Question

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 \(d x^{2}+d y^{2}+d z^{2}=E d u^{2}+2 F d u d v+G d v^{2} .\)

Short answer

Question: Show that the expression \(dx^2 + dy^2 + dz^2\) can be rewritten as \(Edu^2 + 2Fdudv + Gdv^2\) for the parametric equations \(x=x(u,v), y=y(u,v),\) and \(z=z(u,v)\), where \(E, F,\) and \(G\) are obtained from the partial derivatives of these parametric equations. Answer: By taking the total derivative of each parametric equation and summing their squares, we find that \((dx)^2 + (dy)^2 + (dz)^2 = Edu^2 + 2Fdudv + Gdv^2\), where \(E = x_u^2 + y_u^2 + z_u^2\), \(F = x_ux_v + y_uy_v + z_uz_v\), and \(G = x_v^2 + y_v^2 + z_v^2\).

Step-by-step solution

Compute the differentials of x, y, and z

Find the differentials \(dx, dy,\) and \(dz\) by taking the total derivative of each parametric equation with respect to \(u\) and \(v\): \(dx = \frac{\partial x}{\partial u}du + \frac{\partial x}{\partial v}dv = x_udu + x_vdv\) \(dy = \frac{\partial y}{\partial u}du + \frac{\partial y}{\partial v}dv = y_udu + y_vdv\) \(dz = \frac{\partial z}{\partial u}du + \frac{\partial z}{\partial v}dv = z_udu + z_vdv\)

Square and sum the differentials

Square each of the differentials found in Step 1 and sum them together to obtain: \((dx)^2 + (dy)^2 + (dz)^2 = (x_u^2 + y_u^2 + z_u^2)du^2 + 2(x_ux_v+y_uy_v+z_uz_v)dudv + (x_v^2 + y_v^2 + z_v^2)dv^2\)

Identify E, F, and G

Now, we just need to identify the given expressions for \(E, F\), and \(G\): - \(E = x_u^2 + y_u^2 + z_u^2\) - \(F = x_ux_v + y_uy_v + z_uz_v\) - \(G = x_v^2 + y_v^2 + z_v^2\)

Substitute E, F, and G into the expression

Replace the expressions in the result from Step 2 with \(E, F\), and \(G\): \((dx)^2 + (dy)^2 + (dz)^2 = E du^2 + 2F dudv + G dv^2\) This confirms that the given relationship holds as desired.

Key concepts

Parametric Equations

At the heart of understanding surfaces in differential geometry are parametric equations. They effectively describe a surface in a three-dimensional space using two parameters, often denoted as u and v. Imagine sliding along two intersecting lines, where your movement along each line is controlled by one of these parameters. The set of points you can reach by varying u and v is the surface defined by the parametric equations.

Consider the parametric equations of a surface given by functions for each coordinate axis, \( x=x(u, v) \), \( y=y(u, v) \), and \( z=z(u, v) \). Here, each point on the surface is a result of plugging in specific values of u and v into these equations. These functions map the parameter space—imagine this as a flat sheet—to the surface, which could be curved. For example, a simple parametric equation of a circle on a plane would depend on a single parameter, representing the angle, while a sphere would need two parameters, like latitude and longitude.

Surface Theory

Surface theory delves into the properties and behavior of surfaces in a multi-dimensional space. It bridges the gap between two-dimensional curves and higher-dimensional manifolds, focusing on properties like curvature, connectivity, and smoothness. In practice, this involves examining how a surface bends or stretches, and how these properties are expressed mathematically.

Studying a surface in this context often involves examining its differential properties, which is how it changes infinitesimally from point to point. This ties into the main elements of differential geometry, as it uses calculus and algebra to study the geometry of curves and surfaces. Fundamental to this are concepts like tangent spaces, normal vectors, and curvature, which provide insight into the 'shape' and 'behavior' of a surface at a given point.

Total Derivative

The total derivative is a vital concept in differential geometry for functions that depend on multiple variables. It extends the idea of a derivative, which measures the rate of change, to functions of several variables. It provides a way to determine how a function changes in response to changes in each of its variables, capturing all the partial changes simultaneously.

For instance, the total derivative of a function representing the position of a point on a surface will tell you how this point moves when you make an infinitesimal change to the parameters. This encompasses not just how it shifts in one direction or another, but the overall effect of both parameters shifting at once. In the context of parametric equations, it allows for the calculation of the differentials \( dx \), \( dy \), and \( dz \), which then help quantify changes on the surface itself.

First Fundamental Form

The first fundamental form is an expression that encapsulates all the metric properties of a surface—basically, how distances and angles are measured on the surface. At its core are the coefficients \( E \), \( F \), and \( G \), derived from dot products of the partial derivatives of the position vector. In essence, these coefficients describe how the infinitesimal distance between nearby points on the surface depends on the chosen parameters.

With the first fundamental form given as \( ds^2 = E du^2 + 2F dudv + G dv^2 \), we can determine lengths, angles, and areas on the surface. The exercise posits that if you know how \( x, y, \) and \( z \) change with respect to \( u \) and \( v \), you can deduce how the lengths of curves on the surface behave. This link between abstract parameters and concrete geometric qualities paves the way to further explorations in differential geometry, such as computing geodesics—the 'straightest' paths on a curved surface.