Derivative of cross-product of two vectors \frac{d}{dt} [\vec{u(t)} \times \vec{v(t)}],

Gregory Jones 2021-12-16 Answered
Derivative of cross-product of two vectors ddt[u(t)×v(t)], is it possible to find the cross-product of the two vectors first before differentiating?
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Expert Answer

xandir307dc
Answered 2021-12-17 Author has 35 answers
You can evaluate this expression in two ways:
- You can find the cross product first, and then differentiate it.
- Or you can use the product rule, which works just fine with the cross product:
ddt(u×v)=dudt×v+u×dvdt
Picking a method depends on the problem at hand. For example, the product rule is used to derive Frenet Serret formulas.

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Esta Hurtado
Answered 2021-12-18 Author has 39 answers

Working from the first principles:
u(t+δt)×v(t+δt)u(t)×v(t)=u(t+δt)×v(t+δt)u(t)×v(t+δt)+
=u(t)×v(t+δt)u(t)×v(t)=
=[u(t+δt)u(t)]×v(t+δt)+
=u(t)×[v(t+δt)v(t)]
Now divide by δ and take limit as δt0
On the other hand
ddt[ijkvxvyvzuxuyuz]=[ijkdvxdtdvydtdvzdtuxuyuz]+[ijkvxvyvzduzdtduydtduzdt]
Using the rule of differentiation of a determinant. One useful application of it is in the proof of Abel's identity (which before Wikipedia was known to me as OstrograZSKi-Liouville formula)

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