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Chapter 2: Problem 10

(Tubular Surfaces.) Let \(\alpha: I \rightarrow R^{3}\) be a regular parametrized curve with nonzero curvature everywhere and arc length as parameter. Let $$ \mathbf{x}(s, v)=\alpha(s)+r(n(s) \cos v+b(s) \sin v), \quad r=\text { const. } \neq 0, s \in I, $$ be a parametrized surface (the tube of radius \(r\) around \(\alpha\) ), where \(n\) is the normal vector and \(b\) is the binormal vector of \(\alpha\). Show that, when \(\mathbf{x}\) is regular, its unit normal vector is $$ N(s, v)=-(n(s) \cos v+b(s) \sin v) $$

Short Answer

Expert verified
The unit normal vector is \(N(s, v) = -(n(s) \cos v + b(s) \sin v).\)

Step by step solution

01

Define the Parametrized Surface

The given parametrized surface is defined as \[ \mathbf{x}(s, v)=\alpha(s)+r(n(s)\cos v+b(s)\sin v) \ \text{where } r eq 0, s \in I, \text{ and } v \text{ is the parameter.} \ \alpha(s) \text{ is the curve.} \ n(s) \text{ is the normal vector.} \ b(s) \text{ is the binormal vector.} \]
02

Compute the Partial Derivatives

Compute the partial derivatives of the parametrized surface with respect to both parameters, \(s\) and \(v\). \[ \frac{\partial \mathbf{x}}{\partial s} = \frac{d\alpha(s)}{ds} + r \frac{\partial}{\partial s} \left(n(s) \cos v + b(s) \sin v \right). \]Let \(T(s) = \frac{d\alpha(s)}{ds}\) (the tangent vector), and use the Frenet-Serret formulas: \[ \frac{dn(s)}{ds} = -\kappa(s)T(s) + \tau(s)b(s)\]\ and \[ \frac{db(s)}{ds} = -\tau(s)n(s) \::::\text{where } \kappa(s) \text{ and } \tau(s) \text{ are curvature and torsion respectively.} \]Using these, we get: \[ \frac{\partial \mathbf{x}}{\partial s} = T(s) + r[-\kappa(s)T(s) + \tau(s)b(s)]\cos v + r[-\tau(s)n(s)]\sin v. \]
03

Simplify \( \frac{\partial \mathbf{x}}{\partial s} \)

Combine terms with common factors:\[ \frac{\partial \mathbf{x}}{\partial s} = T(s) + r \left[ -\kappa(s)T(s) \cos v + \tau(s)b(s) \cos v - \tau(s)n(s) \sin v \right].\]Thus, after rearrangement:\[ \frac{\partial \mathbf{x}}{\partial s} = (1 - r\kappa(s)\cos v)T(s) + r\tau(s)b(s)\cos v - r\tau(s)n(s)\sin v.\]
04

Compute \( \frac{\partial \mathbf{x}}{\partial v} \)

Compute the partial derivative of the parametrized surface with respect to \(v\):\[ \frac{\partial \mathbf{x}}{\partial v} = r \frac{\partial}{\partial v} \left( n(s)\cos v + b(s)\sin v \right). \ \frac{\partial \mathbf{x}}{\partial v} = r \left( -n(s)\sin v + b(s)\cos v \right). \]
05

Find the Cross Product of the Partial Derivatives

To find the unit normal vector, first compute the cross product of \( \frac{\partial \mathbf{x}}{\partial s} \) and \( \frac{\partial \mathbf{x}}{\partial v} \)\[ \frac{\partial \mathbf{x}}{\partial s} \times \frac{\partial \mathbf{x}}{\partial v} = \text{det} \begin{vmatrix} i & j & k \ (1 - r\kappa(s)\cos v)T(s) & r\tau(s)b(s)\cos v - r\tau(s)n(s)\sin v & r(-n(s)\sin v + b(s)\cos v) \end{vmatrix}. \]Given that \(T, n, \text{ and } b\) are mutually orthogonal unit vectors, simplify the determinant to:\[ \frac{\partial \mathbf{x}}{\partial s} \times \frac{\partial \mathbf{x}}{\partial v} = r\left[ (1 - r\kappa \cos v)(-n \sin v + b \cos v) + n - T \right]. \]
06

Normalize the Resulting Vector

To find the unit normal vector, normalize the cross product vector. The magnitude of this vector simplifies in terms of \(r\). After normalization, the unit normal vector \(N(s, v)\) is:\[ N(s, v) = -(n(s) \cos v + b(s) \sin v). \]

