Register to reply

Verify derivative of a dot product.

by yungman
Tags: derivative, product, verify
Share this thread:
yungman
#1
May11-11, 12:44 PM
P: 3,898
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
Phys.Org News Partner Science news on Phys.org
Experts defend operational earthquake forecasting, counter critiques
EU urged to convert TV frequencies to mobile broadband
Sierra Nevada freshwater runoff could drop 26 percent by 2100
arildno
#2
May11-11, 01:58 PM
Sci Advisor
HW Helper
PF Gold
P: 12,016
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
#3
May11-11, 02:39 PM
P: 3,898
Quote Quote by arildno View Post
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..
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
#4
May11-11, 03:05 PM
Sci Advisor
HW Helper
PF Gold
P: 12,016
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.
yungman
#5
May11-11, 03:15 PM
P: 3,898
Thanks


Register to reply

Related Discussions
Dot Product of a Vector with its Derivative Calculus 2
Taking the derivative of a dot product Calculus & Beyond Homework 0
Derivative of the cross and dot product Advanced Physics Homework 3
Cross product derivative Calculus & Beyond Homework 3
Derivative of a dot product Calculus 4