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Chapter 14: Q 14. (page 1106)

Write ${鈭�}_{C} F (x, y, z) 路 d r$ explicitly as an integral of $t$ , where $F (x, y, z) = {z y - x, x z - y, x y - z 鉄�$ and $r (t) = 鉄� t, \cos t, \sin t 鉄�$ for $a 鈮� t 鈮� b$

Short Answer

Expert verified

The required answer is ${鈭�}_{a}^{b} (\frac{1}{2} \sin 2 t - t + t \cos 2 t) d t$ .

Step by step solution

01

Given Information

It is given that integral is $F (x, y, z) = 鉄� z y - x, x z - y, x y - z 鉄�$

$r (t) = 鉄� t, \cos t, \sin t 鉄�$ for $a 鈮� t 鈮� b$

02

Determination of F(x(t),y(t),z(t))

The line Integral formula is ${鈭�}_{C} F 路 d r = {鈭�}_{C} F (x (t), y (t), z (t)) 路 r^{'} (t) d t$

Parametric Equations are $x (t) = t, y (t) = \cos t, z (t) = \sin t$ for $a 鈮� t 鈮� b$

Hence, $r^{'} (t) = <x^{'} (t), y^{'} (t), z^{'} (t)>$

$= 鉄� 1, - \sin t, \cos t 鉄�$

$鈬� F (x (t), y (t), z (t)) = 鉄� \sin t \cos t - t, t \sin t - \cos t, t \cos t - \sin t 鉄�$

03

Expression of Line Integral

The line integral is

${鈭�}_{C} F 路 d r = {鈭�}_{C} F (x (t), y (t), z (t)) 路 r^{'} (t) d t$

$= {鈭�}_{a}^{b} 鉄� \sin t \cos t - t, t \sin t - \cos t, t \cos t - \sin t 鉄� 路鉄� 1, - \sin t, \cos t 鉄� d t$

$= {鈭�}_{a}^{b} [(\sin t \cos t - t) (1) + (t \sin t - \cos t) (- \sin t) + (t \cos t - \sin t) (\cos t)] d t$

$= {鈭�}_{a}^{b} (\sin t \cos t - t - t \sin^{2} t + \sin t \cos t + t \cos^{2} t - \sin t \cos t) d t$

$= {鈭�}_{a}^{b} (\sin t \cos t - t + t (\cos^{2} t - \sin^{2} t)) d t$

$= {鈭�}_{a}^{b} (\frac{1}{2} \sin 2 t - t + t \cos 2 t) d t$

Hence, required line integral is $= {鈭�}_{a}^{b} (\frac{1}{2} \sin 2 t - t + t \cos 2 t) d t$

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Most popular questions from this chapter

What are the inputs of a vector field in ${鈩�}^{3}$ ?

Find

$\begin{aligned} {鈭�}_{S} 鈥� F (x, y, z) 鈰� n d S if \\ F (x, y, z) = 2 x z i + 2 y z j 鈭� 18 k \end{aligned}$

and S is the portion of the hyperboloid $x^{2} + y^{2} - 9 = z^{2}$ that lies between the planes

z = 鈭�4 and z = 0, with n pointing outwards.

Find ${鈭�}_{S} 鈥� 1 d S$ , where S is the portion of the surface determined by $z = x^{2} 鈭� \sqrt{3} y$ that lies above the region in the xy-plane bounded by the x-axis and the lines with equations $y = 2 x, x = 3$ .

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.

Compute the curl of the vector fields:

$F (x, y) = - 4 x^{2} y i + 4 x y^{2} j$ .

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