I am looking at line integrals and work..
According to my notes: $$\frac{\overrightarrow{R}}{dt}=\hat{T} \cdot |\frac{d\overrightarrow{R}}{dt}|=\hat{T} \sqrt{x'(t)^2+y'(t)^2+z'(t)^2}$$
So,the work is given by the formula: $$W=\int_A^B \overrightarrow{F}(s) \frac{d\overrightarrow{R}}{ds}ds=\int_a^b \overrightarrow{F}(s) \overrightarrow{T}\sqrt{x'(t)^2+y'(t)^2+z'(t)^2} $$
But... why is it like that: $\displaystyle{\frac{\overrightarrow{R}}{dt}=\hat{T} \cdot |\frac{d\overrightarrow{R}}{dt}|}$ ?
And what is $\hat{T}$ ? What does it represent?
This is the unit tangent vector. See here for some more information about it. It is the unit direction vector of the tangent line to each point of the parametric curve $\vec{R}(t)$.