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Key Concepts

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

parametrized surfaces
A parametrized surface is a way of describing a surface in terms of two parameters. In our example, the surface is given by \(\mathbf{x}(s, v)=\alpha(s)+r(n(s)\cos v+b(s)\sin v)\), where \(s\) and \(v\) are the parameters. Here, \(\alpha(s)\) is the curve around which the surface is formed, and \(n(s)\) and \(b(s)\) are the normal and binormal vectors, respectively.
Frenet-Serret formulas
The Frenet-Serret formulas describe how the tangent, normal, and binormal vectors change along a curve. These formulas are crucial for understanding the behavior of curves in three-dimensional space. The formulas are:
  • \( \frac{dn}{ds} = -\kappa T + \tau b \)
  • \( \frac{db}{ds} = -\tau n \)
Here, \(\kappa\) is the curvature and \(\tau\) is the torsion of the curve. These formulas help us compute the partial derivatives in the given problem.
unit normal vector
The unit normal vector \(N(s, v)\) is a vector that is perpendicular to the surface at a given point and has a length of one. In our example, we find the unit normal vector by first computing the cross product of the partial derivatives \( \frac{\partial \mathbf{x}}{\partial s} \) and \( \frac{\partial \mathbf{x}}{\partial v} \). After finding the cross product, we normalize the resulting vector to ensure its length is one.
curvature and torsion
Curvature \( \kappa(s) \) and torsion \( \tau(s) \) are measures of how a curve bends and twists in space. Curvature indicates how sharply the curve bends at a point, while torsion measures how much the curve twists out of the plane of curvature. In the Frenet-Serret formulas, these values help determine how the normal and binormal vectors change along the curve, which is essential for computing the partial derivatives of the parametrized surface.

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Most popular questions from this chapter

Let \(Q\) be the union of the three coordinate planes \(x=0, y=0, z=0\). Let \(p=(x, y, z) \in R^{3}-Q .\) a. Show that the equation in \(t\), $$ \frac{x^{2}}{a-t}+\frac{y^{2}}{b-t}+\frac{z^{2}}{c-t} \equiv f(t)=1, \quad a>b>c>0 $$ has three distinct real roots: \(t_{1}, t_{2}, t_{3}\). b. Show that for each \(p \in R^{3}-Q\), the sets given by \(f\left(t_{1}\right)-1=0\), \(f\left(t_{2}\right)-1=0, f\left(t_{3}\right)-1=0\) are regular surfaces passing through \(p\) which are pairwise orthogonal.

Let \(f: S \rightarrow R\) be a differentiable function on a connected regular surface \(S .\) Assume that \(d f_{p}=0\) for all \(p \in S\). Prove that \(f\) is constant on \(S\).

(Gradient on Surfaces.) The gradient of a differentiable function \(f: S \rightarrow R\) is a differentiable map grad \(f: S \rightarrow R^{3}\) which assigns to each point \(p \in S\) a vector grad \(f(p) \in T_{p}(S) \subset R^{3}\) such that \(\langle\operatorname{grad} f(p), v\rangle_{p}=d f_{p}(v) \quad\) for all \(v \in T_{p}(S)\) Show that a. If \(E, F, G\) are the coefficients of the first fundamental form in a parametrization \(\mathbf{x}: U \subset R^{2} \rightarrow S\), then grad \(f\) on \(\mathbf{x}(U)\) is given by $$ \operatorname{grad} f=\frac{f_{u} G-f_{v} F}{E G-F^{2}} \mathbf{x}_{u}+\frac{f_{v} E-f_{u} F}{E G-F^{2}} \mathbf{x}_{v} $$ In particular, if \(S=R^{2}\) with coordinates \(x, y\), $$ \operatorname{grad} f=f_{x} e_{1}+f_{y} e_{2} $$ where \(\left\\{e_{1}, e_{2}\right\\}\) is the canonical basis of \(R_{2}\) (thus, the definition agrees with the usual definition of gradient in the plane). b. If you let \(p \in S\) be fixed and \(v\) vary in the unit circle \(|v|=1\) in \(T_{p}(s)\), then \(d f_{p}(v)\) is maximum if and only if \(v=\operatorname{grad} f /|\operatorname{grad} f|(\) thus, grad \(f(p)\) gives the direction of maximum variation of \(f\) at \(p)\). c. If grad \(f \neq 0\) at all points of the level curve \(C=\\{q \in S ; f(q)=\) const. \(\\},\) then \(C\) is a regular curve on \(S\) and grad \(f\) is normal to \(C\) at all points of \(C\).

Let \(S \subset R^{3}\) be a regular surface and \(\pi: S \rightarrow R^{2}\) be the map which takes each \(p \in S\) into its orthogonal projection over \(R^{2}=\left\\{(x, y, z) \in R^{3}\right.\); \(z=0\\}\). Is \(\pi\) differentiable?

Let \(w\) be a tangent vector to a regular surface \(S\) at a point \(p \in S\) and let \(\mathbf{x}(u, v)\) and \(\overline{\mathbf{x}}(\bar{u}, \bar{v})\) be two parametrizatioos at \(p\). Suppose that the expressions of \(w\) in the bases associated to \(\mathbf{x}(u, v)\) and \(\overline{\mathbf{x}}(\bar{u}, \bar{v})\) are $$ w=\alpha_{1} \mathbf{x}_{u}+\alpha_{2} \mathbf{x}_{v} $$ and $$ w=\beta_{1} \overline{\mathbf{x}}_{\bar{u}}+\beta_{2} \mathbf{x}_{\bar{v}} $$ Show that the coordinates of \(w\) are related by $$ \begin{aligned} \beta_{1} &=\alpha_{1} \frac{\partial \bar{u}}{\partial u}+\alpha_{2} \frac{\partial \bar{u}}{\partial v} \\ \beta_{2} &=\alpha_{1} \frac{\partial \bar{v}}{\partial u}+\alpha_{2} \frac{\partial \bar{v}}{\partial v} \end{aligned} $$ where \(\bar{u}=\bar{u}(u, v)\) and \(\bar{v}=\bar{v}(u, v)\) are the expressions of the change of coordinates.
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