Chapter 24

Show that the area of an (infinitesimal) triangle with vertices \((x, y),(x+d x, y+d y),(x+\delta x, y+\delta y)\) is equal to \(\frac{1}{2}(d x \delta y-d y \delta x) .\)

Show that if a surface is given in the form \(z=z(x, y)\), then the measure of curvature \(k\) can be expressed as $$ k=\frac{z_{x x} z_{y y}-z_{x y}^{2}}{\left(1+z_{x}^{2}+z_{y}^{2}\right)^{2}} $$ Hint: Show first that if \(X, Y, Z\) are coordinates on the unit sphere corresponding to the point \((x, y, z(x, y))\) on the given surface, then $$ \begin{aligned} &X=\frac{-z_{x}}{\sqrt{1+z_{x}^{2}+z_{y}^{2}}}, \quad Y=\frac{-z_{y}}{\sqrt{1+z_{x}^{2}+z_{y}^{2}}} \\ &Z=\frac{1}{\sqrt{1+z_{x}^{2}+z_{y}^{2}}} \end{aligned} $$

If \(x=x(u, v), y=y(u, v), z=z(u, v)\) are the parametric equations of a surface and if \(E=x_{u}^{2}+y_{u}^{2}+z_{u}^{2}, F=x_{u} x_{v}+\) \(y_{u} y_{v}+z_{u} z_{v}\), and \(G=x_{v}^{2}+y_{v}^{2}+z_{v}^{2}\), show that \(d x^{2}+d y^{2}+d z^{2}=E d u^{2}+2 F d u d v+G d v^{2} .\)

Calculate \(E, F, G\) on the unit sphere parametrized by \(x=\cos u \cos v, y=\cos u \sin v, z=\sin u\), and show that \(d s^{2}=d u^{2}+\cos ^{2} u d v^{2}\)

Derive the formula \(\cos a=\cos b \cos c+\sin b \sin c \cos A\) for an arbitrary spherical triangle with sides \(a, b, c\) and op-posite angles \(A, B, C\) on a sphere of radius 1 by dividing the triangle into two right triangles and applying the formulas of Chapter 5 .

By using power series, show that Taurinus's "log-spherical" formula $$ \cosh \frac{a}{K}=\cosh \frac{b}{K} \cosh \frac{c}{K}-\sinh \frac{b}{K} \sinh \frac{c}{K} \cos A $$ reduces to the law of cosines as \(K \rightarrow \infty\).

Show that Taurinus's formula for an asymptotic right triangle on a sphere of imaginary radius \(i\), namely, \(\sin B=\) \(1 / \cosh x\), is equivalent to Lobachevsky's formula for the angle of parallelism, \(\tan \frac{B}{2}=e^{-x}\).

Given that tan \(\frac{1}{2} \Pi(x)=e^{-x}\), where \(\Pi(x)\) is Lobachevsky's angle of parallelism, derive the formulas $$ \sin \Pi(x)=\frac{1}{\cosh x} \quad \text { and } \quad \cos \Pi(x)=\tanh x $$ and show that their power series expansions up to degree 2 are \(\sin \Pi(x)=1-\frac{1}{2} x^{2}\) and \(\cos \Pi(x)=x\), respectively.

Describe geometrically Beltrami's parametrization of the sphere of radius \(k\) given by $$ \begin{aligned} &x=\frac{u k}{\sqrt{a^{2}+u^{2}+v^{2}}}, \quad y=\frac{v k}{\sqrt{a^{2}+u^{2}+v^{2}}} \\ &z=\frac{a k}{\sqrt{a^{2}+u^{2}+v^{2}}} \end{aligned} $$

Show that Beltrami's formulas for the lengths \(\rho, s, t\) of the sides of a right triangle on his pseudosphere transform into $$ \begin{aligned} &\frac{r}{a}=\tanh \frac{\rho}{k}, \quad \frac{r}{a} \cos \theta=\tanh \frac{s}{k}, \quad \text { and } \\ &\frac{v}{\sqrt{a^{2}-u^{2}}}=\tanh \frac{t}{k} \end{aligned} $$ and then show that $$ \cosh \frac{s}{k} \cosh \frac{t}{k}=\cosh \frac{\rho}{k} $$

Demonstrate how a central projection can transform parallel lines into intersecting lines.

