### Total differential

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Calculus |
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Integral calculus |

Specialized calculi |

In the mathematical field of differential calculus, the term **total derivative** has a number of closely related meanings.

- The total derivative (full derivative) of a function $f$, of several variables, e.g., $t$, $x$, $y$, etc., with respect to one of its input variables, e.g., $t$, is different from its partial derivative ($\backslash partial$). Calculation of the total derivative of $f$ with respect to $t$ does not assume that the other arguments are constant while $t$ varies; instead, it allows the other arguments to depend on $t$. The total derivative adds in these
*indirect dependencies*to find the overall dependency of $f$ on $t$. For example, the total derivative of $f(t,x,y)$ with respect to $t$ is- $\backslash frac\{\backslash operatorname\; df\}\{\backslash operatorname\; dt\}=\backslash frac\{\backslash partial\; f\}\{\backslash partial\; t\}\; \backslash frac\{\backslash operatorname\; dt\}\{\backslash operatorname\; dt\}\; +\; \backslash frac\{\backslash partial\; f\}\{\backslash partial\; x\}\; \backslash frac\{\backslash operatorname\; dx\}\{\backslash operatorname\; dt\}\; +\; \backslash frac\{\backslash partial\; f\}\{\backslash partial\; y\}\; \backslash frac\{\backslash operatorname\; dy\}\{\backslash operatorname\; dt\}.$

- $\backslash frac\{\backslash operatorname\; df\}\{\backslash operatorname\; dt\}=\backslash frac\{\backslash partial\; f\}\{\backslash partial\; t\}\; +\; \backslash frac\{\backslash partial\; f\}\{\backslash partial\; x\}\; \backslash frac\{\backslash operatorname\; dx\}\{\backslash operatorname\; dt\}\; +\; \backslash frac\{\backslash partial\; f\}\{\backslash partial\; y\}\; \backslash frac\{\backslash operatorname\; dy\}\{\backslash operatorname\; dt\}.$

- $\{\backslash operatorname\; df\}=\backslash frac\{\backslash partial\; f\}\{\backslash partial\; t\}\backslash operatorname\; dt\; +\; \backslash frac\{\backslash partial\; f\}\{\backslash partial\; x\}\; \backslash operatorname\; dx\; +\; \backslash frac\{\backslash partial\; f\}\{\backslash partial\; y\}\; \backslash operatorname\; dy.$

It can also refer to differential operates that computes the total derivate.

- It refers to the (total) differential d
*f*of a function, either in the traditional language of infinitesimals or the modern language of differential forms.

- A differential of the form
- $\backslash sum\_\{j=1\}^k\; f\_j(x\_1,\backslash dots,\; x\_k)\; \backslash operatorname\; d\{x\_j\}$

**total differential**or an**exact differential**if it is the differential of a function. Again this can be interpreted infinitesimally, or by using differential forms and the exterior derivative.

- It is another name for the derivative as a linear map, i.e., if
*f*is a differentiable function from**R**^{n}to**R**^{m}, then the (total) derivative (or differential) of*f*at*x*∈**R**^{n}is the linear map from**R**^{n}to**R**^{m}whose matrix is the Jacobian matrix of*f*at*x*.

- It is a synonym for the gradient, which is essentially the derivative of a function from
**R**^{n}to**R**.

- It is sometimes used as a synonym for the material derivative, $\backslash frac\{D\backslash mathbf\{u\}\}\{Dt\}$, in fluid mechanics.

## Contents

## Differentiation with indirect dependencies

Suppose that *f* is a function of two variables, *x* and *y*. Normally these variables are assumed to be independent. However, in some situations they may be dependent on each other. For example *y* could be a function of *x*, constraining the domain of *f* to a curve in *R*^{2}. In this case 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*. The **total derivative** takes such dependencies into account.

For example, suppose

- $f(x,y)=xy$.

