In two dimensions, a change of coordinates is conveniently described by a surjective transformation \[ T:S\to R,\qquad (u,v)\mapsto (x,y) \] where \[ x=x(u,v), \qquad y=(u,v). \]

We shall assume that $T$ and its first-order partial derivatives are continuous on $S$.

A key property of $T$ is that it maps any boundary of the region $S$ in the $u$-$v$ plane to a boundary of $R$ in the $x$-$y$ plane. Such a transformation is particularly useful if we can restrict the coordinates $u,v$ to take values on a rectangle. After possibly applying a second transformation between this rectangle and the unit square \[ \left\{ (u,v)\in \R^2~|~ 0\leq u,v\leq 1 \right\}, \] this prompts us to focus on the case where $S$ itself is the unit square.

If $T(u,v) = (x,y)$, then the point $(x,y)$ is called the image of the point $ (u,v)$. We extend this definition to regions.

We can view the change of variables in two dimensions as a parameterisation \[ \r(u,v) = x(u,v)~\i+y(u,v)~\j, \qquad (u,v)\in S, \] of the region $R$ in the $x$-$y$ plane. This can then be used to analyse how the change of variables affects a double integral over $R$.

To this end, we recall that a double integral over $R$ arises as the limit of a sum over 'larger and larger' families of 'smaller and smaller' patches making up $R$: \[ \sum_{\text{patches}} f\left(x^*, y^*\right)\Delta S^* \to \iint_R f(x,y) ~dS. \] Here $\Delta S^* $ is the area of the patch containing the point $\left(x^*, y^*\right)$.

To this end, we recall that a double integral over $R$ arises as the limit of a sum over 'larger and larger' families of 'smaller and smaller' patches making up $R$: \[ \sum_{\text{patches}} f\left(x^*, y^*\right)\Delta S^* \to \iint_R f(x,y) ~dS. \] Here $\Delta S^* $ is the area of the patch containing the point $\left(x^*, y^*\right)$.

By working out an approximation of the patch area $\Delta S^*,$ and expressing it in terms of the $u,$ $v$ variables, we thus arrive at the formula \[ \iint_R f(x,y)~dx~dy =\iint_S f\big(x(u,v), y(u,v)\big)\left|\frac{\partial (x,y)}{\partial (u,v)}\right|~du~dv \] where \[ \frac{\partial (x,y)}{\partial (u,v)} =\text{det} \left( \begin{array}{cc} \dfrac{\partial x}{\partial u} & \dfrac{\partial x}{\partial v}\\ \dfrac{\partial y}{\partial u} & \dfrac{\partial y}{\partial v}\\ \end{array} \right) = \dfrac{\partial x}{\partial u} \dfrac{\partial y}{\partial v} - \dfrac{\partial x}{\partial v} \dfrac{\partial y}{\partial u} . \] is called the Jacobian of the transformation $T.$

Question: Can one always find a variable transformation that maps from a rectangular or a unit square?

Consider a type I region $R=\left\{ (x,y)~|~ a\leq x\leq b , g_1(x)\leq y\leq g_2(x)\right\}$

\begin{array}{rclc} x(u, v) & = &(1 - u)~a + ub, & 0\leq u \leq 1\\ y(u,v) & = &(1 - v) ~ g_1\big((1-u)~a +ub \big) + v g_2\big((1-u)~a+ ub \big),& 0\leq v \leq 1 \end{array}

\begin{array}{rclc} x(u, v) & = &(1 - u)~a + ub, & 0\leq u \leq 1\\ y(u,v) & = &(1 - v) ~ g_1\big((1-u)~a +ub \big) + v g_2\big((1-u)~a+ ub \big),& 0\leq v \leq 1 \end{array}

What is the Jacobian then? Because $\dfrac{\partial x}{\partial v}=0$ the corresponding Jacobian is simply given by

Of course, a similar change of variables applies if $R$ is of type II.

Using the change of vaiable $x = u + v,$ $y = u - v$, we have \[ \iint_R f\left(x,y\right)dA=\iint_S f\left(u+v,u-v\right)\left|\frac{\partial(x,y)}{\partial(u,v)}\right|du~dv. \]

Here the Jacobian is

Then

Now, to identify the region $S$ we substitute $x = u + v,$ $y = u - v$ into $x^2 +2xy+y^2 = 2(x-y)\,$ and $y=0$. Then we obtain \[ v = u\quad \text{and} \quad v = u^2. \]

So the region $S$ in the $u$-$v$ plane is bounded by the parabola $v=u^2$ and a straightline $v=u$.

Therefore, considering the region $S$ as type I we have that

Therefore, considering the region $S$ as type I we have that