Chapter 14: Q 13. (page 1154)

Evaluate the line integral of the given function over the specified curve.
$\int_{C} F (x, y) \cdot d r$ 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

$\int_{C} F (x, y) \cdot d r = 0$

Step by step solution

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.

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 $0 \leq t \leq 2 π$

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" $\Rightarrow F (x (t), y (t)) = (\cos t) i - (s i n t) j$ (As $\sin^{2} t + \cos^{2} t = 1$ )

Line integral is evaluated as

$\int_{C} F (x, y) \cdot d r = \int_{C} F (x (t), y (t)) \cdot r^{'} (t) d r$

role="math" localid="1652330298802" $= \int_{0}^{2 π} [(\cos t) i - (s i n t) j] \cdot [(- \sin t) i + (\cos t) j] d t$

Solving dot product

$= \int_{0}^{2 π} [(\cos t) (- \sin t) - (s i n t) (\cos t)] d t$

$= \int_{0}^{2 π} {[\cos^{2} t]}_{0}^{2 π}$

$= \int_{0}^{2 π} [\cos^{2} 2 π] - \cos 0$

$\Rightarrow 1^{2} - 1^{2} = 0$

Hence, line integral $\int_{C} F (x, y) \cdot d r = 0$

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Evaluate the line integral of the given function over the specified curve.
$\int_{C} F (x, y) \cdot d r$ 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

Step by step solution

Given Information

Evaluating Line Integral

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Evaluate the line integral of the given function over the specified curve.∫CFx,y·drwhere Fx,y=xx2+y2i-yx2+y2j and C is unit circle centered at origin.

Short Answer

Step by step solution

Given Information

Evaluating Line Integral

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Applied Mathematics

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Study anywhere. Anytime. Across all devices.

Evaluate the line integral of the given function over the specified curve.
$\int_{C} F (x, y) \cdot d r$ 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.