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l_notes06 - Lecture VI Calculus of Vector Functions dR dt 2...

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Lecture VI Calculus of Vector Functions R d 2 R Recall that d denotes the first-order derivative of R ( t ), and that dt 2 de- dt notes the second-order derivative of R ( t ). We introduce new notations for these R ˙ R ¨ R ( t ) = a 1 ( t ) ˆ i + a 2 ( t ) ˆ + a 3 ( t ) ˆ functions: d = R ( t ) and d 2 = R ( t ). Let j k . dt dt 2 Then the following differentiation rules stand: ˙ j + ˙ a 3 ( t ) ˆ 1. R ( t ) = a ˙ 1 ( t ) ˆ i + ˙ a 2 ( t ) ˆ k . ¨ a 1 ( t ) ˆ i + ¨ j a 3 ( t ) ˆ 2. R ( t ) = ¨ a 2 ( t ) ˆ + ¨ k . A ( t ) and Let B ( t ) be differentiable vector functions, and let a ( t ) be a differentiable scalar function. The following differentiation rules stand: dt ( ˙ ˙ 3. d A ( t ) + B ( t )) = A + B . 4. d A ( t )) = a ˙( t ) ˙ A ( t ) + a ( t ) A ( t ). dt ( a ( t ) dt ( ˙ A · B . 5. d A ( t ) · B ( t )) = A · B + ˙ 6. d A ( t ) × ˙ B + B ( t )) = A × A × B . dt ( ˙ R ( t ) be a unit vector function, Theorem 1 (Unit-Vector Theorem) Let R ( t ) = ˆ u ( t ) for all t . Then: u is perpendicular to that of ˆ 1. The direction of ˆ
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