Partial derivative

In the differential calculus a partial derivative is the derivative of a function with several arguments by any of these arguments ( in the direction of coordinate axis ). The values ​​of the remaining arguments are so held.

• 3.1 related derivative, partial derivative, continuity
• 3.2 Set of black

• 5.1 Example 1
• 5.2 Example 2
• 5.3 Partial and total time derivative

Definition

1st order

Be an open subset of Euclidean space, and a function. Be still obtain an element in. If the natural number with the following limit exists:

Then it is called the partial derivative of after -th variable in point. The function is then called differentiable at the point. The symbol ∂ ( it resembles the italicized section of the Cyrillic minuscule д ) is pronounced as d or even to distinguish del. The notation was known by the use of CGJ Jacobi.

Higher order

The partial derivative with respect to is itself a function from to, if is differentiable in all the way. As a shorthand notation for the partial derivatives is also often, or find.

If the function at each point of its domain differentiable, so are the partial derivatives

Again in accordance with features of which can be tested in turn on differentiability. This gives higher partial derivatives

Geometric interpretation

In a three-dimensional coordinate system of the function graph of a function is considered. The domain is an open subset of the xy plane. Is differentiable, then the graph of the function is a surface over the domain.

For a fixed value of then is a function in. For fixed points would stretch parallel to the axis. This distance is projected by a curved line on the graph of FIG. The partial derivative of after under these conditions corresponds to the slope of the tangent to this curve at the point.

Sets and properties

Related derivative, partial derivative, continuity

• Total differentiable functions are continuous.
• Total differentiable functions are differentiable.
• Partially differentiable functions are not necessarily continuous, then also not totally differentiable.
• Continuously partial differentiable functions, ie functions whose partial derivatives are continuous, however, are ever totally differentiable.

Set of black

• It is the set of Black: When the second partial derivatives are continuous, so you can change the order of the derivative:

Use

• The first partial derivatives can be arranged in a vector, the gradient of:

• The second partial derivatives can be arranged in a matrix, the Hessian matrix
• It is the Taylor formula: If the function - times continuously differentiable, it can be approximated in the vicinity of each point by its Taylor polynomials:

• Easy extreme value problems can be found in the Analysis in the calculation of maxima and minima of a function of a real variable (see the article on the differential calculus ). The generalization of the derivative functions of several variables ( variables, parameters) allows the determination of their extreme values ​​, and for calculating the partial derivatives are needed.

• In differential geometry one needs partial derivatives to determine a total differential. Applications for total differentials are found largely in thermodynamics.

• Partial derivatives are also an essential component of vector analysis. Forming the components of the gradient, the Laplacian, the divergence and rotation in the scalar and vector fields. They occur also in the Jacobian matrix.

Examples

Example 1

As an example The function is considered, which depends on the two variables.

Is considered as a constant, for example, the function depends only on the variable with:

Therefore applies to the new feature and you can form the differential quotient

The same result is obtained when forming the partial derivative of the function by:

The partial derivative of after is accordingly:

This example demonstrates how the partial derivative of a function is determined that is dependent on several variables:

Up to a variable, all other variables are assumed to be constant with respect to these variables of a differential quotient is determined. As a result, we obtain the partial derivative of the function by this one variable.

Example 2

Since the partial derivative with respect to a variable of the ordinary derivation corresponds to held values ​​of all other variables, all rules of inference can be used for the calculation as for functions of one variable. For example, if

As follows with product and chain rule:

Partial and total time derivative

In physics (especially in theoretical mechanics) often occurs in the following situation: One size depends by a totally differentiable function of the spatial coordinates, and the time. One can thus make the partial derivatives, and. The coordinates of the moving point are given by the functions, and. The development over time of the value of the variable at the respective path point is then linked with the function

Described. This function depends only on one variable, namely time. We can therefore form the ordinary derivative. This is called the total or total derivative of with respect to time and also writes for short. It is calculated according to the multidimensional chain rule as follows:

While in the partial derivative with respect to time, only the explicit dependence of the function is taken into account and all other variables are held constant, taken into account the total dissipation and the indirect (or implicit), depending on which this does, that along the path of movement of the location coordinates dependent on time.

( Thus, by taken into account the implicit time dependence, we speak in the jargon of physics by " substantial " time derivative, or, in the jargon of the fluid mechanics of the Euler - Lagrange derivative in contrast to the derivation. )

→ For a more detailed description see Totales differential

Generalization: directional derivative

A generalization of the partial discharge, the directional derivative dar. The derivative is considered in the direction of any vector, not only in the direction of the coordinate axes.