Show that if the point \(p^{\prime}\) lies on the polar \(\pi\) of a point \(p\) with respect to a conic \(C\), then \(\pi^{\prime}\), the polar of \(p^{\prime}\), goes through \(p\). (Hint: Assume first that \(C\) is a circle.)

Determine homogeneous coordinates of the points \((3,4)\) and \((-1,7)\).

Write the homogeneous coordinates of the point at infinity on the line \(2 x-y=0\)

Determine rectangular coordinates of the points \((3,1,1)\) and \((4,-2,2)\) given in homogeneous coordinates.

Show that a map with the minimal number of countries that requires at least five colors cannot contain a digon (a) country with only two boundary edges).

Letting \(i, j, k\) be first-order units in three-dimensional space, determine the combinatory product of \(2 i+3 j-4 k\), \(3 i-i+k \cdot i+2 i-k\)

Show that in Grassmann's combinatory multiplication, $$ \begin{aligned} &\left(\sum \alpha_{1 i} \epsilon_{i}\right)\left(\sum \alpha_{2 i} \epsilon_{i}\right) \cdots\left(\sum \alpha_{n i} \epsilon_{i}\right) \\ &\quad=\operatorname{det}\left(\alpha_{i j}\right)\left[\epsilon_{1} \epsilon_{2} \cdots \epsilon_{n}\right] \end{aligned} $$ where each linear combination is of a given set of \(n\) firstorder units and where \(\left[\epsilon_{1} \epsilon_{2} \cdots \epsilon_{n}\right]\) is the single unit of \(n\)th order.

If \(\omega\) is the differential one-form in three dimensions given by \(\omega=A d x+B d y+C d z\), show that $$ \begin{aligned} d \omega=&\left(\frac{\partial C}{\partial y}-\frac{\partial B}{\partial z}\right) d y d z+\left(\frac{\partial A}{\partial z}-\frac{\partial C}{\partial x}\right) d z d x \\ &+\left(\frac{\partial B}{\partial x}-\frac{\partial A}{\partial y}\right) d x d y \end{aligned} $$

Show that the exterior derivative of \(\omega=A d y d z+\) \(B d z d x+C d x d y\) is the three-form $$ \left(\frac{\partial A}{\partial x}+\frac{\partial B}{\partial y}+\frac{\partial C}{\partial z}\right) d x d y d z $$

Show that \(d(d \omega)=0\), where \(\omega\) is a differential one-form or two- form in three-dimensional space.

Let \(\omega\) be the two-form in \(R^{3}-\\{0\\}\) given by $$ \omega=\frac{x d y d z+y d z d x+z d x d y}{\left(x^{2}+y^{2}+z^{2}\right)^{3 / 2}} $$ Show that \(d \omega=0\) but that there is no one-form \(\eta\) such that \(d \eta=\omega\). (Hint: If there were such a one-form, then by Stokes's theorem, with \(T\) being the unit sphere, we would have \(\int_{T} \omega=\int_{T} d \eta=\int_{S} \eta=0\), because the boundary of \(T\) is empty. Then calculate \(\int_{T} \omega\) directly.)

Study several new high school geometry texts. Do they follow Euclid's axioms or Hilbert's axioms or some combination? Comment on the usefulness of using Hilbert's reformulation in teaching a high school geometry class.

Read a complete version of Riemann's lecture "On the Hypotheses Which Lie at the Foundation of Geometry" (see note 10 at the end of the chapter for a reference). Describe Riemann's major new ideas and comment on how they have been followed up in the twentieth century. In particular, comment on the oft- repeated statement that Riemann's work was a precursor of Einstein's general theory of relativity.