Definition 5.17. Jacobian determinant.
The determinant of the Jacobian matrix, \(\det\bigl(J_{\vect g}(\vect y)\bigr)\text{,}\) is called the Jacobian determinant or simply the Jacobian of \(\vect g\) at \(\vect y\text{.}\)
Section 5.3 The Transformation Formula
In this section we want to derive a formula generalising the substitution formula
\begin{equation}
\int_a^bf(g(s))g'(s)\,ds=\int_{g(a)}^{g(b)}f(t)\,dt\tag{5.5}
\end{equation}
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!
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 \(D\text{,}\) over which we integrate our function, as shown in Figure 5.15.
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 \(x_2=g_2(y_1,y_2)\text{.}\) We assume that
\begin{equation*}
\vect g(y_1,y_2)=\bigl(g_1(y_1,y_2),g_2(y_1,y_2)\bigr)
\end{equation*}
defines a function on the closure of \(D\text{.}\) The deformed domain is then the set
\begin{equation*}
\vect g(D):=\bigl\{\vect g(\vect y)\colon\vect y\in D\bigr\}
\end{equation*}
To derive a substitution formula we look at a small rectangle
\begin{equation*}
R:=[y_1,y_1+\Delta y_1]\times[y_2,y_2+\Delta y_2]
\end{equation*}
in \(D\) as done in the construction of the double integral in Section 5.1. We want to estimate the area of its image \(\vect g(R)\text{.}\) The rectangle and its image are shown in Figure 5.16.
We assume that \(\vect g\) has continuous first order derivatives on \(D\text{.}\) For fixed \(y_2\) the image of the map \(y_1\to\vect g(y_1,y_2)\) is a curve containing the lower edge of \(\vect g(R)\text{.}\) According to Remark 4.14 the length of that lower edge is approximately the length of the vector
\begin{equation*}
\vect v_1:=\frac{\partial}{\partial y_1}\vect g(\vect y)\Delta y_1\text{.}
\end{equation*}
Likewise the left edge of \(\vect g(R)\) has approximately the length of
\begin{equation*}
\vect v_2:=\frac{\partial}{\partial y_2}\vect g(\vect y)\Delta y_2\text{.}
\end{equation*}
The surface area of \(\vect g(R)\) is therefore approximately the surface area of the parallelogram spanned by \(\vect v_1\) and \(\vect v_2\text{.}\) In Theorem 1.16 we saw that the
\begin{equation*}
\text{area of the parallelogram spanned by }\vect v_1\text{ and }\vect v_2
=
\biggl|\det
\begin{bmatrix}
\vect v_1 \amp \vect v_2
\end{bmatrix}
\biggr|\text{.}
\end{equation*}
Using the definition of \(\vect v_1\text{,}\) \(\vect v_2\) and the properties of the determinant we have
\begin{align*}
\det
\begin{bmatrix}
\vect v_1 \amp \vect v_2
\end{bmatrix}
\amp=\det
\begin{bmatrix}
\dfrac{\partial g_1}{\partial y_1}(\vect y)\Delta y_1 \amp
\dfrac{\partial g_1}{\partial y_2}(\vect y)\Delta y_2 \\
\dfrac{\partial g_2}{\partial y_1}(\vect y)\Delta y_1 \amp
\dfrac{\partial g_2}{\partial y_2}(\vect y)\Delta y_2
\end{bmatrix}\\
\amp=\det
\begin{bmatrix}
\dfrac{\partial g_1}{\partial y_1}(\vect y) \amp
\dfrac{\partial g_1}{\partial y_2}(\vect y) \\
\dfrac{\partial g_2}{\partial y_1}(\vect y) \amp
\dfrac{\partial g_2}{\partial y_2}(\vect y)
\end{bmatrix}\Delta y_1\Delta y_2\\
\amp=\det\bigl(J_{\vect g}(\vect y)\bigr)\Delta y_1\Delta y_2\text{,}
\end{align*}
where \(J_{\vect g}(\vect y)\) is the Jacobian matrix of \(\vect g\) at \(\vect y\) as introduced in Definition 4.18. Hence the
\begin{align}
\text{area of the parallelogram spanned by }\amp\vect v_1\text{ and }\vect v_2\notag\\
\amp=\bigl|\det\bigl(J_{\vect g}(\vect y)\bigr)\bigr|
\Delta y_1\Delta y_2\text{.}\tag{5.6}
\end{align}
Intuitively, \(\bigl|\det\bigl(J_{\vect g}(\vect y)\bigr)\bigr|\) is the factor by which the area of a small rectangle \(R\) is distorted by the map \(\vect g\text{.}\)
Remark 5.18.
