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Curvature Using Vector Functions 

For the following discussion, we will consider a parameterized curve defined by the vector function \(\displaystyle{ \vec{r}(t) = \langle x(t), y(t), z(t) \rangle }\) which is traversed once on the continuous interval \( a \leq t \leq b \). 
Some books use the Greek letter κ for curvature. 
The curvature of a smooth curve is a measure of how 'tight' or 'sharp' the curve is. If we have a smooth curve \(\vec{r}\) and we have a function s which is the arc length function, the curvature is defined to be
\(\displaystyle{ K(s) = \left\ \frac{d\vec{T}}{ds} \right\}\).
This equation for the curvature is not particularly useful for calculations. So we have several ways to write the equation of the curvature. But first notice, that the curvature is a scalar function, not a vector function. And since it is the norm of a vector, the curvature will always be positive.
Curvature Formula #1
For our first equation to use when calculating the curvature, we will use the chain rule to write \(\displaystyle{ \frac{d\vec{T}}{dt} = \frac{d\vec{T}}{ds} \cdot \frac{ds}{dt} }\). We can solve for \(d\vec{T}/ds\) to get
\(\displaystyle{ K = \left\ \frac{d\vec{T}}{ds} \right\ = \frac{\ d\vec{T}/dt \}{ \ ds/dt \ } = \frac{1}{\ \vec{v} \} \left\ \frac{d\vec{T}}{dt} \right\ }\)
Notice in the previous equation, we used \(ds/dt = \\vec{v}\\) to simplify the equation somewhat.
Now we can write the first curvature formula in a form that we can use for calculations.
\(\displaystyle{ K(t) = \frac{1}{\\vec{v}\} \left\ \frac{d\vec{T}}{dt} \right\ = \frac{\ \vec{T}'(t) \}{\\vec{r}'(t)\} }\)
Curvature Formula #2
If we define a vector \(\vec{a} = d\vec{v}/dt\) as an acceleration vector, a second curvature formula is
\(\displaystyle{ K=\frac{\\vec{v} \times \vec{a} \}{\\vec{v}\^3} }\)
Before working some practice problems, here is a quick video clip for you that should help you understand the curvature a bit better.
MIP4U  Determining Curvature of a Curve Defined by a Vector Valued Function [3min9secs]  
Here is a longer video explaining the derivation of some of the equations.
Louis Saumier  Curvature: Definition, Derivation [25min30secs]  
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Practice Problems 

Instructions   Unless otherwise instructed, find the curvature of these vector functions. Give your answers in exact, completely factored form.