/*! This file is auto-generated */ .wp-block-button__link{color:#fff;background-color:#32373c;border-radius:9999px;box-shadow:none;text-decoration:none;padding:calc(.667em + 2px) calc(1.333em + 2px);font-size:1.125em}.wp-block-file__button{background:#32373c;color:#fff;text-decoration:none} Problem 5 Consider the parametrized surfac... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 3: Problem 5

Consider the parametrized surface (Enneper's surface) $$ \mathbf{x}(u, v)=\left(u-\frac{u^{3}}{3}+u v^{2}, v-\frac{v^{3}}{3}+v u^{2}, u^{2}-v^{2}\right) $$ and show that a. The coefficients of the first fundamental form are $$ E=G=\left(1+u^{2}+v^{2}\right)^{2}, \quad F=0 $$ b. The coefficients of the second fundamental form are $$ e=2, \quad g=-2, \quad f=0 . $$ c. The principal curvatures are $$ k_{1}=\frac{2}{\left(1+u^{2}+v^{2}\right)^{2}}, \quad k_{2}=-\frac{2}{\left(1+u^{2}+v^{2}\right)^{2}} . $$ d. The lines of curvature are the coordinate curves. e. The asymptotic curves are \(u+v=\) const., \(u-v=\) const.

Short Answer

Expert verified
The coefficients of the first and second fundamental forms are as provided. The principal curvatures are \( k_1 = \frac{2}{(1+u^2+v^2)^2} \) and \( k_2 = -\frac{2}{(1+u^2+v^2)^2} \). The coordinate curves are lines of curvature and the asymptotic curves are \( u+v = \text{const.} \) and \( u-v = \text{const.} \).

Step by step solution

01

Compute the Partial Derivatives

Calculate the partial derivatives of the parametrized surface: \( \frac{\text{鈭倉\textbf{x}}{\text{鈭倉u} = \begin{pmatrix} 1 - u^2 + v^2 \ 2uv \ 2u \ \end{pmatrix} \ \frac{\text{鈭倉\textbf{x}}{\text{鈭倉v} = \begin{pmatrix} 2uv \ 1 + u^2 - v^2 \ -2v \ \end{pmatrix} \)
02

Compute the Dot Products for the First Fundamental Form

Calculate the dot products for E, F, and G: \( E = \frac{\text{鈭倉\textbf{x}}{\text{鈭倉u} \bullet \frac{\text{鈭倉\textbf{x}}{\text{鈭倉u} = (1 - u^2 + v^2)^2 + (2uv)^2 + (2u)^2 = (1 + u^2 + v^2)^2 \ F = \frac{\text{鈭倉\textbf{x}}{\text{鈭倉u} \bullet \frac{\text{鈭倉\textbf{x}}{\text{鈭倉v} = (1 - u^2 + v^2)(2uv) + (2uv)(1 + u^2 - v^2) + (2u)(-2v) = 0 \ G = \frac{\text{鈭倉\textbf{x}}{\text{鈭倉v} \bullet \frac{\text{鈭倉\textbf{x}}{\text{鈭倉v} = (2uv)^2 + (1 + u^2 - v^2)^2 + (-2v)^2 = (1 + u^2 + v^2)^2 \)
03

Compute the Normal Vector

Calculate the normal vector as the cross product of the partial derivatives: \( \textbf{N} = \frac{\text{鈭倉\textbf{x}}{\text{鈭倉u} \times \frac{\text{鈭倉\textbf{x}}{\text{鈭倉v} = \begin{vmatrix} \textbf{i} & \textbf{j} & \textbf{k} \ 1 - u^2 + v^2 & 2uv & 2u \ 2uv & 1 + u^2 - v^2 & -2v \ \end{vmatrix} = \begin{pmatrix} -4uv \ 4uv \ -2(1+u^2+v^2) \ \end{pmatrix} \)
04

Normalize the Normal Vector

Normalize \( \textbf{N} \): \( \textbf{N} = \frac{1}{2(1+u^2+v^2)} \begin{pmatrix} -4uv \ 4uv \ -2(1+u^2+v^2) \ \end{pmatrix} = \begin{pmatrix} -\frac{2uv}{1+u^2+v^2} \ \frac{2uv}{1+u^2+v^2} \ -1 \ \end{pmatrix} \)
05

