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