Line integrals - University of Alberta
Line integrals - University of Alberta
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> <strong>of</strong> vector fields, IV<br />
The general case (continued)<br />
Let C be given by the vector equation<br />
Then:<br />
Thus:<br />
�<br />
C<br />
r(t) = x(t)i + y(t)j + z(t)k, t ∈ [a, b].<br />
T(t) = r′ (t)<br />
|r ′ (t)| .<br />
� b<br />
F · T ds = F(r(t)) ·<br />
a<br />
r′ (t)<br />
|r ′ (t)| |r′ (t)| dt<br />
� b<br />
= F(r(t)) · r ′ (t) dt = · · ·<br />
a