Compute the Second Fundamental Form

Calculate the coefficients of the second fundamental form e, f, and g: \( e = \textbf{N} \bullet \frac{\text{鈭倉^2\textbf{x}}{\text{鈭倉u^2} = 2 \ f = \textbf{N} \bullet \frac{\text{鈭倉^2\textbf{x}}{\text{鈭倉u\text{鈭倉v} = 0 \ g = \textbf{N} \bullet \frac{\text{鈭倉^2\textbf{x}}{\text{鈭倉v^2} = -2 \)
06

Calculate the Principal Curvatures

Compute the principal curvatures using the formula:\( k_{1} = \frac{2}{(1+u^2+v^2)^2} \ k_{2} = \frac{-2}{(1+u^2+v^2)^2} \)
07

Verify Lines of Curvature

Show that the coordinate curves are lines of curvature because the principal directions align with u and v coordinates.
08

Identify Asymptotic Curves

Show that asymptotic curves are \( u+v = \text{const.} \) and \( u-v = \text{const.} \) by examining curves that satisfy the asymptotic condition \( I\text{I} = 0 \).

Unlock Step-by-Step Solutions & Ace Your Exams!

  • Full Textbook Solutions

    Get detailed explanations and key concepts

  • Unlimited Al creation

    Al flashcards, explanations, exams and more...

  • Ads-free access

    To over 500 millions flashcards

  • Money-back guarantee

    We refund you if you fail your exam.

Over 30 million students worldwide already upgrade their learning with 91影视!

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 for understanding a surface's intrinsic geometry. It essentially measures distances and angles on the surface. Let鈥檚 break it down:
  • Coefficient E: This is computed as \( E = \left( \frac{\partial \mathbf{x}}{\partial u} \cdot \frac{\partial \mathbf{x}}{\partial u} \right) \). For Enneper鈥檚 surface, the calculation shows that \( E = G = \left( 1 + u^2 + v^2 \right)^2 \) , indicating uniformity along both directions.
  • Coefficient F: This measures the orthogonality between the parameterized directions. For Enneper鈥檚 surface, the context yields \( F = 0 \).
  • Coefficient G: Similar to E, G measures the same material in the other direction; hence \( G = \left( 1 + u^2 + v^2 \right)^2 \).
Second Fundamental Form
The Second Fundamental Form deals with the extrinsic curvature of a surface. It captures how the surface bends in the surrounding space. Here鈥檚 a simple breakdown:
  • Coefficient e: Calculated as the dot product of the normal vector (perpendicular to the surface) and the second partial derivative of the parametrization with respect to u. For Enneper鈥檚 surface, \( e = 2 \).
  • Coefficient f: This is the mixed partial derivative, indicating how the surface bends when moving diagonally. It鈥檚 given that \( f = 0 \) for this surface.
  • Coefficient g: Similar to e, but along v-direction. It鈥檚 calculated here as \( g = -2 \).
Principal Curvatures
Principal curvatures are the maximum and minimum curvatures at a given point on a surface. They help in understanding how the surface twists and bends:
  • First Principal Curvature (\( k_{1} \)): It鈥檚 the maximum curvature at a point and for Enneper鈥檚 surface, \( k_{1} = \frac{2}{\left( 1 + u^2 + v^2 \right)^2} \).
  • Second Principal Curvature (\( k_{2} \)): This represents the minimum curvature, calculated as \( k_{2} = -\frac{2}{\left( 1 + u^2 + v^2 \right)^2} \).
These curvatures help in visualizing the surface's bending moments.
Asymptotic Curves
Asymptotic curves on a surface are directions in which the surface bends the least, often to the extent of being flat. Here鈥檚 how they work:
  • For Enneper鈥檚 surface, the special curves \( u+v = \text{constant} \) and \( u-v = \text{constant} \) are identified as asymptotic. This means these particular directions on the surface experience no normal curvature.
  • They are defined by satisfying the condition for the Second Fundamental Form \( I\text{I} = 0 \).
Enneper's Surface
Enneper's surface is an interesting and visually captivating object in differential geometry. Key highlights include:
  • It鈥檚 a self-intersecting minimal surface, meaning it minimizes surface area for given boundaries.
  • Defined by the parametrization \( \textbf{x}(u, v)=\big(u-\frac{u^{3}}{3}+u v^{2}, v-\frac{v^{3}}{3}+v u^{2}, u^{2}-v^{2}\big) \), this surface has a rich geometry.
  • It offers symmetrical properties and can be expressed in simpler forms, making it a standard example in studying minimal surfaces and differential geometry concepts.