The rate of change of *f* with respect to *x* is usually the partial derivative of *f* with respect to *x*; in this case,

- $\backslash frac\{\backslash partial\; f\}\{\backslash partial\; x\}\; =\; y$.

However, if *y* depends on *x*, the partial derivative does not give the true rate of change of *f* as *x* changes because it holds *y* fixed.

Suppose we are constrained to the line

- $y=x;$

then

- $f(x,y)\; =\; f(x,x)\; =\; x^2$.

In that case, the total derivative of *f* with respect to *x* is

- $\backslash frac\{\backslash mathrm\{d\}f\}\{\backslash mathrm\{d\}x\}\; =\; 2\; x$.

Instead of immediately substituting for *y* in terms of *x*, this can be found equivalently using the chain rule:

- $\backslash frac\{\backslash mathrm\{d\}f\}\{\backslash mathrm\{d\}x\}=\; \backslash frac\{\backslash partial\; f\}\{\backslash partial\; x\}\; +\; \backslash frac\{\backslash partial\; f\}\{\backslash partial\; y\}\backslash frac\{\backslash mathrm\{d\}y\}\{\backslash mathrm\{d\}x\}\; =\; y+x\; \backslash cdot\; 1\; =\; x+x=2x.$

Notice that this is not equal to the partial derivative:

- $\backslash frac\{\backslash mathrm\{d\}f\}\{\backslash mathrm\{d\}x\}\; =\; 2\; x\; \backslash neq\; \backslash frac\{\backslash partial\; f\}\{\backslash partial\; x\}\; =\; y\; =\; x$.

While one can often perform substitutions to eliminate indirect dependencies, the chain rule provides for a more efficient and general technique. Suppose *M*(*t*, *p*_{1}, ..., *p _{n}*) is a function of time

*t*and

*n*variables $p\_i$ which themselves depend on time. Then, the total time derivative of

*M*is

- $\{\backslash operatorname\{d\}M\; \backslash over\; \backslash operatorname\{d\}t\}\; =\; \backslash frac\{\backslash operatorname\; d\}\{\backslash operatorname\; d\; t\}\; M\; \backslash bigl(t,\; p\_1(t),\; \backslash ldots,\; p\_n(t)\backslash bigr).$

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

- $\{\backslash operatorname\{d\}M\; \backslash over\; \backslash operatorname\{d\}t\}$

= \frac{\partial M}{\partial t} + \sum_{i=1}^n \frac{\partial M}{\partial p_i}\frac{\operatorname{d}p_i}{\operatorname{d}t} = \biggl(\frac{\partial}{\partial t} + \sum_{i=1}^n \frac{\operatorname{d}p_i}{\operatorname{d}t}\frac{\partial}{\partial p_i}\biggr)(M).

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. An interesting example concerns the resolution of causality concerning the Wheeler-Feynman time-symmetric theory. The operator in brackets (in the final expression) is also called the total derivative operator (with respect to *t*).

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

- $\backslash frac\{\backslash operatorname\; df\}\{\backslash operatorname\; dt\}\; =\; \{\; \backslash partial\; f\; \backslash over\; \backslash partial\; x\}\{\backslash operatorname\; dx\; \backslash over\; \backslash operatorname\; dt\; \}+\{\; \backslash partial\; f\; \backslash over\; \backslash partial\; y\}\{\backslash operatorname\; dy\; \backslash over\; \backslash operatorname\; dt\; \}.$

Here there is no ∂*f* / ∂*t* term since *f* itself does not depend on the independent variable *t* directly.

