Total derivative

The total differentiability is a fundamental property of functions between finite dimensional vector spaces over the mathematical subfield of Analysis. By means of this property can be many more for the Analysis meaningful statements about functions show that are not valid when using the weaker partial differentiability, which is formally similar to the usual definition of differentiability of a real function as a convergence of the difference quotient. Therefore, many more concepts of analysis are based on the total differentiability.

The total differentiability of a function at a point means that these there locally by a linear transformation to approximate (zoom ) leaves, while the partial differentiability ( in all directions) only the local approximability by straight lines in all directions, but not linear as a single Figure calls.

While the derivative of a function is usually interpreted as a number at a point, one combines in the derivative of the higher dimensional case as precisely those local linear approximation. This is a linear mapping can be represented by a matrix, the derivative matrix, or Jacobian matrix called the fundamental matrix ( in the one-dimensional case is obtained in turn a 1 × 1 matrix, ie a single number ). In the one-dimensional case agree the classic real, the total and the partial Differenzierbarkeitsbegriff.

The concept of Fréchet differentiability generalizes the total differentiability to infinite-dimensional spaces, it takes over the property of the derivative as a local linear approximation.

Motivation / Introduction

Functions for deriving the location in the rule is carried

Defined, with or. In this form, you can not transfer to the definition pictures, because you can not divide by. Therefore, we follow a different path.

Describes the derivation of the slope of the tangent at the point in the function graph. The tangent itself has the equation

So it's the graph of the linear ( affine ) function

This function approximates the function in the following sense:

Or ( with so )

Where the error term is for faster towards 0 as, ie

In this form, can be transferred to images of the concept of differentiability. In this case, a vector in a vector in and a linear mapping from to.

Definition

Given an open subset, a point and a picture. The mapping is called the point ( totally ) differentiable if a linear map

Exists, the figure shows the

Approximated, that is, for the " error function ,"

Applies

It refers to a vector in. The dual modulus signs denote a normal vector in or. As in all or standards are equivalent, it does not matter which standard is chosen.