Q. 14 Generalize your answer to Exerci... [FREE SOLUTION]

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Chapter 14: Q. 14 (page 1119)

Generalize your answer to Exercise 12 to give a parametrization and a normal vector for the extension of any differentiable plane curve y = f(x) through a 鈮� z 鈮� b.

Short Answer

Expert verified

$Therefore, the required normal vector for the above cylinder is n = = \frac{鈭� f}{鈭� u} i 鈭� j .$

Step by step solution

Step 1. Given

The given plane curve is y = f(x) through a 鈮� z 鈮� b.

Step 2. Finding the equation

$The parametrization of the curve y = x^{2} . Let x = u then, y = x^{2} = u^{2} Hence, The parametrization of the curve y = x^{2} is, \{\begin{cases} x = u \\ y = u^{2} \end{cases} Let = v, then the parametrization of a " generalized cylinder " S, defined by extending the curve y = x 虏 upwards and downwards from = = - 2 to z = 3 is, \{\begin{cases} x = u \\ y = u^{2} \\ z = v \end{cases} Where, - 2 鈮� v 鈮� 3 Therefore, the required smooth parametrization is r (u, v) = <u, u^{2}, v>$

Step 3. Find ru and rv.

$By above step, the surface S is parametrized by as follows : r (u, v) = <u, u^{2}, v> . T h e n, \begin{aligned} r_{u} = \frac{鈭�}{鈭� u} r (u, v) \\ = \frac{鈭�}{鈭� u} <u, u^{2}, v> \\ = 鉄� 1, 2 u, 0 鉄� \end{aligned} a n d \begin{aligned} r_{v} = \frac{鈭�}{鈭� v} r (u, v) \\ = \frac{鈭�}{鈭� 谓} <u, u^{2}, v> \\ = 鉄� 0, 0, 1 鉄� \end{aligned}$

Step 4. Find n=rv×rv

$n = r_{v} 脳 r_{v} \begin{aligned} = <1, \frac{鈭� f}{鈭� u}, 0> 脳鉄� 0, 0, 1 鉄� \\ = |\begin{array}{ccc} i & j & k \\ 1 & 鈭� f / 鈭� u & 0 \\ 0 & 0 & 1 \end{array}| \\ = [\frac{鈭� f}{鈭� u} 鈰� 1 鈭� 0 鈰� 0] i 鈭� [1 鈰� 1 鈭� 0 鈰� 0] j + [1 鈰� 0 鈭� \frac{鈭� f}{鈭� u} 鈰� 0] k \\ = \frac{鈭� f}{鈭� u} i 鈭� 1 j + 0 k \\ = \frac{鈭� f}{鈭� u} i 鈭� j . \end{aligned} Therefore, the required normal vector for the above cylinder is n = = \frac{鈭� f}{鈭� u} i 鈭� j .$

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91影视

Generalize your answer to Exercise 12 to give a parametrization and a normal vector for the extension of any differentiable plane curve y = f(x) through a 鈮� z 鈮� b.

Short Answer

Step by step solution

Step 1. Given

Step 2. Finding the equation

Step 3. Find ru and rv.

Step 4. Find n=rv×rv

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