Mixed Hiddenap Calculus

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  • 1Definition
  • 2Domain considerations

Definition

For a function of two variables

Suppose is a function of two variables which we denote and . There are two possible second-order mixed partial derivative functions for , namely and . In most ordinary situations, these are equal by Clairaut's theorem on equality of mixed partials. Technically, however, they are defined somewhat differently.

Often the term mixed partial is used as shorthand for the second-order mixed partial derivative. However, mixed partial may also refer more generally to a higher partial derivative that involves differentiation with respect to multiple variables.

The following are all multiple equivalent notations and definitions of .

Name Notation Definition in terms of first-order partial derivatives
Subscript notation Defined as . More explicitly:
Let . Then, .
Leibniz notation Defined as

Note that the order in which we write the and is different in the subscript and Leibniz notations because in the subscript notation, the differentiations are carried out from left to right (on subscripts) whereas in the Leibniz notation, the differentiations are carried out from right to left while simplifying.

Solver

The following are all multiple equivalent notations and definitions of .

Name Notation Definition in terms of first-order partial derivatives
subscript notation Defined as . More explicitly:
Let . Then, .
Leibniz notation Defined as

For a function of many variables

For a function of more than two variables, we can define the second-order mixed partial derivative with respect to two of the variables (in a particular order) in the same manner as for a function of two variables, where we treat the remaining variables as constant. For instance, for a function of three variables , we can consider the six mixed partials (holding fixed), (holding fixed), (holding fixed).

In general, for a function of variables, there are many second-order mixed partials that we can construct.

Definition as a double limit at a point

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We consider again the case of a function of two variables. In this case, the partial derivatives and at a point can be expressed as double limits:

We now use that:

and:

Plugging (2) and (3) back into (1), we obtain that:

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A similar calculation yields that:

As Clairaut's theorem on equality of mixed partials shows, we can, under reasonable assumptions of existence and continuity, show that these two second-order mixed partials are the same.

Domain considerations

For a function of two variables

Suppose is a function of two variables . Consider a point in the domain of . Suppose we are interested in determining whether exists. We can say the following:

  • A necessary (though not sufficient) condition for to exist is that exist for everywhere in an open interval containing . In other words, exists at and near on the line . Another way of saying this is that needs to exist not only at the point, but also if we perturb slightly.
  • Building on this, a necessary (though not sufficient) condition for to exist is that should exist if we perturb a bit and then perturb a bit. It's tempting to believe that it is necessary that should be defined in an open neighborhood of the point . However, this is not necessarily the case because it is not necessary that there exist a positive lower bound on the radius of the -neighborhoods for definition of for close to .

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For a function of more than two variables

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Suppose is a function of variables . Consider a point in the domain of . Consider the mixed partial at :

  • A necessary (though not sufficient) condition for this second-order mixed partial to exist is that be defined at points close to on the line where we fix all coordinates other than and allow to vary. Another way of saying this is that needs to exist not just at the point, but also if we perturb slightly.
  • A necessary (though not sufficient) condition for this second-order mixed partial to exist is that be defined at points close to in the plane parallel to the -plane that passes through the point. Explicitly, needs to exist not just at the point, but at all points close to it obtained by perturbing slightly and then perturbing slightly.

Facts

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  • Clairaut's theorem on equality of mixed partials states that under assumption of continuity (on an open set) of both the second-order mixed partials of a function of two variables, the two mixed partials are equal.

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