Line integral from vector calculus over a closed curve

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I present an example where I calculate the line integral of a given vector function over a closed curve.. In particular, I the vector function is a
$${bf F}(x,y) := (-y/(x^2 + y^2), x/(x^2 + y^2)$$
and the closed curve is the unit circle, oriented in the anticlockwise direction.

I solve the problem and discuss the significance of the line integral through the mention of specific applications to engineering and physics.

Such an example is seen in second year university mathematics.


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