## The total derivative via differentials

Differentials provide a simple way to understand the total derivative. For instance, suppose $M(t,p\_1,\backslash dots,p\_n)$ is a function of time *t* and *n* variables $p\_i$ as in the previous section. Then, the differential of *M* is

- $\backslash operatorname\; d\; M\; =\; \backslash frac\{\backslash partial\; M\}\{\backslash partial\; t\}\; \backslash operatorname\; d\; t\; +\; \backslash sum\_\{i=1\}^n\; \backslash frac\{\backslash partial\; M\}\{\backslash partial\; p\_i\}\backslash operatorname\{d\}p\_i.$

This expression is often interpreted *heuristically* as a relation between infinitesimals. However, if the variables *t* and *p*_{j} are interpreted as functions, and $M(t,p\_1,\backslash dots,p\_n)$ is interpreted to mean the composite of *M* with these functions, then the above expression makes perfect sense as an equality of differential 1-forms, and is immediate from the chain rule for the exterior derivative. The advantage of this point of view is that it takes into account arbitrary dependencies between the variables. For example, if $p\_1^2=p\_2\; p\_3$ then $2p\_1\backslash operatorname\; dp\_1=p\_3\; \backslash operatorname\; d\; p\_2+p\_2\backslash operatorname\; d\; p\_3$. In particular, if the variables *p*_{j} are all functions of *t*, as in the previous section, then

- $\backslash operatorname\; d\; M$

= \frac{\partial M}{\partial t} \operatorname d t + \sum_{i=1}^n \frac{\partial M}{\partial p_i}\frac{\partial p_i}{\partial t}\,\operatorname d t.

## The total derivative as a linear map

Let $U\backslash subseteq\; \backslash mathbb\{R\}^\{n\}$ be an open subset. Then a function $f:U\backslash rightarrow\; \backslash mathbb\{R\}^m$ is said to be (**totally**) **differentiable** at a point $p\backslash in\; U$, if there exists a linear map $\backslash operatorname\; df\_p:\backslash mathbb\{R\}^n\; \backslash rightarrow\; \backslash mathbb\{R\}^m$ (also denoted D_{p}*f* or D*f*(p)) such that

- $\backslash lim\_\{x\backslash rightarrow\; p\}\backslash frac\{\backslash |f(x)-f(p)-\backslash operatorname\; df\_p(x-p)\backslash |\}\{\backslash |x-p\backslash |\}=0.$

The linear map $\backslash operatorname\; d\; f\_p$ is called the (**total**) **derivative** or (**total**) **differential** of $f$ at $p$. A function is (**totally**) **differentiable** if its total derivative exists at every point in its domain.

Note that *f* is differentiable if and only if each of its components $f\_i:U\backslash rightarrow\; \backslash mathbb\{R\}$ is differentiable. For this it is necessary, but not sufficient, that the partial derivatives of each function *f*_{j} exist. However, if these partial derivatives exist and are continuous, then *f* is differentiable and its differential at any point is the linear map determined by the Jacobian matrix of partial derivatives at that point.

## Total differential equation

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*.

## Application of the total differential to error estimation

In measurement, the total differential is used in estimating the error Δ*f* of a function *f* based on the errors Δ*x*, Δ*y*, ... of the parameters *x, y, ...*. Assuming that the interval is short enough for the change to be approximately linear:

- Δ
*f*(*x*) =*f'*(*x*) × Δ*x*

and that all variables are independent, then for all variables,

- $\backslash Delta\; f\; =\; f\_x\; \backslash Delta\; x\; +\; f\_y\; \backslash Delta\; y\; +\; \backslash cdots$

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 Δ*x*. As they are assumed to be independent, the analysis describes the worst-case scenario. The absolute values of the component errors are used, because after simple computation, the derivative may have a negative sign. From this principle the error rules of summation, multiplication etc. are derived, e.g.:

- Let f(
*a*,*b*) =*a*×*b*;

- Δ
*f*=*f*_{a}Δ*a*+*f*_{b}Δ*b*; evaluating the derivatives

- Δ
*f*=*b*Δ*a*+*a*Δ*b*; dividing by*f*, which is*a*×*b*

- Δ
*f*/*f*= Δ*a*/*a*+ Δ*b*/*b*

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

## 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. ISBN 1-58488-297-2

- From thesaurus.maths.org total derivative

## External links

- MathWorld.
- http://www.sv.vt.edu/classes/ESM4714/methods/df2D.htmlja:偏微分#全微分