In the more traditional literature the Jacobian determinant is often denoted by
\begin{equation*}
\frac{\partial(g_1,g_2)}{\partial(y_1,y_2)}
:=\det\bigl(J_{\vect g}(\vect y)\bigr).
\end{equation*}
We will occasionally use this notation.
Now we partition \(D\) into small rectangles. Let us denote the collection of rectangles covering \(D\) by \(\mathcal R\text{.}\) If \(\vect y\) denotes the left lower corner of each \(R\in\mathcal R\) the sum
\begin{equation*}
\sum_{R\in\mathcal R}f(\vect g(\vect y))\area(\vect g(R))
\end{equation*}
is an approximation for the volume of the region between \(\vect g(D)\) and the graph of \(f\text{.}\) If we replace \(\area(\vect g(R))\) by the approximation (5.6) this volume is approximately
\begin{equation*}
\sum_{R\in\mathcal R}f(\vect g(\vect y))
\bigl|\det\bigl(J_{\vect g}(\vect y)\bigr)\bigr|
\Delta y_1\Delta y_2.
\end{equation*}
The last expression is a Riemann sum for the function \(\vect y\mapsto f(\vect g(\vect y))\bigl|\det\bigl(J_{\vect g}(\vect y)\bigr)\bigr|\text{.}\) Passing to the limit this suggests that the volume of the region between \(\vect g(D)\) and the graph of \(f\) is given by
\begin{equation*}
\int_Df(\vect g(\vect y))
\bigl|\det\bigl(J_{\vect g}(\vect y)\bigr)\bigr|\,d\vect y\text{.}
\end{equation*}
By the way we constructed double integrals we know that the volume of the region between \(\vect g(D)\) and the graph of \(f\) is also given by
\begin{equation*}
\int_{\vect g(D)}f(\vect x)\,d\vect x,
\end{equation*}
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 \(\vect g\) 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.
Theorem 5.19. Transformation formula.
Suppose that \(D\subset\mathbb R^2\) is a closed set on which the double integral for every continuous function is well defined. Assume that \(\vect g\colon D\to\mathbb R^2\) is a one-to-one function with continuous first order derivatives. Then for every continuous function \(f\colon\vect g(D)\to\mathbb R\) we have
\begin{equation*}
\int_Df(\vect g(\vect y))\bigl|
\det\bigl(J_{\vect g}(\vect y)\bigr)\bigr|\,d\vect y
=\int_{\vect g(D)}f(\vect x)\,d\vect x\text{,}
\end{equation*}
where \(J_{\vect g}(\vect y)\) is the Jacobian matrix of \(\vect g\) at \(\vect y\text{.}\)
Remark 5.20.
In more traditional notation the transformation formula reads
\begin{equation*}
\iint_Df(x_1,x_2)
\Bigl|\frac{\partial(x_1,x_2)}{\partial(y_1,y_2)}
\Bigr|\,dy_1\,dy_2
=\iint_{\vect g(D)}f(x_1,x_2)\,dx_1\,dx_2\text{,}
\end{equation*}
where \(x_1,x_2\) are considered to be functions of \(y_1\) and \(y_2\text{.}\)
Example 5.21. Transformation formula for an affine map.
Let us write down the transformation formula if \(\vect g\) is an affine map, that is, a linear map up to a translation. If \(\vect a\) is a vector and \(A\) an invertible \(2\times 2\)-matrix, then the map
\begin{equation*}
\vect g(\vect y):=\vect a+A\vect y
\end{equation*}
is an affine map. Then (check this as an exercise), \(J_{\vect g}(\vect y)=A\) for all \(\vect y\text{.}\) Hence the Jacobian determinant is given by \(\det J_{\vect g}(\vect y)=\det A\text{.}\) As \(\det A\) is a constant the transformation formula reduces to
\begin{equation}
\int_{\vect a+A(D)}f(\vect x)\,d\vect x
=|\det A|\int_Df(\vect a+A\vect y)d\vect y\tag{5.7}
\end{equation}
