Question

Find the parameterization of the surface area given by `z = x^2 - 2x + y^2`?

A possible parameterization is `**r**(p, q) = (1 + p * cos(q), p sin(q), p^2 - 1)`, but why?

Answer 1

Looking at your suggested parameterization, rather than actually finding one:

• `bb r (p, q) = (:underbrace(1 + p cos(q))_(=x(p,q)),underbrace( p sin(q))\_(=y(p,q)), underbrace(p^2 - 1)\_(=z(p,q)) :)`

Looking at individual terms of `z(x,y)= x^2 - 2x + y^2` in terms of the parameterization:

• `{(x^2 = 1 + 2 p cosq + p^2 cos^2 q , qquad bbb(A)) ,(- 2x = -2 - 2 p cosq , qquad bbb(B)), (y^2 = p^2 sin^2q , qquad bbb(C) ):}`

`bbbA + bbbB + bbbC = 1 + cancel(2 p cosq) + p^2 cos^2 q -2 - cancel(2 p cosq) + p^2 sin^2q`

`= p^2 (cos^2 q + sin^2q) -1`

`= p^2 -1 color(blue)( = z(p,q))`

So that seems to work