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Plotting change of variables

GreenGoblin

Member
Feb 22, 2012
68
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

MHB Oldtimer
Staff member
Feb 7, 2012
2,725
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.

images.jpeg
 

HallsofIvy

Well-known member
MHB Math Helper
Jan 29, 2012
1,151
Yes, that's right. The "family" [tex]x^2- y^2= C[/tex] satisfies [tex]2x- 2y y'= 0[/tex] or [tex]y'= x/y[/tex] and so the "orthogonal family", the family of all curves that are perpendicular to any curves in the first family must satisfy [tex]y'= -y/x[/tex]. That is the same as [tex]dy/dx= -y/x[/tex] or [tex]dy/y= -dx/x[/tex]. Integrating, [tex]ln(y)= -ln(x)+ C[/itex] which is the same [tex]ln(y)+ ln(x)= ln(xy)= C[/tex] so that [tex]xy= e^C= C'[/tex]. To graph those, let C be a number of different values. Perhaps positive and negative integers would be simplest.