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Chapter 3: Problem 4

Determine the asymptotic curves and the lines of curvature of \(z=x y\).

Short Answer

Expert verified
The asymptotic curves are the x-axis and the y-axis. The lines of curvature are the intersections with the coordinate planes.

Step by step solution

01

- Understand the surface

Given the surface equation: \(z = xy\). This represents a hyperbolic paraboloid surface.
02

- Compute the first fundamental form

We compute the first fundamental form coefficients: \(E = 1 + (\frac{\text{d}z}{\text{d}x})^2\), \(F = (\frac{\text{d}z}{\text{d}x})(\frac{\text{d}z}{\text{d}y})\), \(G = 1 + (\frac{\text{d}z}{\text{d}y})^2\). For \(z = xy\), we find that \(z_x = y\) and \(z_y = x\). Thus: \(E = 1 + y^2\), \(F = xy\), \(G = 1 + x^2\).
03

- Compute the second fundamental form

To compute the second fundamental form coefficients, calculate: \(z_{xx} = 0\), \(z_{xy} = 1\), \(z_{yy} = 0\). Using these, we get: \(L = 0\), \(M = 1\), \(N = 0\).
04

- Use the Gauss-Weingarten equations

The equations of the asymptotic curves can be derived using the Gauss-Weingarten relations: \(L\text{d}x^2 + 2M\text{d}x\text{d}y + N\text{d}y^2 = 0\). Substituting our values gives: \(2\text{d}x\text{d}y = 0\) which results in \(\text{d}x\text{d}y = 0\).
05

- Solve for asymptotic curves

Solving \(\text{d}x\text{d}y = 0\) provides the solutions \(\text{d}x = 0\) or \(\text{d}y = 0\). Hence, the asymptotic curves are the x-axis and the y-axis.
06

- Identify the lines of curvature

By analyzing the fundamental forms, it is observed that the lines of curvature for the given surface are simply the intersections of the surface with planes parallel to the coordinate planes, corresponding to the principal curvatures.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

first fundamental form
The first fundamental form is essential in understanding how surfaces bend and stretch in space. It encapsulates the metric properties of the surface, such as lengths and angles. For a given surface parameterized by coordinates \(u\) and \(v\), the first fundamental form is given by the expression \(E \, du^2 + 2F \, du \, dv + G \, dv^2\). Here: \ E = \langle \frac{\partial \mathbf{r}}{\partial u}, \frac{\partial \mathbf{r}}{\partial u} \rangle \ and \ G = \langle \frac{\partial \mathbf{r}}{\partial v}, \frac{\partial \mathbf{r}}{\partial v} \rangle \ describe the stretching in the \ u \ and \ v \ directions respectively. F is the indication of how the directions are related and is defined as \ F = \langle \frac{\partial \mathbf{r}}{\partial u}, \frac{\partial \mathbf{r}}{\partial v} \rangle \.
second fundamental form
The second fundamental form helps evaluate the local curvature of a surface. It is calculated through the coefficients \(L, M,\) and \(N\). These coefficients are derived from the second partial derivatives of the surface function \(z=xy\). The second fundamental form is expressed as \(L \, du^2 + 2M \, du \, dv + N \, dv^2 \). Here:
\ L = \frac{\partial^2 \mathbf{r}}{\partial u^2} \ and similarly for \ N = \frac{\partial^2 \mathbf{r}}{\partial v^2} \.
hyperbolic paraboloid
A hyperbolic paraboloid is a unique, saddle-shaped surface, defined by the equation \(z=xy\). It is named this way because its cross-sections are hyperbolas and parabolas. Specifically:
  • The parabolic sections occur when the plane cuts parallel to either axis (x or y).
  • The hyperbolic sections occur when the plane is perpendicular to the z-axis but not parallel to the xy-plane.
  • Understanding this shape is crucial in architecture and structural engineering because of its stability attributes and aesthetic appeal.

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Most popular questions from this chapter

(Surfaces of Revolution with Constant Curvature.) \((\varphi(v) \cos u\), \(\varphi(v) \sin u, \psi(v)), \varphi \neq 0\) is given as a surface of revolution with constant Gaussian curvature \(K\). To determine the functions \(\varphi\) and \(\psi\), choose the parameter \(v\) in such a way that \(\left(\varphi^{\prime}\right)^{2}+\left(\psi^{\prime}\right)^{2}=1\) (geometrically, this means that \(v\) is the arc length of the generating curve \((\varphi(v), \psi(v)))\). Show that a. \(\varphi\) satisfies \(\varphi^{\prime \prime}+K \varphi=0\) and \(\psi\) is given by \(\psi=\int \sqrt{1-\left(\varphi^{\prime}\right)^{2}} d v\); thus, \(01, C<1\). Observe that \(C=1\) gives a sphere (Fig. 3-23). c. All surfaces of revolution with constant curvature \(K=-1\) may be given by one of the following types: 1\. \(\varphi(v)=C \cosh v\), \(\psi(v)=\int_{0}^{v} \sqrt{1-C^{2} \sinh ^{2} v} d v .\) 2\. \(\varphi(v)=C \sinh v\), \(\psi(v)=\int_{0}^{v} \sqrt{1-C^{2} \cosh ^{2} v} d v .\) 3\. \(\varphi(v)=e^{v}\), \(\psi(v)=\int_{0}^{v} \sqrt{1-e^{2 v}} d v\) Determine the domain of \(v\) and draw a rough sketch of the profile of the surface in the \(x z\) plane. d. The surface of type 3 in part \(\mathrm{c}\) is the pseudosphere of Exercise \(6 .\) e. The only surfaces of revolution with \(K \equiv 0\) are the right circular cylinder, the right circular cone, and the plane.

Show that a surface which is compact (i.e., it is bounded and closed in \(R^{3}\) ) has an elliptic point.

If the surface \(S_{1}\) intersects the surface \(S_{2}\) along the regular curve \(C\), then the curvature \(k\) of \(C\) at \(p \in C\) is given by $$ k^{2} \sin ^{2} \theta=\lambda_{1}^{2}+\lambda_{2}^{2}-2 \lambda_{1} \lambda_{2} \cos \theta $$ where \(\lambda_{1}\) and \(\lambda_{2}\) are the normal curvatures at \(p\), along the tangent line to \(C\), of \(S_{1}\) and \(S_{2}\), respectively, and \(\theta\) is the angle made up by the normal vectors of \(S_{1}\) and \(S_{2}\) at \(p\).

Determine the asymptotic curves and the lines of curvature of the helicoid \(x=v \cos u, y=v \sin u, z=c u\), and show that its mean curvature is zero.

Describe the region of the unit sphere covered by the image of the Gauss map of the following surfaces: a. Paraboloid of revolution \(z=x^{2}+y^{2}\). b. Hyperboloid of revolution \(x^{2}+y^{2}-z^{2}=1\). c. Catenoid \(x^{2}+y^{2}=\cosh ^{2} z\).
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