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Verify derivative of a dot product. 
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#1
May1111, 12:44 PM

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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 


#2
May1111, 01:58 PM

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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.. 


#3
May1111, 02:39 PM

P: 3,883

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 


#4
May1111, 03:05 PM

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Verify derivative of a dot product.
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. 


#5
May1111, 03:15 PM

P: 3,883

Thanks



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