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Problem 532

a) Define a surface patch surrounding the point \((0,0,1)\) on the sphere \(x^{2}+y^{2}+z^{2}=1\) b) Show that the paraboloid \(z=x^{2}+y^{2}\) is a surface patch.

Expert verified

a) A surface patch surrounding the point \((0,0,1)\) on the sphere \(x^2+y^2+z^2=1\) is given by:
$$
\begin{cases}
x = \sin\theta \cos\phi \\
y = \sin\theta \sin\phi \\
z = \cos\theta
\end{cases}
, \quad 0 \leq \theta \leq \epsilon, \quad 0 \leq \phi \leq 2\pi,
$$
where \(\epsilon\) is a small positive number.
b) The paraboloid \(z=x^2+y^2\) is a surface patch with parameterization:
$$
\begin{cases}
x = u \\
y = v \\
z = u^2 + v^2
\end{cases}
, \quad a \leq u \leq b, \quad c \leq v \leq d.
$$
where \(a, b, c, d\) are constants.

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Chapter 18

Compute the coefficients of the first fundamental form in each of the following cases. (1) the plane \(z=0\) (ii) the cylinder \(\mathrm{x}^{2}+\mathrm{y}^{2}=1\) (iii) the sphere \(\mathrm{x}^{2}+\mathrm{y}^{2}+z^{2}=1\).

Chapter 18

Find equations for the asymptotic lines for the hyperbolic paraboloid whose equations are $$ \begin{gathered} b x+a y=2 a b u, b x-a y=2 a b v, z=2 c u v \\ (a, b, c \neq 0) \end{gathered} $$

Chapter 18

Compute the first fundamental form for the following surfaces: (i) The paraboloid \(z=x^{2}+y^{2}\) (ii) The cone \(z^{2}=x^{2}+y^{2} \quad z>0\) (iii) The hyperboloid \(z=x^{2}-y^{2}\) (iv) $\Sigma: \mathrm{x}^{-}(\mathrm{u}, \mathrm{v})=\left(\mathrm{u}+\mathrm{v}^{2}, \mathrm{v}+\mathrm{u}^{2}, \mathrm{uv}\right)$

Chapter 18

Let \(\mathrm{d} s^{2}=\mathrm{E} \mathrm{du}^{2}+2 \mathrm{~F}\) dudv \(+\mathrm{G} \mathrm{dv}^{2}\) be the first fundamental form of a surface patch. Show that the family of curves orthogonal to the family defined by \(\mathrm{M} \mathrm{du}+\mathrm{N} \mathrm{dv}=0\) is determined by $$ (E N-F M) d u+(F N-G M) d v=0 $$

Chapter 18

Given the circular helix \(x^{-}(t)=(\) cost, \(a\) sint, \(b t)\) find the tangent vector the normal vector, the curvature, the binormal vector, and the torsion along this curve.

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