Second-order mixed partial derivative
Contents
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 ![]() Let ![]() ![]() |
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.
The following are all multiple equivalent notations and definitions of .
Name | Notation | Definition in terms of first-order partial derivatives |
---|---|---|
subscript notation | ![]() |
Defined as ![]() Let ![]() ![]() |
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
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:
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
.
For a function of more than two variables
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
- 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.