In this libre text chapter, in example 15.7.1B, the author illustrates the change of variables using the following example:
The statement near it is written:
For the vertical length $A:u=0,0≤v≤1$ transforms to $x=−v^2,y=0$ so this is the horizontal length $A′ $ that joins $(−1,0)$ and $ (0,0)$
I find it confusing that regardless of whatever function of $x$ could have been on $v$, it seems that all of them would only have a straight line of $A'$ like suppose it was $x=-v^3$, that'd also denote the same line
If I understood correctly, we derive $A'$ from $A$ by feeding constraints of $A$ into the dependencies of the new variables on the old. Under that idea, does the kind of function $x$ is on the old coordinates not matter in sketching the region(only the bounds does)?
Best Answer
Note the mapping \begin{align*} x&=u^2-v^2\tag{1}\\ y&=uv\tag{2} \end{align*} from the $u,v$-plane in the $x,y$-plane is also given in (2) by $y=uv$. This means whenever we set $\color{blue}{u=0}$, we have $y=0$ regardless of the setting of $v$.
Nevertheless the mapping to parabolic coordinates given by (1) and (2) is often used since it is a one-to-one mapping with very nice geometrical properties:
The curves $x = \mathrm{constant}$ give rise in the $u,v$-plane to the rectangular hyperbolas $u^2-v^2=\mathrm{constant}$. They have asymptotes which are $u=v$ and $u=-v$. The lines $y = \mathrm{constant}$ also correspond to a family of rectangular hyperbolas. Here the asymptotes are the coordinate axes. The hyperbolas of each family cut those of the other family at right angles.
The lines parallel to the axes in the $u,v$-plane correspond to two families of parabolas in the $x,y$-plane. The parabolas $y^2=c^2(c^2-x)$ correspond to the lines $u=c$ and the parabolas $y^2=k^2(k^2+x)$ correspond to the lines $v=k$. All these parabolas have the origin as focus and the $x$-axis as axis. They form a famliy of confocal and coaxial parabolas.
This explanation can be found for instance in the classic Introduction to Calculues and Analysis II by R. Courant and F. John.
I also like the following application of this kind of bijective mapping given by the authors. In the following we denote in OP's example the region bounded by the line segments $A,B$ and $C$ with $R$ and the region bounded by the curve segments $A^{\prime}, B^{\prime}$ and $C^{\prime}$ with $R^{\prime}$.