Total derivative

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In mathematics, a total derivative is a combination of partial derivatives. Specifically, it may mean either of the following:

a derivative which takes indirect dependencies into account;
a differential operator involving the sum of all the partial derivatives with respect to all variables.

Derivative taking indirect dependencies into account

Suppose, for instance, that [M(p_1,\dots,p_n, t)] is a function of time t and n variables [p_i] which themselves depend on time. Then, the total time derivative of M is

[ M \over \mathrmt} = \frac M \big( p_1(t), \ldots, p_n(t), t \big). ]

The chain rule for differentiating a function of several variables implies that

[ M \over \mathrmt} = \sum_^n \left (p_i \over \mathrmt} \right) + = \sum_^n \left (\dot \right) + .]

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 n generalized coordinates lead to the same equations of motion.

For example, the total derivative of f(x(t), y(t)) is

[ = +.]

There is no [] term since f itself does not depend on an independent variable t directly.

Sum of all partial derivatives

Additionally, total derivative may mean either

a differential operator involving the sum of all the partial derivatives with respect to all variables in a problem, or be used compatibly
to express the exterior derivative d, as applied to differential forms, and in particular as applied to a function F considered as a 0-form, so that

dF(x₁, x₂, ..., x_n) = Σ F_idx_i,

where F_i is the partial derivative with respect to x_i.

A total differential equation is a differential equation expressed in terms of total derivatives. Since the exterior derivative is a natural operator, in a sense that can be given a technical meaning, such equations are intrinsic and geometric.

In measurement, the total differential is used in estimating the error Δf of a function f based on the errors Δx, Δy, ... of the parameters x, y, ...:

Δf = f_x |Δx| + f_y |Δy| +...

This is because the derivative f_x with respect to the particular parameter x gives the sensitivity of the function f to a change in x, in particular the error Δx. As they are assumed to be independent, the analysis describes the worst-case scenario. From this principle the error rules of summation, multiplication etc. are derived, e.g.:

Let f(a,b) = a × b;

Δf = f_a|Δa| + f_b|Δb|; evaluating the derivatives

Δf = b|Δa| + a|Δb|; dividing by f, which is a × b

Δf/f = |Δa|/a + |Δb|/b

That is to say, in multiplication, the total relative error is the sum of the relative errors of the parameters.

Bibliography

A. D. Polyanin and V. F. Zaitsev, Handbook of Exact Solutions for Ordinary Differential Equations (2nd edition), Chapman & Hall/CRC Press, Boca Raton, 2003. ISBN 1-58488-297-2

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