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Line integrals - University of Alberta

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MATH 209—<br />

Calculus,<br />

III<br />

Volker Runde<br />

<strong>Line</strong> <strong>integrals</strong><br />

in R 2<br />

Types <strong>of</strong> line<br />

<strong>integrals</strong><br />

<strong>Line</strong> <strong>integrals</strong><br />

in R 3<br />

<strong>Line</strong> <strong>integrals</strong><br />

<strong>of</strong> vector fields<br />

<strong>Line</strong> <strong>integrals</strong>, II<br />

The construction<br />

Divide [a, b] into n subintervals [tj−1, tj].<br />

Set xj = x(tj) and yj = y(tj). The corresponding points<br />

Pj(xj, yj) divide C into n subcurves with lengths ∆s1, . . . , ∆sn.<br />

) on each subcurve.<br />

Pick a support point Pj(x ∗ j<br />

Definition<br />

, y ∗<br />

j<br />

The line integral <strong>of</strong> f along C is defined as<br />

�<br />

C<br />

f (x, y) ds = lim<br />

n→∞<br />

n�<br />

j=1<br />

f (x ∗ j , y ∗<br />

j )∆sj.

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