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10 Pages

### vDerivative.slides

Course: MATH 316, Fall 2009
School: Cal Poly Pomona
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Word Count: 635

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Derivative Vector expressed in terms of its smoothly varying (time dependent) magnitude &amp; direction (unit vector) functions. u t = u t u t du = u = uu + uu dt A Vector's Derivative is determined by its current magnitude &amp; direction, as well as their 1st rates of change. u = u u d u 2 = d u u dt dt Magnitude Variation 2 2 u u = u u + u u 2 u u = 2 u u 2 u u = u u u...

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Derivative Vector expressed in terms of its smoothly varying (time dependent) magnitude & direction (unit vector) functions. u t = u t u t du = u = uu + uu dt A Vector's Derivative is determined by its current magnitude & direction, as well as their 1st rates of change. u = u u d u 2 = d u u dt dt Magnitude Variation 2 2 u u = u u + u u 2 u u = 2 u u 2 u u = u u u = u u 2 u = u u ~ Vector's Strain Rate u = u uu u = u u ~ Vector's Strain Rate u = u uu u "increasing magnitude" u > 0 uu > 0 acute u "decreasing magnitude" u < 0 uu < 0 u obtuse u "constant (or extreme) magnitude" u = 0 uu = 0 u u In particular, for a (constant unit magnitude) unit vector function u=u t u = 1 = const. u = u u = 0 u u // to u to u u = u = = uu uu u + + uu uu u ; // - decomposition with = u ; u = 1u u u u u + u u u u u u u = 12 u u u + 12 u u u u u = uu u uu + uu u uu ; u = uu 2 u = u u + Qu u ; u = u u = u uu u Q u u u = ? uu = u = u u u uu check + + Qu u Qu u Qu u = u u ? ; u = uu u u = Qu u ? We will confirm independently! Algebraic expansion of the Q (?) vector: u u = 1 uu uu + uu Qu = 2 uu u = 1 uu 1 uu + uu u u u u + u = u u = u uu + u = u u u + u u u u = u u Qu = uu zero Conclude: Q-vector is INDEPENDENT of u's magnitude!! Moreover, Q u u = uu u uu 2bl X-product switch DXP Identity = u = = uu u - uu u 1 u- 0 u Q u u = u So that, & u = Qu u Q u u u = u u uu which leads to the desired confirmation, = viz u u u = u Qu u = Qu u u = Qu u ; check So that, Summarize: u u + uu u= u u + Qu u ; u = u u = u uu u uu = uu = Q u uu ? Next Issue: Geometric Interpretation of this Q-vector Geometric Interpretation of the Q-vector u(t) u(t + t) = 11sin N 1 N ~ RHRotation Rule Normal to u's rotation plane !" ~ angle of rotation (radians) u t + !t = u + !u 1 !" 1 u t =u u u + u = sin N sin u u + u u = N sin 0 + u u = N sin u u = N t t Pla t Rotation t ! u t+" u ne lim u u t 0 t Direction Change = lim t 0 sin 1 N t u u = n ~ u's instantaneous (inst.) angular rotation speed n ~ RHRotation Rule Normal to u's inst. rotation plane This establishes the physical meaning of the Q-vector uu = uu = n Q u = uu as u's instantaneous ANGULAR VELOCITY VECTOR as indeed it is normal to u's inst. plane of rotation, has a RHR rule sense, and a magnitude equal to u's inst. angular rotation speed (rad/sec). This crucial observ...

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