How Do You Graphically Represent a Change of Variables in a Coordinate System?

[GreenGoblin]

hi,
i need to know how to plot the change of variables
am i just to take x, and y as different constants, and treat it as normally on the (u,v) axis? so for the first, it will be like plotting on the (x,y) axis something like y^2 = x^2 - c, for different c, how many lines would i need to plot? and how would i mark it? would i say what c is each time
just to be sure these are the right lines, they are curvy like a hourglass, and they meet the axis at sqrt(c) on each side, so for each c we get two of these curves, so i need to plot all this twice. and then, also for the y=2uv, do i also plot this many times (for the same c) would i need to put all this lines on to the same plot? and mark each time which is c?

OR DO I NEED TO SHOW JUST WHERE X AND Y ARE BOTH CONSTANT? the wording has ambiguity.

 

[Opalg]

hi,
i need to know how to plot the change of variables
am i just to take x, and y as different constants, and treat it as normally on the (u,v) axis? so for the first, it will be like plotting on the (x,y) axis something like y^2 = x^2 - c, for different c, how many lines would i need to plot? and how would i mark it? would i say what c is each time
just to be sure these are the right lines, they are curvy like a hourglass, and they meet the axis at sqrt(c) on each side, so for each c we get two of these curves, so i need to plot all this twice. and then, also for the y=2uv, do i also plot this many times (for the same c) would i need to put all this lines on to the same plot? and mark each time which is c?

OR DO I NEED TO SHOW JUST WHERE X AND Y ARE BOTH CONSTANT? the wording has ambiguity.

I think this is the sort of thing they are looking for (it's an image I found online by searching for "orthogonal hyperbola family"). The family of "y=2uv" hyperbolas looks like the family of "hourglass" hyperbolas rotated through 45º. The important thing about the two families of curves is that wherever a curve from one family meets a curve from the other family, they intersect at right angles to each other.

View attachment 122

 

[HallsofIvy]

Yes, that's right. The "family" $$x^2- y^2= C$$ satisfies $$2x- 2y y'= 0$$ or $$y'= x/y$$ and so the "orthogonal family", the family of all curves that are perpendicular to any curves in the first family must satisfy $$y'= -y/x$$. That is the same as $$dy/dx= -y/x$$ or $$dy/y= -dx/x$$. Integrating, [tex]ln(y)= -ln(x)+ C[/itex] which is the same $$ln(y)+ ln(x)= ln(xy)= C$$ so that $$xy= e^C= C'$$. To graph those, let C be a number of different values. Perhaps positive and negative integers would be simplest.