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Chapter 11: Problem 15

Find a parametric representation of the surface. $$x^{2}+y^{2}-z^{2}=1$$

Short Answer

Expert verified
The parametric representation of the surface \(x^{2}+y^{2}-z^{2}=1\) is \(x = \cosh(u) \cdot \cos(v)\), \(y = \cosh(u) \cdot \sin(v)\), and \(z = i \cdot \sinh(u)\).

Step by step solution

01

Identification of Surface Shape

The provided equation \(x^{2}+y^{2}-z^{2}=1\) is similar to the general form of a hyperboloid of one sheet. The general formula for such a surface in three-dimensions is \(x^{2}/a^{2} + y^{2}/b^{2} - z^{2}/c^{2} = 1\), where \(a\), \(b\), \(c\) are the semi-axes lengths. In our problem, a=b=1 and c is imaginary.
02

Apply Parametric Equations for Hyperboloid

The standard parametric equations for a hyperboloid of one sheet are: \(x = a \cdot \cosh(u) \cdot \cos(v)\), \(y = b \cdot \cosh(u) \cdot \sin(v)\), and \(z = c \cdot \sinh(u)\), where \(\cosh\) is the hyperbolic cosine and \(\sinh\) is the hyperbolic sine. For our problem since a=b=1 and c is imaginary, the parametric equations then become \(x = \cosh(u) \cdot \cos(v)\), \(y = \cosh(u) \cdot \sin(v)\), and \(z = i \cdot \sinh(u)\).
03

Conclude

The required parametric representation of the surface \(x^{2}+y^{2}-z^{2}=1\) is \(x = \cosh(u) \cdot \cos(v)\), \(y = \cosh(u) \cdot \sin(v)\), and \(z = i \cdot \sinh(u)\).

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