u t = u t u t d u = u u + u u d t = u A Vectors Derivative - - PowerPoint PPT Presentation

Vector Derivative expressed in terms of its smoothly varying

(time dependent) magnitude & direction (unit vector) functions.

u t = u t u t du dt = u

• = u
• u + uu
• A Vector’s Derivative is determined by its current magnitude

& direction, as well as their 1st rates of change.

u

2= u•u

d dt u

2 = d

dt u•u 2uu

• = u
• •u + u•u
• 2uu
• = 2u•u
• ÷ 2

uu

• = u•u
• ÷ u

2= u•u

• u = u

u

• = u•u
• u•u ~ Vector's Strain Rate

Magnitude Variation

• u = u

u

• = u•u
• u•u ~ Vector's Strain Rate

"increasing magnitude"

• u > 0
• u•u
• > 0
• "decreasing magnitude"
• u < 0
• u•u
• < 0
• "constant (or extreme) magnitude"
• u = 0
• u•u
• = 0
• In particular, for a (constant unit magnitude) unit vector function

u = u t

• u = 1 = const.

u

• = u•u
• = 0

u

• u

u u

• acute

acute

u u

• btuse

u

u

u

• =

u

• u

+ uu

• u
• =

u•u

• u

+ uu

• u

; // - decomposition with = u

= u u •u

• u

u + u u u

• u

u

; u = 1 uu

= 1 u

2 u•u

• u +

1 u

2 uu

• u

= u•u

• u•u u

+ uu

• u•u u

; u

2 = u•u

u

• =
• uu

+ Quu ;

• u= u•u
• u•u = u
• u

Qu uu

• u•u =?

= u

• u u

+ Quu u

• =

u

• u

+ Quu

; u = uu

// to u to u

check

We will confirm independently!

Quu =

? uu

• ÷ u

u

• =

? Quu

Algebraic expansion of the Q (?) vector: Conclude: Q-vector is INDEPENDENT of u’s magnitude!!

zero

Qu = uu

• u•u= 1

u

2 uu u

• u + uu
• = 1

u uu 1 u u

• u + uu
• = u u
• u

u + u

• = u
• uu + u
• =
• u uu + uu
• Qu = uu
• u•u= uu

Quu= uu

• u

= u u

• u

= u•u u

• u
• •u u

= 1 u

• 0 u

Quu= u

• Moreover,

2bl X-product switch DXP Identity

Qu uu

• u•u= uu
• &

u

• = Quu

So that, which leads to the desired confirmation, viz

u

• = uu
• = u Quu = Qu uu = Quu ;

check

Summarize:

u

• =

u

• u + uu
• uu + Quu ;
• u= u•u
• u•u = u
• u

Qu uu

• u•u = uu
• =?

Next Issue: Geometric Interpretation of this Q-vector

1

u(t)u(t+t)= 11sin N u u + u = sin N uu + uu= sin

• N

0 + uu= sin

• N

u u t = sin

• t N

lim

t 0 u u

t = lim

t 0

sin

• t N

uu

• =
• n
• ~ u's instantaneous (inst.) angular rotation speed

n~ RHRotation Rule Normal to u's inst. rotation plane

Geometric Interpretation of the Q-vector

Direction Change

1 1

• 1

N ~ RHRotation Rule Normal to u's rotation plane

~ angle of rotation (radians)

u t

• u

t +

• t

R

• t

a t i

• n

P l a n e

u t = u u t+t = u+u

Qu= uu

• u•u = uu
• =
• n

This establishes the physical meaning of the Q-vector 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 observation justifies the adoption of the formal name change

Qu u

and the consequent resummarization:

u

• =

u

• u + uu
• uu + uu ;
• u= u•u
• u•u = u
• u ~ u's strain rate

u= uu

• u•u = uu
• =
• n ~ u's

) velocity vector

u's instantaneous rotation plane

• ~ u's inst. angular rotation speed (rad/sec)
• u

n RHON triad ~ u's perpendicular (in-plane) "turning direction" ~ u's TURN SIGNAL u

• u
• u ~ u's inst. angular velocity vector

u//

• =
• uu = u
• u

u

• = uu

=

• n uu

= u

• nu

u

• = u

u

• =
• uu + uu = u
• u + u

u = u(t) = uu

1

u n ~ RHRotation Rule normal to u's inst. rotation plane

1 1

nu

u

• u

• u

n RHON triad

• u
• 1

u

1 1

nu

• ~ u's TURN SIGNAL

u's instantaneous rotation plane

• ~ u's inst. angular rotation speed (rad/sec)

u ~ u's inst. angular velocity vector n ~ RHRotation Rule normal to u's inst. rotation plane

u

• ~ u's perpendicular (in-plane) "turning direction"

Rate of change of a Smoothly Varying Unit Vector Function

u

• = uu =