verify derivative of a dot product.

yungman

Let [tex] \vec w(t) \;,\; \vec v (t) [/tex] be 3 space vectors that is a function of time t. I want to verify that:

[tex] \frac {d(\vec w \cdot \vec v)}{dt} = \vec v \cdot \frac { d\vec w}{dt} \;+\; \vec w \cdot \frac { d\vec v}{dt} [/tex]

I work through the verification by splitting w and v into x, y, z components, do the dot product and take the derivative to verify already. Just want to run this by the expert to confirm.

Thanks

Alan

arildno

That will work.

Note, however, that by definition, the dot product has the distributive property of multiplication:
[tex](u+du)\cdot(v+dv)=u\cdot{v}+u\cdot{dv}+v\cdot{du}+du\cdot{dv}[/tex]
For all vectors u,du,v and dv.

Thus, the result for the derivative ought to be apparent..

yungman

Thanks

But I don't see how

[tex](u+du)\cdot(v+dv)=u\cdot{v}+u\cdot{dv}+v\cdot{du}+du\cdot{dv}[/tex]


relate to my original question. Please explain.

Thanks

Alan

arildno

It shows that a dot product works "just like" a normal product, and thus, the differentiation rule "ought" to be the same (i.e, your result).

However, for rigorous verification, you should do as you've done.

yungman

Thanks