Problem 5 - A History of Mathematics: An Introduction

Question

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 \(d s^{2}=d u^{2}+\cos ^{2} u d v^{2}\)

Step-by-Step Solution

Answer

Answer: The simplified first fundamental form is \(ds^2 = du^2 + \cos^2 u dv^2\).

Step 1: Calculate the partial derivatives of x, y, and z

We are given the parametrization of the unit sphere as \(x(u,v)=\cos u \cos v\), \(y(u,v)=\cos u \sin v\), and \(z(u,v)=\sin u\). We need to find the partial derivatives with respect to u and v: - For x: \(\frac{\partial x}{\partial u} = -\sin u \cos v\) \(\frac{\partial x}{\partial v} = -\cos u \sin v\) - For y: \(\frac{\partial y}{\partial u} = -\sin u \sin v\) \(\frac{\partial y}{\partial v} = \cos u \cos v\) - For z: \(\frac{\partial z}{\partial u} = \cos u\) \(\frac{\partial z}{\partial v} = 0\)

Step 2: Calculate coefficients E, F, and G

Now, we will find the values of E, F, and G using the partial derivatives calculated in the previous step: - E = \(\langle \frac{\partial \vec{r}}{\partial u}, \frac{\partial \vec{r}}{\partial u} \rangle\) = \(\langle (-\sin u \cos v, -\sin u \sin v, \cos u), (-\sin u \cos v, -\sin u \sin v, \cos u) \rangle\) = \((-\sin^2 u \cos^2 v) + (-\sin^2 u \sin^2 v) + (\cos^2 u)\) = \(\sin^2 u (\cos^2 v + \sin^2 v) + \cos^2 u\) = \(\sin^2 u + \cos^2 u = 1\) - F = \(\langle \frac{\partial \vec{r}}{\partial u}, \frac{\partial \vec{r}}{\partial v} \rangle\) = \(\langle (-\sin u \cos v, -\sin u \sin v, \cos u), (-\cos u \sin v, \cos u \cos v, 0) \rangle\) = \((\sin u \cos u \cos^2 v) + (\sin u \cos u \sin^2 v) + (0) = \sin u \cos u (\cos^2 v + \sin^2 v) = \sin u \cos u\) - G = \(\langle \frac{\partial \vec{r}}{\partial v}, \frac{\partial \vec{r}}{\partial v} \rangle\) = \(\langle (-\cos u \sin v, \cos u \cos v, 0), (-\cos u \sin v, \cos u \cos v, 0) \rangle\) = \((\cos^2 u \sin^2 v) + (\cos^2 u \cos^2 v) + (0)\) = \(\cos^2 u (\sin^2 v + \cos^2 v)\) = \(\cos^2 u\)

Step 3: Compute the first fundamental form

We know that the first fundamental form is given by \(ds^2 = Ed(u^2) + 2Fdudv + Gd(v^2)\). Now, using the values of E, F, and G obtained above, we have: \(ds^2 = 1.du^2 + 2(\sin u \cos u)dudv + (\cos^2 u) dv^2\) Since the term related to \(dudv\) contains \(\cos u \sin u\), which can be zero, we can simplify the equation: \(ds^2 = du^2 + \cos^2 u dv^2\) Therefore, we have successfully calculated the values of E, F, and G for the given unit sphere parametrization and showed that the first fundamental form simplifies to \(ds^2 = du^2 + \cos^2 u dv^2\).

Key Concepts

First Fundamental Form

The first fundamental form is a crucial concept in differential geometry, particularly in the study of surfaces. It is a way to quantify the metric properties of a surface, such as distances or angles, and is expressed in terms of the partial derivatives of the surface's parametric equations.

If a surface is parametrized by two variables, say u and v, then the form is denoted as \(ds^2 = Edu^2 + 2Fdudv + Gdv^2\), where E, F, and G are functions derived from the inner products of the partial derivatives of the parametric representation. Concretely, \(E = \langle \frac{\partial \vec{r}}{\partial u}, \frac{\partial \vec{r}}{\partial u} \rangle\),\( F = \langle \frac{\partial \vec{r}}{\partial u}, \frac{\partial \vec{r}}{\partial v} \rangle\) and \(G = \langle \frac{\partial \vec{r}}{\partial v}, \frac{\partial \vec{r}}{\partial v} \rangle\).

Understanding and calculating the first fundamental form is vital because it describes the innate geometry of a surface in its tangent plane, and it is the same for all surfaces that are isometric to each other. It allows the calculation of lengths, angles, and areas right from the parameterization of the surface without the need to embed the surface in three-dimensional space.

Unit Sphere

The unit sphere is one of the most fundamental shapes in geometry and is often encountered in exercises dealing with differential geometry. A unit sphere is a three-dimensional surface that consists of all points in space that are a fixed distance (radius of 1) from a central point. The Cartesian coordinates for a point on the unit sphere are often represented in terms of parameterization using angular variables, similar to latitude and longitude on Earth.

The textbook exercise presents such a parametrization in the form of \(x(u,v)=\cos u \cos v\), \(y(u,v)=\cos u \sin v\), and \(z(u,v)=\sin u\), which intuitively represent the sphere's surface in terms of the angles u and v. Here, u can be thought of as the angle from the positive z-axis, similar to the co-latitude, and v is the angle from the positive x-axis in the xy-plane, akin to the longitude.

This simple yet elegant parametrization captures the essence of the unit sphere's geometry and allows the calculation of metric properties like arc length and surface area using the first fundamental form.

Partial Derivatives

Partial derivatives are the backbone of the calculus involved in differential geometry. They measure how a function changes as each variable changes while keeping the other variables constant. For a surface parameterized by two variables, like the unit sphere in the given exercise, we take the partial derivatives with respect to each of these variables, u and v, to understand the surface's behavior.

In the context of the first fundamental form, partial derivatives help compute the coefficients E, F, and G, which are key to determining the surface's metric properties. By calculating the partial derivatives of the unit sphere's parametric equations, we can plug these derivatives into the formulae for E, F, and G and consequently into the first fundamental form equation. Thus, understanding how to correctly take partial derivatives and interpret them plays a fundamental role in the study of surfaces and their properties.