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How to find a parametric equation to the surface S that is bounded between $z=x^2-y^2$ inside the cylinder $x^2+y^2=1$, and while C be the the Boundary of that surface.

While reading the solution of one of the questions they said that a parametric equation for surface S is:

$\vec r(\rho$,$\theta$) = $\rho$cos$\theta$$\hat i$ + $\rho$sin$\theta$$\hat j$ + $\rho^2$cos$2\theta$$\hat k$ while $0\le$ $\theta \le 2\pi$ , $0\le$ $\rho$ $ \le 1$

and

a parametric equation for curve/boundary C is:

$\vec r(\theta$)= cos$\theta$$\hat i$ + sin$\theta$$\hat j$ + cos$2\theta$$\hat k$ while $0\le$ $\theta \le 2\pi$

but how did they reach these equations? I would appreciate any kind of help.

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any point inside the cylinder $x^2 + y^2 = 1$ can be represented by $$x = r \cos t, y = r \sin t,\space 0 \le r \le 1, 0 \le t \le 2\pi.$$ then $$z = x^2 - y^2 = r^2 (\cos^2 t - \sin^2 t) = r^2 \cos (2t). $$

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