3.8: Optional— Integrals in General Coordinates

One of the most important tools used in dealing with single variable integrals is the change of variable (substitution) rule

Equation 3.8.1

See Theorems 1.4.2 and 1.4.6 in the CLP-2 text. Expressing multivariable integrals using polar or cylindrical or spherical coordinates are really multivariable substitutions. For example, switching to spherical coordinates amounts replacing the coordinates with the coordinates by using the substitution

where

We'll now derive a generalization of the substitution rule 3.8.1 to two dimensions. It will include polar coordinates as a special case. Later, we'll state (without proof) its generalization to three dimensions. It will include cylindrical and spherical coordinates as special cases.

Suppose that we wish to integrate over a region, in and that we also wish¹ to use two new coordinates, that we'll call and in place of and The new coordinates are related to the old coordinates by the functions²

To make formulae more compact, we'll define the vector valued function by

As an example, if the new coordinates are polar coordinates, with renamed to and renamed to then and

Note that if we hold fixed and vary then sweeps out a curve. For example, if and then, if we hold fixed and vary sweeps out a straight line (that makes the angle with the -axis), while, if we hold fixed and vary sweeps out a circle (of radius centred on the origin).

            

We start by cutting (the shaded region in the figure below) up into small pieces by drawing a bunch of curves of constant (the blue curves in the figure below) and a bunch of curves of constant (the red curves in the figure below).

Concentrate on any one of the small pieces. Here is a greatly magnified sketch.

For example, the lower red curve was constructed by holding fixed at the value varying and sketching and the upper red curve was constructed by holding fixed at the slightly larger value varying and sketching So the four intersection points in the figure are

Now, for any small constants and we have the linear approximation³

Applying this three times, once with (to approximate ), once with (to approximate ), and once with (to approximate ),

We have dropped all Taylor expansion terms that are of degree two or higher in The reason is that, in defining the integral, we take the limit Because of that limit, all of the dropped terms contribute exactly to the integral. We shall not prove this. But we shall show, in the optional §3.8.1, why this is the case.

The small piece of surface with corners is approximately a parallelogram with sides

Here the notation, for example, refers to the vector whose tail is at the point and whose head is at the point Recall, from 1.2.17 that

So the area of our small piece of is essentially

Equation 3.8.2

Recall that denotes the determinant of the matrix Also recall that we don't really need determinants for this text, though it does make for nice compact notation.

The formula (3.8.2) is the heart of the following theorem, which tells us how to translate an integral in one coordinate system into an integral in another coordinate system.

Theorem 3.8.3

Let the functions and have continuous first partial derivatives and let the function be continuous. Assume that provides a one-to-one correspondence between the points of the region in the -plane and the points of the region in the -plane. Then

The determinant

that appears in (3.8.2) and Theorem 3.8.3 is known as the Jacobian⁴.

Example 3.8.4. for

We'll start with a pretty trivial example in which we simply rename to and to That is

Since

(3.8.2), but with renamed to and renamed to gives

which should really not be a shock.

Example 3.8.5. for Polar Coordinates

Polar coordinates have

Since

(3.8.2), but with renamed to and renamed to gives

which is exactly what we found in 3.2.5.

Example 3.8.6. for Parabolic Coordinates

Parabolic⁵ coordinates are defined by

Since

(3.8.2) gives

In practice applying the change of variables Theorem 3.8.3 can be quite tricky. Here is just one simple (and rigged) example.

Example 3.8.7

Evaluate

Solution

We can simplify the integrand considerably by making the change of variables

Of course to evaluate the given integral by applying Theorem 3.8.3 we also need to know

• [] the domain of integration in terms of and and
• [] in terms of

By (3.8.2), recalling that and

To determine what the change of variables does to the domain of integration, we'll sketch and then reexpress the boundary of in terms of the new coordinates and Here is the sketch of in the original coordinates

The region is a quadrilateral. It has four sides.

• The left side is part of the line Recall that So, in terms of and this line is
• The right side is part of the line In terms of and this line is
• The bottom side is part of the line or Recall that So, in terms of and this line is
• The top side is part of the line or In terms of and this line is

Here is another copy of the sketch of But this time the equations of its four sides are expressed in terms of and

So, expressed in terms of and the domain of integration is much simpler:

As and the integrand the integral is, by Theorem 3.8.3,

There are natural generalizations of (3.8.2) and Theorem 3.8.3 to three (and also to higher) dimensions, that are derived in precisely the same way as (3.8.2) was derived. The derivation is based on the fact, discussed in the optional Section 1.2.4, that the volume of the parallelepiped (three dimensional parallelogram)

determined by the three vectors and is given by the formula

where the determinant of a matrix can be defined in terms of some determinants by

If we use

to change from old coordinates to new coordinates then

Equation 3.8.8

Example 3.8.9. for Cylindrical Coordinates

Cylindrical coordinates have

Since

(3.8.8), but with renamed to and renamed to gives

which is exactly what we found in (3.6.3).

Example 3.8.10. for Spherical Coordinates

Spherical coordinates have

Since

(3.8.8), but with renamed to renamed to and renamed to gives

which is exactly what we found in (3.7.3).

Optional — Dropping Higher Order Terms in

In the course of deriving (3.8.2), that is, the formula for

we approximated, for example, the vectors

where is bounded⁶ by a constant times and is bounded by a constant times That is, we assumed that we could just ignore the errors and drop and by setting them to zero.

So we approximated

where the length of the vector is bounded by a constant times We'll now see why dropping terms like does not change the value of the integral at all⁷. Suppose that our domain of integration consists of all 's in a rectangle of width and height as in the figure below.

Subdivide the rectangle into a grid of small subrectangles by drawing lines of constant (the red lines in the figure) and lines of constant (the blue lines in the figure). Each subrectangle has width and height Now suppose that in setting up the integral we make, for each subrectangle, an error that is bounded by some constant times

Because there are a total of subrectangles, the total error that we have introduced, for all of these subrectangles, is no larger than a constant times

When we define our integral by taking the limit of the Riemann sums, this error converges to exactly As a consequence, it was safe for us to ignore the error terms when we established the change of variables formulae.

 

1. We'll keep our third wish in reserve.
2. We are abusing notation a little here by using and both as coordinates and as functions. We could write and but it is easier to remember and
3. Recall 2.6.1.
4. It is not named after the Jacobin Club, a political movement of the French revolution. It is not named after the Jacobite rebellions that took place in Great Britain and Ireland between 1688 and 1746. It is not named after the Jacobean era of English and Scottish history. It is named after the German mathematician Carl Gustav Jacob Jacobi (1804 – 1851). He died from smallpox.
5. The name comes from the fact that both the curves of constant and the curves of constant are parabolas.
6. Remember the error in the Taylor polynomial approximations. See 2.6.13 and 2.6.14.
7. See the optional § 1.1.6 of the CLP-2 text for an analogous argument concerning Riemann sums.