﻿ Symmetry of second derivatives

# Symmetry of second derivatives

The set of black (after Hermann Amandus Schwarz, is also set of Clairaut called ) is a set of mathematics in the differential calculus of several variables. He says that at several times continuously differentiable functions of several variables, the order in which the partial differentiations (outlets ) can be performed by each variable, is not critical to the outcome.

In fact, he has also headed forth from the existence of, for example, the partial first derivatives and a partial second derivative of the existence and value of another partial second derivative.

The set of black is not to be confused with the Schwarz lemma.

## Statement

Be an open set and at least p times differentiable and all p - th derivatives in U are still at least continuous, then the order of differentiation in all q-th partial derivatives with negligible.

Especially for and therefore applies

The sentence is true even under slightly weaker assumptions: It is sufficient that the first partial derivatives in the considered point are totally differentiable.

### Other spellings

Possible without parentheses are

When the partial differentiation itself as a representation of gradually perceives from to, you can even shorter write:

## Other formulations

The set of black says also that the Hessian matrix is symmetric.

Summing as differentiable 0 - form and writes for the exterior derivative, the rate of black has the shape or even just.

For can also express this as follows: The rotation of the Gradientenvektorfelds is equal to zero: , or written with nabla symbol: . The Gradientenvektorfeld is therefore irrotational.

## Example

Consider the function by table results for the first partial derivatives

And the two second partial derivatives, and

It will be appreciated that the following applies

## Counterexample

Without the continuity of the second derivatives of the sentence actually not true in general. A counter- example in which the commutativity does not hold, the function is available with or

With this function, the second partial derivatives exist at all, but it is

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