What is the Derivation of Work in Classical Dynamics?

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Homework Statement


I'm working through a derivation for work in Thornton's Classical Dynamics but I'm stuck at one step.
\begin{align}<br /> \vec{F} \cdot d\vec{r} &amp;= m\frac{d\vec{v}}{dt} \cdot \frac{d\vec{r}}{dt}dt = m\frac{d\vec{v}}{dt} \cdot \vec{v}dt \\ &amp;= \frac{m}{2}\frac{d}{dt}(\vec{v}\cdot \vec{v})dt \end{align}

I'm having trouble getting from the last equality on the first line to the second line.

Homework Equations


The Attempt at a Solution


I don't know what the mathematical reasoning is.
 
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I don't get it too. Might be a mistake.
 
sloane729 said:
Okay I understand now from here: http://en.wikipedia.org/wiki/Kinetic_energy#Derivation
But I've never seen differentials like d(\vec{v}\cdot\vec{v}) before. Why am I not allowed to say d(\vec{v}\cdot\vec{v}) = dv^2?
You can't say that because you haven't defined "v^2". Since v is a vector, what you really mean is \vec{v}\cdot\vec{v}= ||\vec{v}||^2.
But, the basic idea is correct:
\frac{d(\vec{v}\cdot\vec{v})}{dt}= 2\vec{v}\cdot\frac{d\vec{v}}{dt}
 
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