Set of black

The set of black (by Hermann Schwarz ; is also set of Clairaut called; or Young -Theorem ) is a set of the mathematics in the differential calculus of several variables. It says that in the case of multiple, continuously differentiable functions of several variables, the order in which the partial differentiations (derivatives) are carried out according to the individual variables is not decisive for the result.

In fact, he also derives the existence and the value of a further partial second derivative from the existence of the partial first derivatives, for example, and a partial second derivative.

Schwarz's theorem should not be confused with Schwarz's lemma .

statement

If an open set as well as at least -times are partially differentiable and if all -th partial derivatives are at least still continuous , then -time is totally differentiable and in particular the order of the differentiation in all -th partial derivatives is irrelevant. ${\ displaystyle U \ subseteq \ mathbb {R} ^ {n}}$${\ displaystyle f \ colon U \ to \ mathbb {R}}$${\ displaystyle k}$${\ displaystyle k}$${\ displaystyle U}$${\ displaystyle f}$ ${\ displaystyle k}$${\ displaystyle l}$${\ displaystyle l \ leq k}$

In particular for and is therefore true ${\ displaystyle n = 2}$${\ displaystyle k \ geq 2}$

${\ displaystyle {\ frac {\ partial} {\ partial x}} \ left ({\ frac {\ partial} {\ partial y}} f (x, y) \ right) = {\ frac {\ partial} { \ partial y}} \ left ({\ frac {\ partial} {\ partial x}} f (x, y) \ right).}$

The theorem already applies under slightly weaker conditions: It is sufficient that the first partial derivatives in the point under consideration are totally differentiable . To be precise, the following geometric formulation of the theorem also applies, for example: If and are Banach spaces over a commutative field or and is an open subset of on which a function is defined that is twice (totally) differentiable at the point , then is the second derivative of there, which by definition an element of , so a self-on is bilinear and continuous function, symmetrical: that is, for all true ${\ displaystyle E}$${\ displaystyle F}$${\ displaystyle \ mathbb {K} = \ mathbb {R}}$${\ displaystyle \ mathbb {K} = \ mathbb {C}}$${\ displaystyle U}$${\ displaystyle E}$${\ displaystyle f: U \ to F}$${\ displaystyle a \ in U}$${\ displaystyle f ^ {\ prime \ prime}}$${\ displaystyle f}$${\ displaystyle L (E; L (E; F)) \ simeq L (E, E; F)}$${\ displaystyle E \ times E}$${\ displaystyle h, k \ in E}$

${\ displaystyle (f ^ {\ prime \ prime} (a) \ cdot h) \ cdot k = (f ^ {\ prime \ prime} (a) \ cdot k) \ cdot h.}$

If it breaks down into a finite, -fold product of Banach spaces , that is, if the norm of is compatible with the product topology, then the existence and symmetry of both the existence of the second partial derivatives with and for in the point , the elements by definition of are as well as their symmetry by interchanging the variables and arguments. That is, for everyone and applies ${\ displaystyle E}$${\ displaystyle n}$${\ displaystyle E_ {i}}$${\ displaystyle E = E_ {1} \ times \ cdots \ times E_ {n}}$${\ displaystyle E}$${\ displaystyle f ^ {\ prime \ prime} (a)}$${\ displaystyle {\ frac {\ partial ^ {2} f} {\ partial x_ {i} \ partial x_ {j}}}}$${\ displaystyle x_ {i} \ in E_ {i}}$${\ displaystyle x_ {j} \ in E_ {j}}$${\ displaystyle i, j = 1, \ ldots, n}$${\ displaystyle a \ in U}$${\ displaystyle L (E_ {i}; L (E_ {j}; F)) \ simeq L (E_ {i}, E_ {j}; F)}$${\ displaystyle k_ {i} \ in E_ {i}}$${\ displaystyle h_ {j} \ in E_ {j}}$

${\ displaystyle \ left ({\ frac {\ partial ^ {2} f} {\ partial x_ {i} \ partial x_ {j}}} (a) \ cdot k_ {i} \ right) \ cdot h_ {j } = \ left ({\ frac {\ partial ^ {2} f} {\ partial x_ {j} \ partial x_ {i}}} (a) \ cdot h_ {j} \ right) \ cdot k_ {i} }$.

Notes: As is well known, the continuity of all 2nd partial derivatives follows . But this is not a prerequisite for the sentence. The classic formulation corresponds to the special case and , since all standards are equivalent to (and ), these are automatically compatible with the product topology, so that this requirement is then not applicable. The geometric formulation generalizes the classical to not necessarily finite-dimensional, real or complex Banach spaces and . Without their arguments and the given formula would be wrong in general, because and act on different spaces. So the Banach spaces and even if they are finite dimensional, could be of different dimensions. Since the multilinear mappings on products of Banach spaces (with the operator norm) themselves form Banach spaces again, the (complete) symmetry is transferred (by complete induction) to all higher derivatives, so that the arbitrary interchangeability of the partial derivatives in this sense up to and including the order of differentiability (the function at this point) applies. ${\ displaystyle f ^ {\ prime \ prime}}$${\ displaystyle E = \ mathbb {R} ^ {n}}$${\ displaystyle E_ {i} = F = \ mathbb {R}}$${\ displaystyle \ mathbb {R} ^ {n}}$${\ displaystyle \ mathbb {C} ^ {n}}$${\ displaystyle E_ {i}}$${\ displaystyle F}$${\ displaystyle k_ {i}}$${\ displaystyle h_ {j}}$${\ displaystyle {\ frac {\ partial ^ {2} f} {\ partial x_ {i} \ partial x_ {j}}} (a)}$${\ displaystyle {\ frac {\ partial ^ {2} f} {\ partial x_ {j} \ partial x_ {i}}} (a)}$${\ displaystyle E_ {i}}$${\ displaystyle E_ {j}}$

