partial derivative formula_31 exists at formula_8. The converse is false: It can happen that all of the partial derivatives of formula_7 at formula_8 exist, but formula_7 is not differentiable at formula_8. This means that the function is very "rough" at formula_8, to such an extreme that its behavior cannot be adequately described by its behavior in the coordinate directions. When formula_7 is not so rough, this cannot happen. More precisely, if all the partial derivatives of formula_7 at formula_8 exist and are continuous in a neighborhood of formula_8, then formula_7 is differentiable at formula_8. When this happens, then in addition, the total derivative of formula_7 is the linear transformation corresponding to the Jacobian matrix of partial derivatives at that point.
The total derivative as a differential form.
When the function under consideration is real-valued, the total derivative can be recast using differential forms. For example, suppose that formula_45 is a differentiable function of variables formula_46. The total derivative of formula_7 at formula_8 may be written in terms of its Jacobian matrix, which in this instance is a row matrix:
The linear approximation property of the total derivative implies that if
is a small vector (where the formula_51 denotes transpose, so that this vector is a column vector), then
Heuristically, this suggests that if formula_53 are infinitesimal increments in the coordinate directions, then
In fact, the notion of the infinitesimal, which is merely symbolic here, can be equipped with extensive mathematical structure. Techniques, such as the theory of differential forms, effectively give analytical and algebraic descriptions of objects like infinitesimal increments, formula_55. For instance, formula_55 may be inscribed as a linear functional on the vector space formula_57. Evaluating formula_55 at a vector formula_59 in formula_57 measures how much formula_59 points in the formula_62th coordinate direction. The total derivative formula_6 is a linear combination of linear functionals and hence is itself a linear functional. The evaluation formula_64 measures how much formula_59 points in the direction determined by formula_7 at formula_8, and this direction is the gradient. This point of view makes the total derivative an instance of the exterior derivative.
Suppose now that formula_7 is a vector-valued function, that is, formula_69. In this case, the components formula_70 of formula_7 are real-valued functions, so they have associated differential forms formula_72. The total derivative formula_73 amalgamates these forms into a single object and is therefore an instance of a vector-valued differential form.
The chain rule for total derivatives.
The chain rule has a particularly elegant statement in terms of total derivatives. It says that, for two functions formula_7 and formula_75, the total derivative of the composite formula_76 at formula_8 satisfies
If the total derivatives of formula_7 and formula_75 are identified with their Jacobian matrices, then the composite on the right-hand side is simply matrix multiplication. This is enormously useful in applications, as it makes it possible to account for essentially arbitrary dependencies among the arguments of a composite function.
Example: Differentiation with direct dependencies.
Suppose that "f" is a function of two variables, "x" and "y". If these two variables are independent, so that the domain of "f" is formula_81, then the behavior of "f" may be understood in terms of its partial derivatives in the "x" and "y" directions. However, in some situations, "x" and "y" may be dependent. For example, it might happen that "f" is constrained to a curve formula_82. In this case, we are actually interested in the behavior of the composite function formula_83. The partial derivative of "f" with respect to "x" does not give the true rate of change of "f" with respect to changing "x" because changing "x" necessarily changes "y". However, the chain rule for the total derivative takes such dependencies into account. Write formula_84. Then, the chain rule says
By expressing the total derivative using Jacobian matrices, this becomes:
Suppressing the evaluation at formula_87 for legibility, we may also write this as
This gives a straightforward formula for the derivative of formula_83 in terms of the partial derivatives of formula_7 and the derivative of formula_91.
For example, suppose
The rate of change of "f" with respect to "x" is usually the partial derivative of "f" with respect to "x"; in this case,
However, if "y" depends on "x", the partial derivative does not give the true rate of change of "f" as "x" changes because the partial derivative assumes that "y" is fixed. Suppose we are constrained to the line
Then
and the total derivative of "f" with respect to "x" is
which we see is not equal to the partial derivative formula_97. Instead of immediately substituting for "y" in terms of "x", however, we can also use the chain rule as above:
Example: Differentiation with indirect dependencies.
While one can often perform substitutions to eliminate indirect dependencies, the chain rule provides for a more efficient and general technique. Suppose formula_99 is a function of time formula_100 and formula_101 variables formula_102 which themselves depend on time. Then, the time derivative of formula_103 is
The chain rule expresses this derivative in terms of the partial derivatives of formula_103 and the time derivatives of the functions formula_102:
This expression is often used in physics for a gauge transformation of the Lagrangian, as two Lagrangians that differ only by the total time derivative of a function of time and the formula_101 generalized coordinates lead to the same equations of motion. An interesting example concerns the resolution of causality concerning the Wheeler–Feynman time-symmetric theory. The operator in brackets (in the final expression above) is also called the total derivative operator (with respect to formula_100).
For example, the total derivative of formula_110 is
Here there is no formula_112 term since formula_7 itself does not depend on the independent variable formula_100 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 formula_117 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.