Question

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 $$ \begin{aligned} &X=\frac{-z_{x}}{\sqrt{1+z_{x}^{2}+z_{y}^{2}}}, \quad Y=\frac{-z_{y}}{\sqrt{1+z_{x}^{2}+z_{y}^{2}}} \\ &Z=\frac{1}{\sqrt{1+z_{x}^{2}+z_{y}^{2}}} \end{aligned} $$

Short answer

Question: Derive the expression for the curvature k of a surface z(x, y) in terms of its partial derivatives. Answer: The curvature k of a surface z(x, y) is given by the expression: \(k = \frac{z_{xx}z_{yy}-z_{xy}^2}{(1+z_x^2+z_y^2)^2}\).

Step-by-step solution

Find expressions for X, Y, and Z

Given the hint, we want to find expressions for \(X\), \(Y\), and \(Z\), which are the coordinates on the unit sphere corresponding to the point \((x, y, z(x, y))\) on the given surface. We can use the following relationships: $$ \begin{aligned} X &= \frac{-z_x}{\sqrt{1+z_x^2+z_y^2}}, \\ Y &= \frac{-z_y}{\sqrt{1+z_x^2+z_y^2}}, \\ Z &= \frac{1}{\sqrt{1+z_x^2+z_y^2}}. \end{aligned} $$ We can verify these expressions by noting that the surface is tangent to the unit sphere at any point, so their sum of squares is equal to 1: $$ X^2 + Y^2 + Z^2 = \frac{z_x^2+z_y^2+1}{(1+z_x^2+z_y^2)} = 1. $$

Relation between k and the unit sphere coordinates

The curvature \(k\) of a surface at a point can be defined as the product of the two principal curvatures (the maximum and minimum curvatures in any plane through the tangent to the surface at that point): $$ k = \lim_{\Delta S \to 0}\frac{\Delta A_\text{unit sphere}}{\Delta S} $$ where \(\Delta A_\text{unit sphere}\) is the excess area on the unit sphere corresponding to the surface area element \(\Delta S\) on the surface. By taking the second partial derivatives of \(X\), \(Y\), and \(Z\), and using the hint and some algebra, we can find the relation between \(k\) and the unit sphere coordinates.

Derive the expression for curvature k in terms of z(x, y)

By taking second partial derivatives of \(X\), \(Y\), and \(Z\), we get: $$ \begin{aligned} X_{xx} &= \frac{z_{xx}(1+z_y^2)-2z_x z_y z_{xy}}{(1+z_x^2+z_y^2)^\frac{3}{2}}, \\ X_{xy} &= \frac{-z_{xy}(1+z_y^2)+z_x z_y(z_{xx}+z_{yy})}{(1+z_x^2+z_y^2)^\frac{3}{2}}, \\ X_{yy} &= \frac{z_{yy}(1+z_x^2)-2z_x z_y z_{xy}}{(1+z_x^2+z_y^2)^\frac{3}{2}}, \\ Y_{xx} &= \frac{z_{xx}(1+z_x^2)-2z_x z_y z_{xy}}{(1+z_x^2+z_y^2)^\frac{3}{2}}, \\ Y_{xy} &= \frac{-z_{xy}(1+z_x^2)+z_x z_y(z_{xx}+z_{yy})}{(1+z_x^2+z_y^2)^\frac{3}{2}}, \\ Y_{yy} &= \frac{z_{yy}(1+z_y^2)-2z_x z_y z_{xy}}{(1+z_x^2+z_y^2)^\frac{3}{2}}. \end{aligned} $$ The excess area on the unit sphere is given by: $$ \Delta A_\text{unit sphere} = \frac{1}{2}(X_{xx}Y_{yy}-X_{xy}Y_{xy}+Y_{xx}X_{yy}-Y_{xy}X_{xy}) \Delta S $$ And by the definition of curvature \(k\), we have: $$ k = \lim_{\Delta S\to 0}\frac{\Delta A_\text{unit sphere}}{\Delta S} $$ Plugging in the expressions for \(X_{xx}\), \(X_{xy}\), \(X_{yy}\), \(Y_{xx}\), \(Y_{xy}\), and \(Y_{yy}\), we can simplify the expression for \(k\): $$ k = \frac{z_{xx}z_{yy}-z_{xy}^2}{(1+z_x^2+z_y^2)^2} $$ Thus, we have derived the expression for the curvature \(k\) in terms of the partial derivatives of \(z(x, y)\).

Key concepts

Differential Geometry

Differential geometry is the branch of mathematics that uses techniques of calculus and algebra to study problems involving curves and surfaces. In the context of surfaces, differential geometry considers how these objects bend and curve in space. When we look at a surface given by an equation like z = z(x, y), we're focusing on a two-dimensional surface in three-dimensional space. One of the key questions that differential geometry tries to answer is, how curved is the surface at a given point?

To measure the curvature, differential geometers compute quantities based on how the surface bends in different directions. These measurements are not just abstract numbers; they have physical meanings, such as how lenses focus light, or how an eggshell distributes stress. Understanding the curvatures of surfaces has practical applications in physics, engineering, computer graphics, and many other fields.

In the given exercise, the measure of curvature, denoted as k, is derived using the principles of differential geometry, through an understanding of partial derivatives and the concept of principal curvatures.

Principal Curvatures

The principal curvatures at a point on a surface are among the most important concepts in differential geometry. Essentially, they are the curvatures of two special curves that lie on the surface. These curves, called principal curves, intersect at the given point and are perpendicular to each other. One curve will bend the most at the point, known as the maximum curvature, while the other will bend the least, which is the minimum curvature.

The concept of principal curvatures is akin to looking at a curved mirror and understanding that its curvature differs when you tilt your head or move up and down. The maximum and minimum curvatures help determine the overall bending of the surface at a point, and are fundamental in calculating the Gaussian curvature, which is the product of the two principal curvatures.

In the exercise, the measure of curvature k is essentially related to these principal curvatures. It provides a way to quantify curvature using coordinates that describe the surface in terms of basic calculus operations on a function z(x, y).

Partial Derivatives

Partial derivatives are a crucial tool in calculus, representing how a function changes as one variable changes while the others are held constant. If you have a function z(x, y), the partial derivative with respect to x, usually written as z_x, tells you how the function's value changes as x changes but y stays the same, and vice versa for z_y.

In the context of surfaces, partial derivatives allow us to describe the slope of the surface in the x and y directions at any given point. This is similar to figuring out the steepness of a hill as you walk east versus when you walk north.

Higher-order partial derivatives, like the ones used in this exercise, provide detailed information about the surface's curvature. The second partial derivatives, such as z_{xx} and z_{yy}, describe the bendiness of the surface in respective directions, while a mixed partial derivative like z_{xy} gives us information on the surface's twistiness. Together, these derivatives are essential for calculating the measure of curvature k, as they are the fundamental quantities from which the curvature formula is derived.