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Chapter 14: Q 13. (page 1154)

Evaluate the line integral of the given function over the specified curve.
${鈭�}_{C} F (x, y) 路 d r$ where $F (x, y) = \frac{x}{x^{2} + y^{2}} i - \frac{y}{x^{2} + y^{2}} j$ and C is unit circle centered at origin.

Short Answer

Expert verified

${鈭�}_{C} F (x, y) 路 d r = 0$

Step by step solution

01

Given Information

$F (x, y) = \frac{x}{x^{2} + y^{2}} i - \frac{y}{x^{2} + y^{2}} j$ and C is unit circle centered at origin.

02

Evaluating Line Integral

For unit circle, $x = \cos t, y = \sin t$

The curve will be parametrized as

$r (t) = (\cos t) i + (\sin t) j$ for $0 鈮� t 鈮� 2 蟺$

Differentiating, we get

role="math" localid="1652330269655" $r (t) = (- \sin t) i + (\cos t) j$

Substituting values of $(x, y)$

role="math" localid="1652330056674" $F (x (t), y (t)) = \frac{\cos t}{\cos^{2} t + \sin^{2} t} i - \frac{\sin t}{\cos^{2} t + \sin^{2} t} j$

role="math" localid="1652330234276" $鈬� F (x (t), y (t)) = (\cos t) i - (s i n t) j$ (As $\sin^{2} t + \cos^{2} t = 1$ )

Line integral is evaluated as

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

role="math" localid="1652330298802" $= {鈭�}_{0}^{2 蟺} [(\cos t) i - (s i n t) j] 路 [(- \sin t) i + (\cos t) j] d t$

Solving dot product

$= {鈭�}_{0}^{2 蟺} [(\cos t) (- \sin t) - (s i n t) (\cos t)] d t$

$= {鈭�}_{0}^{2 蟺} {[\cos^{2} t]}_{0}^{2 蟺}$

$= {鈭�}_{0}^{2 蟺} [\cos^{2} 2 蟺] - \cos 0$

$鈬� 1^{2} - 1^{2} = 0$

Hence, line integral ${鈭�}_{C} F (x, y) 路 d r = 0$

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Most popular questions from this chapter

Suppose that an electric field is given by

$E = 2 y i + 2 x y j + y z k$

Compute the flux ${鈭�}_{S} 鈥� E 鈰� n d A$ of the field through the unit cube $[0, 1] 脳 [0, 1] 脳 [0, 1]$ .

Compute a general formula for dS for any plane $a x + b y + c z = k i f c 鈮� 0 .$

Make a chart of all the new notation, definitions, and theorems in this section, including what each new item means in terms you already understand.

Why do surface integrals of multivariate functions not include an n term, whereas surface integrals of vector fields do include this term?

$F (x, y, z) = i + j + k$ , where S is the lower half of the unit sphere, with n pointing outwards.

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