Other spellings

Possible spellings without brackets are

${\ displaystyle {\ frac {\ partial ^ {2} f} {\ partial x \ partial y}} (x, y) = {\ frac {\ partial ^ {2} f} {\ partial y \ partial x} } (x, y)}$or also .${\ displaystyle {f_ {xy} \; = \; f_ {yx}}}$

If one understands the partial differentiation itself as a mapping from to and from to , one can write even more briefly: ${\ displaystyle C ^ {2} (U, \ mathbb {R})}$${\ displaystyle C ^ {1} (U, \ mathbb {R})}$${\ displaystyle C ^ {1} (U, \ mathbb {R})}$${\ displaystyle C ^ {0} (U, \ mathbb {R})}$

${\ displaystyle {\ frac {\ partial ^ {2}} {\ partial x \ partial y}} = {\ frac {\ partial ^ {2}} {\ partial y \ partial x}}}$or also .${\ displaystyle \ partial _ {1} \ partial _ {2} = \ partial _ {2} \ partial _ {1}}$

Other formulations

Schwarz's theorem also states that the Hessian matrix is symmetric .

If one understands as a differentiable 0-form and writes for the outer derivation , then the theorem of Schwarz has the form or simply only . ${\ displaystyle f \ in C ^ {2} (U, \ mathbb {R})}$${\ displaystyle d}$${\ displaystyle d (df) = 0}$${\ displaystyle dd = 0}$

For can also express this as follows: The rotation of Gradientenvektorfelds is zero: or with Nabla written symbol: . The gradient vector field is therefore free of eddies . ${\ displaystyle U \ subseteq \ mathbb {R} ^ {3}}$${\ displaystyle \ operatorname {red} (\ operatorname {grad} f) = 0}$${\ displaystyle {\ vec {\ nabla}} \ times {\ vec {\ nabla}} f = {\ vec {0}}}$

example

The function is given by It results for the first partial derivatives ${\ displaystyle f \ colon \ mathbb {R} ^ {2} \ to \ mathbb {R}}$${\ displaystyle f (x, y) = e ^ {x ^ {2}} \ sin {y}.}$

${\ displaystyle f_ {x} = 2xe ^ {x ^ {2}} \ sin {y} \ qquad f_ {y} = e ^ {x ^ {2}} \ cos {y}}$

and for the two second partial derivatives and${\ displaystyle f_ {yx}}$${\ displaystyle f_ {xy}}$

${\ displaystyle f_ {yx} = 2xe ^ {x ^ {2}} \ cos {y} \ qquad f_ {xy} = 2xe ^ {x ^ {2}} \ cos {y}.}$

It can be seen that it is true ${\ displaystyle f_ {xy} = f_ {yx}.}$

Counterexample

Without the continuity of the second derivatives, the theorem does not hold in general. A counterexample, in which the interchangeability does not apply, is the function with and ${\ displaystyle f \ colon \ mathbb {R} ^ {2} \ to \ mathbb {R}}$${\ displaystyle f (0,0) = 0}$

${\ displaystyle f (x, y) = {\ frac {x ^ {3} y-xy ^ {3}} {x ^ {2} + y ^ {2}}}}$for .${\ displaystyle (x, y) \ neq (0,0)}$

With this function, the second partial derivatives exist entirely , but it holds ${\ displaystyle \ mathbb {R} ^ {2}}$

${\ displaystyle {\ frac {\ partial ^ {2}} {\ partial x \ partial y}} f (0,0) = 1}$and .${\ displaystyle \ displaystyle {\ frac {\ partial ^ {2}} {\ partial y \ partial x}} f (0,0) = - 1}$

Relation to exact differential equations

A differential equation of the form is given

${\ displaystyle a (x, y) + b (x, y) \ cdot y '= 0}$.

This is called exact if there is a continuously partially differentiable function such that : ${\ displaystyle \ Phi \ colon U \ to \ mathbb {R}}$${\ displaystyle (x, y) \ in U \ subseteq \ mathbb {R} ^ {2}}$

${\ displaystyle {\ frac {\ partial} {\ partial y}} \ Phi (x, y) = b (x, y)}$and .${\ displaystyle {\ frac {\ partial} {\ partial x}} \ Phi (x, y) = a (x, y)}$

If and are continuously partially differentiable functions , then, according to Schwarz's theorem, a necessary condition for this is that ${\ displaystyle a}$${\ displaystyle b}$${\ displaystyle U}$

${\ displaystyle {\ frac {\ partial} {\ partial y}} a (x, y) = {\ frac {\ partial} {\ partial x}} b (x, y) \ qquad (*)}$

applies.

If the open set is simply connected , then the condition also implies the existence of (e.g. for star-shaped sets this follows from the Poincaré lemma ). ${\ displaystyle U \ subset \ mathbb {R} ^ {2}}$ ${\ displaystyle (*)}$${\ displaystyle \ Phi}$ ${\ displaystyle U}$