Let
be a function of
and
;
for example
The partial derivative of
with respect to
is the function
obtained by differentiating
with respect to
,
treating
as a constant; in this case
The partial derivative of
with respect to
is the function
obtained by differentiating
with respect to
,
treating
as a constant; in this case
These partial derivatives are formally defined as limits:
Example 1. For
we have
and
Example 2. For the function
we have
The number
gives the rate of change with respect to
of the function
the number
gives the rate of change with respect to
of the function
Through the surface
we pass a plane
parallel to the
-plane.
The plane
intersects the surface in a curve, the
-section
of the surface.
The
-section
of the surface is the graph of the function
Differentiating with respect to
,
we have
and in particular
The number
is thus the slope of the
-section
of the surface
at the point
The other partial derivative
can be given a similar interpretation. The same surface
is sliced by a plane
parallel to the
-plane.
The plane
intersects the surface in a curve,
-section
of the surface.
The
-section
of the surface is the graph of the function
Differentiating, this time with respect to
,
we have
and thus
The number
is the slope of the
-section
of the surface
at the point
In the case of a function of three variables, there are three partial
derivatives: the partial with respect to
,
the partial with respect to
,
and also the partial with respect to
.
These partials
are defined as follows:
Each partial can be found by differentiating with respect to the subscript
variable, treating the other two variables as constants.
Example 3. For the function
the partial derivatives are
In particular
Example 4. For
we have
Example 5. For function of the form
we can write
The number
gives the rate of change with respect to
of
at
;
gives the rate of change with respect to
of
at
and
gives the rate of change with respect to
of
at
Example 6. The function
has partial derivatives
The numbers
gives the rate of change with respect to
of the function
gives the rate of change with respect to
of the function
gives the rate of change with respect to
of the function
There is obviously no need to restrict ourselves to the variables
Where convenient we can use other letters.
Example 7. The volume of the frustum of a cone is given by
the function
At time
,
Find the rate of change of the volume with respect to each of its dimensions
at time
if the other dimensions are held constant.
Solution. The partial derivatives of
are as follows:
At time
,
the rate of change of
with respect to
is
the rate of change of
with respect to
is
the rate of change of
with respect to
is
The subscript notation is not the only one used for partial differentiation. A
variant of Leibniz's double-d notation is also commonly used. In this notation
the partials
are denoted by
Thus, for
we have
or more simply,
We can also write
The ``double-decker'' notation is not restricted to the letters
For we can write
For the function
we have
Let
The function is said to have a limit
at
if for each
there exists
such that, if
then
In this case we write
Example 1. We will show that the function
does not have a limit at
Along the obvious paths to
,
the coordinate axes, the limiting value is
:
along the
-axis,
and thus
tends to
;
along the
-axis,
and thus
tends to
.
However, along the line
the limiting value is
We have shown that not all paths to
yield the same limiting value. It follows that
does not have a limit at
As in the one-variable case, the limit (if it exists) is unique. Moreover, if
then
and
To say that
is continuous at
is to say that
or, equivalently, that
For two variables we can write
and for three variables
To say that
is continuous on
is to say that
is continuous at all points of
.
Polynomials in several variables, for example,
are everywhere continuous. In the two-variable case, that means continuity at
each point of the
-plane;
in the three-variable case, continuity at each point of three-space.
Rational functions (quotients of polynomials) are continuous everywhere except
where the denominator is zero. Thus
is continuous at each point of the
-plane
other than the origin
is continuous except on the line
is continuous except on the parabola
is continuous at each point of three-space other than the origin
is continuous except on the plane
.
More elaborate continuous functions can be constructed by forming composites:
take, for example,
The first function is continuous except along the vertical plane
.
The other two functions are continuous at each point of space. The continuity
of such composites follows from a simple theorem that we state and prove
below. In the theorem,
is a function of several variables, but
is a function of a single variable.
Theorem. If
is continuous at the point
and
is continuous at the number
then the composition
is continuous at the point
.
Proof. We begin with
.
We must show that there exists
0 such that
From the continuity of
at
we know that there exists
such that
From the continuity of
at
,
we know that there exists
such that
This last
obviously works; namely,
A continuous function of several variables is continuous in each of its
variables separately. In the two-variable case, this means that, if
then
The converse is false.
Example 2. We set
Since
we have
Thus, at the point
,
is continuous in
and continuous in
.
However, as a function of two variables,
is not continuous at
One way to see this is to note that we can approach
as closely as we wish by points of the form
with
.
At such points
takes on the value
:
Hence,
cannot tend to
as required.
For functions of a single variable the existence of the derivative guarantees
continuity. For functions of several variables the existence of partial
derivatives fails to guarantee continuity. To show this, we can use the same
function
Since both
and
are constantly zero, both partials exist (and are zero) at
,
and yet, the function is discontinuous at
It is not hard to understand how a function can have partial derivatives and
yet fail to be continuous. The existence of
at
depends on the behavior of
only at points of the form
.
Similarly, the existence of
at
depends on the behavior of
only at points of the form
.
On the other hand, continuity at
depends on the behavior of
at points of the more general form
.
More briefly, we can put it this way: the existence of a partial
derivative depends on the behavior of the function along a line segment (two
directions), whereas continuity depends on the behavior of the function in all
directions.
Suppose that
is a function of
and
with first partials
These are again functions of
and
and may themselves possess partial derivatives:
These last functions are called the second-order partials. Note that
there are two ``mixed'' partials
The first of these is obtained by differentiating first with respect to
and then with respect to
.
The second is obtained by differentiating first with respect to
and then with respect to
.
Example 3. The function
has first partials
The second-order partials are
Example 4. Setting
we have
The second-order partials are
Notice that in both examples we had
Since in neither case was
symmetric in
and
,
this equality of the mixed partials was not due to symmetry. Actually it was
due to continuity.
Theorem. If
and its partials
are continuous then the mixed partials
Proof. Fix
Assume that
Then there exists
such that
for
From the fundamental theorem of calculus we have
Similarly,
Since the two iterated integrals are equal
we have
a contradiction.
In the case of a function of three variables we look for three first partials
and nine second partials
Here again, there is equality of the mixed partials
provided that
and its first and second partials are continuous.
Example 5. For
we have
