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Unformatted text preview: > (t) (dot) V> '(t) (5) d/dt [M> (t) x V> (t)] = M> '(t) x V> (t) + M> (t) x V> (t) (6) d/dt [ M> (f(t))] = M> '(f(t)) (f'(t)) chain rule What's a smooth curve? A curve w/ no sharp turns and gaps.  We say a curve is smooth if r> '(t) != 0> for any t Integrals  ~ r> (t) dt = R> (t) + C > note the constant is a vector! Definite Integral:  t=b t=a r> (t) dt = R> (t)  t=b t=a = R> (b)R> (a) Ex: (cospit i > + sinpi + j> + tk> ) dt = < (1/pi)sinpit, 1/pi cospit, t 2 /2 > C>  < cospit, sinpit, t > dt = sinpit/pi i> cospit/pi j> + t 2 /2 k> + C> 1 <0, 4/1+t 2 , 2t/(1+t 2 ) > dt  1 (4/(1+t 2 )j> + 2t/(1+t 2 ) k> ) dt = 4 tan1 t + ln(1+t 2 )k>  1 = [4tan1 j> + ln(2)k> ]  [4tan 01 (0) j> + ln(1) K> ] 4(pi/4) j> + ln(2) k > = pi j> + ln2 k> = <0, pi, ln2> du/(1+u 2 ) = tan1 u + C 2t/(1+t 2 ) dt = du/u = lnu+C ^ u = 1+t 2 , du = 2tdt...
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 Fall '05
 Riggs
 Derivative, Integrals, Vector Space, Manifold, unit tangent vector, function vectorval func

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