# Chapter 18: Chapter 18

Problem 535

Find the length of the arc $\mathrm{u}=\mathrm{e}^{\mathrm{k} \theta}, \theta=\theta, 0 \leq \theta \leq \pi$, \(k=\) constant on the cone \(X^{-}=(u \cos \theta, u \sin \theta, u) .\)

Problem 536

Let \(\sum\) be a surface. Define the notion of orthogonality of two curves at a point p on \(\sum\). Derive necessary and sufficient conditions for the parametric curves to be orthogonal.

Problem 537

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 \(\mathrm{x}^{2}+\mathrm{y}^{2}+z^{2}=1\).

Problem 538

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) $\Sigma: \mathrm{x}^{-}(\mathrm{u}, \mathrm{v})=\left(\mathrm{u}+\mathrm{v}^{2}, \mathrm{v}+\mathrm{u}^{2}, \mathrm{uv}\right)$

Problem 539

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 $$ (E N-F M) d u+(F N-G M) d v=0 $$

Problem 540

Let \(\mathrm{f}\) be a \(\mathrm{C}^{1}\) function defined in a domain \(\mathrm{D}\) in \(\mathrm{R}^{2}\). (a) Show that \(\Sigma:\\{z=\mathrm{f}(\mathrm{x}, \mathrm{y})\\}\) is a surface patch with coordinate \(\mathrm{x}, \mathrm{y}\). (b) Compute the first fundamental form.

Problem 541

Let u and \(\mathrm{v}\) be longitude and latitude on a sphere. Find the angle at which the curve \(\mathrm{v}=\mathrm{u}\) cuts the equator. Choose units so that the radius is unity.

Problem 543

Compute the total area of the torus $\mathrm{x}=(\mathrm{a}+\mathrm{b} \cos \varphi) \cos \theta$, $\mathrm{y}=(\mathrm{a}+\mathrm{b} \cos \varphi) \sin \theta, \mathrm{z}=\mathrm{b} \sin \varphi, 0<\mathrm{b}<\mathrm{a}$ by the first fundamental form.

Problem 544

Determine the second fundamental form of the surface represented by $$ \mathrm{X}^{-}=\mathrm{ue}^{-}{ }_{1}+\mathrm{ve}^{-}{ }_{2}+\left(\mathrm{u}^{2}-\mathrm{v}^{2}\right) \mathrm{e}_{3} $$ where \(\mathrm{e}^{-} 1, \mathrm{e}^{-}{ }_{2}, \mathrm{e}^{-} 3\) are the standard basis: $$ \mathrm{e}^{-}{ }_{1}=(1,0,0), \mathrm{e}^{-}{ }_{2}=(0,1,0), \mathrm{e}^{-}{ }_{3}=(0,0,1) $$

Problem 545

Show that the surface $$ \mathrm{X}^{-}=\mathrm{ue}^{-}_{1}+\mathrm{ve}^{-}{ }_{2}+\left(\mathrm{u}^{2}+\mathrm{v}^{3}\right) \mathrm{e}^{-}{ }_{3} $$ is elliptic when \(\mathrm{v}>0\), hyperbolic when \(\mathrm{v}<0\), and parabolic for \(\mathrm{v}=0\).