The Transformation Formula

Section 5.3 The Transformation Formula

In this section we want to derive a formula generalising the substitution formula

\begin{matrix} (5.5) & \int_{a}^{b} f (g (s)) g^{'} (s) d s = \int_{g (a)}^{g (b)} f (t) d t \end{matrix}

you know from first year calculus. In the context of multiple integral such a formula is usually called the transformation formula or the area formula for multiple integrals. Make sure you understand the ideas presented in this section. A similar procedure can be applied to triple integrals. More importantly the ideas are essential when defining and discussing surface integrals!

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The function

g

in the above formula transforms the interval

[a, b]

into another interval, hence the different limits. When looking at double integrals this corresponds to a deformation of the domain

D,

over which we integrate our function, as shown in Figure 5.15.

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Figure 5.15. Deformation of a domain into another domain
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The function deforming one domain into the other is a vector valued function, taking a point

(y_{1}, y_{2})

to the point

x_{1} = g_{1} (y_{1}, y_{2})

and

x_{2} = g_{2} (y_{1}, y_{2}) .

We assume that

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g (y_{1}, y_{2}) = (g_{1} (y_{1}, y_{2}), g_{2} (y_{1}, y_{2}))

defines a function on the closure of

D .

The deformed domain is then the set

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g (D) := {g (y) : y \in D}

To derive a substitution formula we look at a small rectangle

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R := [y_{1}, y_{1} + Δ y_{1}] \times [y_{2}, y_{2} + Δ y_{2}]

D

as done in the construction of the double integral in Section 5.1. We want to estimate the area of its image

g (R) .

The rectangle and its image are shown in Figure 5.16.

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Figure 5.16. Deformation of a small rectangle by $g$
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We assume that

g

has continuous first order derivatives on

D .

For fixed

y_{2}

the image of the map

y_{1} \to g (y_{1}, y_{2})

is a curve containing the lower edge of

g (R) .

According to Remark 4.14 the length of that lower edge is approximately the length of the vector

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v_{1} := \frac{\partial}{\partial y_{1}} g (y) Δ y_{1} .

Likewise the left edge of

g (R)

has approximately the length of

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v_{2} := \frac{\partial}{\partial y_{2}} g (y) Δ y_{2} .

The surface area of

g (R)

is therefore approximately the surface area of the parallelogram spanned by

v_{1}

and

v_{2} .

In Theorem 1.16 we saw that the

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area of the parallelogram spanned by v_{1} and v_{2} = | det [\begin{matrix} v_{1} & v_{2} \end{matrix}] | .

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Using the definition of

v_{1},

v_{2}

and the properties of the determinant we have

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\begin{aligned} det [\begin{array}{c} v_{1} & v_{2} \end{array}] & = det [\begin{array}{c} \frac{\partial g_{1}}{\partial y_{1}} (y) Δ y_{1} & \frac{\partial g_{1}}{\partial y_{2}} (y) Δ y_{2} \\ \frac{\partial g_{2}}{\partial y_{1}} (y) Δ y_{1} & \frac{\partial g_{2}}{\partial y_{2}} (y) Δ y_{2} \end{array}] \\ = det [\begin{array}{c} \frac{\partial g_{1}}{\partial y_{1}} (y) & \frac{\partial g_{1}}{\partial y_{2}} (y) \\ \frac{\partial g_{2}}{\partial y_{1}} (y) & \frac{\partial g_{2}}{\partial y_{2}} (y) \end{array}] Δ y_{1} Δ y_{2} \\ = det (J_{g} (y)) Δ y_{1} Δ y_{2}, \end{aligned}

where

J_{g} (y)

is the Jacobian matrix of

g

y

as introduced in Definition 4.18. Hence the

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\begin{aligned} area of the parallelogram spanned by & v_{1} and v_{2} \\ (5.6) & = | det (J_{g} (y)) | Δ y_{1} Δ y_{2} . \end{aligned}

Intuitively,

| det (J_{g} (y)) |

is the factor by which the area of a small rectangle

R

is distorted by the map

g .

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Definition 5.17. Jacobian determinant.

The determinant of the Jacobian matrix,

det (J_{g} (y)),

is called the Jacobian determinant or simply the Jacobian of

g

y .

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Remark 5.18.

In the more traditional literature the Jacobian determinant is often denoted by

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\frac{\partial (g_{1}, g_{2})}{\partial (y_{1}, y_{2})} := det (J_{g} (y)) .

We will occasionally use this notation.

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Now we partition

D

into small rectangles. Let us denote the collection of rectangles covering

D

R .

y

denotes the left lower corner of each

R \in R

the sum

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\sum_{R \in R} f (g (y)) area (g (R))

is an approximation for the volume of the region between

g (D)

and the graph of

f .

If we replace

area (g (R))

by the approximation (5.6) this volume is approximately

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\sum_{R \in R} f (g (y)) | det (J_{g} (y)) | Δ y_{1} Δ y_{2} .

The last expression is a Riemann sum for the function

y \mapsto f (g (y)) | det (J_{g} (y)) | .

Passing to the limit this suggests that the volume of the region between

g (D)

and the graph of

f

is given by

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\int_{D} f (g (y)) | det (J_{g} (y)) | d y .

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By the way we constructed double integrals we know that the volume of the region between

g (D)

and the graph of

f

is also given by

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\int_{g (D)} f (x) d x,

and so must be equal to the previous integral. The above procedure is not a proof, but with some effort all arguments can be made rigorous. In contrast to (5.5) we need that

g

is one-to-one. The reason is that double integrals are not `oriented.’ Hence we have the following generalisation of the substitution formula, called transformation formula or area formula for double integrals.

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