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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 tan-1 t + ln(1+t 2 )k-> | 1 = [4tan-1 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 ) = tan-1 u + C |2t/(1+t 2 ) dt = |du/u = ln|u|+C ^ u = 1+t 2 , du = 2tdt...

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- Fall '05
- Riggs
- Derivative, Integrals, Vector Space, Manifold, unit tangent vector, function vector-val func