# Partial Derivatives

## Two Variables

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

## Geometric Interpretation

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

## Three Variables

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

## Other Notations

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

# Limits and Continuity; Equality of Mixed Partials

## Definition of the limit of a function of Several Variables

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 .

# Some Examples of Continuous Functions

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.

# Continuity of Composite Functions

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,

# Continuity in Each Variable Separately

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.

# Continuity and Partial Differentiability

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.

# Equality of Mixed Partials

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

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

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