Hello chiro.
I've tried in two ways.
Please read the picture below.
Does any one of my two attempts correct?
I don't know the reason why we need to do parametrization.
sometimes, when I encounter some simple system of equations, I can just let x=t, y=s and then write z in terms of t and s.
but most of the time, it is not the case.
how do I know or how can I check if my parametrization is correct?
Thank you.
by the way, since I'm afraid if I wrote a long question then no one would answer me, I raise a question as short as possible.
It's happy to heard from you that outlining my scratch work!
x- 1= 0 only when x= 1 so as long as x is not 1, you have x+1- y= 0 or y= x+1.
From the original equations, you already have z= 1- xy= 1- x(x+ 1)= 1- x- x^2.
Now take x itself as parameter: x= t, y= t+ 1, z= 1- t- t^2.
You need to remember that there are many different "parameterizations" for the same curve so at some point you need to make a "choice" as I did in setting x= t.
oh...
can't find any clue to mfind the parametrimization of the curves.