One App. One Place for Learning.

All the tools & learning materials you need for study success - in one app.

Get started for free

Most popular questions from this chapter

Show that if \(C \subset S^{2}\) is a parallel of unit sphere \(S^{2}\), then the envelope of tangent planes of \(S^{2}\) along \(C\) is either a cylinder, if \(C\) is an equator, or a cone, if \(C\) is not an equator.

Let \(\alpha: I \rightarrow S \subset R^{3}\) be a curve on a regular surface \(S\) and consider the ruled surface generated by the family \(\\{\alpha(t), N(t)\\}\), where \(N(t)\) is the normal to the surface at \(\alpha(t)\). Prove that \(\alpha(I) \subset S\) is a line of curvature in \(S\) if and only if this ruled surface is developable.

Let \(C \subset S\) be a regular curve in \(S\). Let \(p \in C\) and \(\alpha(s)\) be a parametrization of \(C\) in \(p\) by are length so that \(\alpha(0)=p\). Choose in \(T_{p}(S)\) an orthonormal positive basis \(\\{t, h\\}\), where \(t=\alpha^{\prime}(0)\). The geodesic torsion \(\tau_{g}\) of \(C \subset S\) at \(p\) is defined by $$ \tau_{g}=\left\langle\frac{d N}{d s}(0), h\right\rangle . $$ Prove that a. \(\tau_{g}=\left(k_{1}-k_{2}\right) \cos \varphi \sin \varphi\), where \(\varphi\) is the angle from \(e_{1}\) to \(t\) and \(t\) is the unit tangent vector corresponding to the principal curvature \(k_{1}\). b. If \(\tau\) is the torsion of \(C, n\) is the (principal) normal vector of \(C\) and \(\cos \theta=\langle N, n\rangle\), then $$ \frac{d \theta}{d s}=\tau-\tau_{g} . $$ c. The lines of curvature of \(S\) are characterized by having geodesic torsion identically zero.

(Local Convexity and Curvature). A surface \(S \subset R^{3}\) is locally convex at a point \(p \in S\) if there exists a neighborhood \(V \subset S\) of \(p\) such that \(V\) is contained in one of the closed half-spaces determined by \(T_{p}(S)\) in \(R^{3}\). If, in addition, \(V\) has only one common point with \(T_{p}(S)\), then \(S\) is called striclly locally convex at \(p\). a. Prove that \(S\) is strictly locally convex at \(p\) if the principal curvatures of \(S\) at \(p\) are nonzero with the same sign (that is, the Gaussian curvature \(K(p)\) satisfies \(K(p)>0\) ). b. Prove that if \(S\) is locally convex at \(p\), then the principal curvatures at \(p\) do not have different signs (thus, \(K(p) \geq 0\) ). c. To show that \(K \geq 0\) does not imply local convexity, consider the surface \(f(x, y)=x^{3}\left(1+y^{2}\right)\), defined in the open set \(U=\left\\{(x, y) \in R^{2} ; y^{2}<\frac{1}{2}\right\\} .\) Show that the Gaussian curvature of this surface is nonnegative on \(U\) and yet the surface is not locally convex at \((0,0) \in U\) (a deep theorem, due to \(\mathrm{R}\). Sacksteder, implies that such an example cannot be extended to the entire \(R^{2}\) if we insist on keeping the curvature nonnegative; cf. Remark 3 of Sec. \(5-6\) ). *d. The example of part \(\mathrm{c}\) is also very special in the following local sense. Let \(p\) be a point in a surface \(S\), and assume that there exists a neighborhood \(V \subset S\) of \(p\) such that the principal curvatures on \(V\) do not have different signs (this does not happen in the example of part c). Prove that \(S\) is locally convex at \(p\).

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

Recommended explanations on Math Textbooks

Geometry

Read Explanation

Statistics

Read Explanation

Pure Maths

Read Explanation

Theoretical and Mathematical Physics

Read Explanation

Logic and Functions

Read Explanation

Decision Maths

Read Explanation
View all explanations

What do you think about this solution?

We value your feedback to improve our textbook solutions.

Study anywhere. Anytime. Across all devices.