Total differential

The total differential (also complete differential ) is a term from the differential calculus and called the differential of a function, especially in functions of several variables. Given a totally differentiable function is called with the total differential, for example:

This is an open subset of the real vector space, or more generally a differentiable manifold. To distinguish between full and partial differentials, different symbols are used here: a "non- italic d" the total differential and a " cursive d" () for the partial derivatives. Note that in the following always total differentiation of the function is required, and not only the existence of the partial derivatives, represented by the above formula.

Traditionally, and still today, often in the natural sciences and economics, is meant by a differential as infinitesimal differences. In today's mathematics is understood as differential forms (more precisely, 1- forms). This can either be regarded as purely formal terms, or as linear maps. The differential of a function at the point is then the linear mapping ( linear form ) that each vector the directional derivative of at the point in the direction of associates. With this meaning, the (total ) differential is also called the total derivative. This importance can be the term on pictures with values ​​in, generalize in a different vector space or in a manifold.

• 2.1 real vector space
• 2.2 manifold
• 2.3 Chain Rule

• 4.1 Fundamental Theorem of Calculus

Simple case

For a function of two independent variables is meant by the total differential expression

Total differential is, the expression, because it contains all the information on the derivative, while the partial derivatives only contain information on the derivative in the direction of the coordinate axes. The summands and are sometimes called partial differentials.

Application (concatenation)

Hanging and (for example, when describing the path of a point in the plane as a function of time) of a size from, that are functions and, so, the discharge of the composite function

Are calculated as follows:

The derivatives of and can be used as and write. It puts this into the above expression and thus receives

Or in the usual in physics notation

So

Formally, therefore, the total differential is simply divided by. Mathematically, this is an application of the multi-dimensional chain rule ( see below).

Deviation from the intended use of the terms partial and total derivative in physics

In the mechanism, typically situations are handled, in which the function not only depends on the location coordinates and, but also on time. As above, the case is considered that the location coordinates and a moving point, are. In this situation, the composite function depends

In two ways on the time:

Now This is called the partial derivative of with respect to time, if you think the partial derivative of the first function, so

In fixed and. So here only the explicit time dependence is taken into account.

In contrast, it is called the total derivative of with respect to time, if you think the derivative of the composite function, ie

The two are related as follows:

So here the explicit and the implicit time dependence are considered.

An example from fluid mechanics: With 'll denotes the temperature at the time at the place. Then describes the partial derivative of the time of temperature change at a fixed location. The temperature change experienced by a moving with the flow of particles, but also is dependent upon the change in location. The total derivative of the temperature can be then as above described with the help of the total differential:

Or

The total differential as a linear map

Real vector space

In the event that an open subset of the real vector space and is a differentiable function from to, is at any point, the total differential of a linear mapping that assigns to each vector the directional derivative in the direction of this vector, ie:

Since the total differential is a linear mapping after, so a linear form, it can be written in the following form

Wherein the linear shape, a vector that associates his -th component, i.e., ( dual basis ).

With the help of the gradient can be the total differential also be written as follows:

Where on the right side of the dot stands.

Plurality

For the general case is at any point, the total differential of a linear mapping that maps each tangential direction the directional derivative in that direction. If the tangent vector of a curve in with, so is

The total differential is thus an element of Kotangentialraums of the point.

For an account of in coordinates, consider a map a neighborhood of the point with. With the standard basis is denoted the. The different curves represent a basis of the tangent space and by

We obtain the partial derivatives. Similar to the real vector space then applies

Wherein the total differential of the function, that is the element of the cotangent, which is dual to the base vector.

Considering tangent vectors as derivations, the following applies.

Chain rule

If a differentiable function and is a differentiable path (for example, the description of a moving point ), then for the concatenated function derivative:

The analogous statement is true for manifolds.

Differential and linear approximation

The derivative of a totally differentiable function at the point is a linear mapping (function), the function of the

Approximated, ie

For small changes.

In modern mathematics is called the ( total ) differential of the point just this function ( see above). The terms " total differential " and " total derivative " are thus equivalent. The representation

So an equation between functions. The differentials are functions, namely the coordinate functions that map the vector, the ith component: The Approximierungseigenschaft thus written as

In the traditional, common in many science perspective, the differentials for the small changes itself, the total differential of are then available for the value of said linear mapping, and the approximation property written as

Or:

Examples of this view show the picture on the left and the image above.

Integrability

Each total differential is a shape, that is has the following representation

In the calculus of differential forms the Cartan derivative is described as the following form:

Is it really at a total differential of a function, that is true, then

According to the set of black.

Local also always applies the inverse: Meets the 1-form the condition, then there exists at least in a neighborhood of any given point of a primitive function, that is, a differentiable function, then that.

This is called the condition therefore integrability. Formulated in detail it is:

Or:

Which is referred to in terms of physical applications as a generalized rotation condition.

In many cases, then there is even a global primitive and is actually a total differential. This is for example the case when the domain of the differential form of the Euclidean space, or more generally when it is in a star shape, or simply connected.

The statement that, on a manifold has any 1-form that satisfies the integrability condition a primitive function (ie, a total differential is ), is equivalent to the fact that the first de Rham cohomology group is trivial.

Main theorem of differential and integral calculus

Looking at and any form. Then dimensional reasons over and the integrability condition is satisfied for valid. Thus, there is a function that satisfies the equation or. This is precisely the main theorem of differential and integral calculus for functions of one variable.

Generalizations

Quite analogously (in principle component-wise ) can be defined as the total derivative for vector-valued functions. As a generalization for pictures in a differentiable manifold is obtained push forwards.

In the functional analysis one can generalize the concept of total derivative in an obvious way for Fréchet derivatives in the calculus of variations for the so-called variation leads.