Consider a change of variables that is given by a transformation from the $uv$-plane to the $xy$-plane: \[ T(u,v)=(x,y) \] where $x$ and $y$ are related to $u$ and $v$ by the equations \begin{eqnarray}\label{eq:01} x=f(u,v),\qquad y=g(u,v) \end{eqnarray} or \[ x=x(u,v),\qquad y=y(u,v) \] We usually assume that $T$ is a $C^1$ transformation, which means that $f$ and $g$ have continuous first-order partial derivatives.
A transformation $T$ is really just a function whose domain and range are both subsets of $\mathbb R^2$. If $T(u_1,v_1)=(x_1,y_1)$, then the point $(x_1,y_1)$ is called the image of the point $(u_1,v_1)$. If no two points have the same image, $T$ is called one-to-one.
One way to understand the geometry of a transformation $T$ is to see how it deforms or changes a particular region. The figure below shows the effect of a transformation $T$ on a region $A$ in the $uv$-plane. $T$ transforms a region $A$ into a region $B$ in the $xy$-plane called the image of $A$, consisting of the images of all points in $A$.
If $T$ is a one-to-one transformation, then it has an inverse transformation $T^{-1}$ from the $xy$-plane to the $uv$-plane and it may be possible to solve equations (\ref{eq:01}) for $u$ and $v$ in terms of $x$ and $y$: \[ u=F(x,y),\qquad v=G(x,y). \]
A parabolic transformation is defined by the equations $x=u^2-v^2$ and $y=2uv$.
Things to try:
A polar transformation is defined by the equations $x=r \cos \theta$, $y=r\sin \theta$.
Things to try:
The following file contains activities and problems associated with the simulations.
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