Definition of the normal vector to a curve I'm just a bit confused.
I know that given a curve $r(s)$,
$$T'(s) = k(s)N(s) $$
where $k(s)$ is the curvature. 
And I know that $N(s)$ (the normal) is in the direction of $r''(s)$ so why is the normal not defined as 
$$N(s) = \frac{r''(s)}{|r''(s)|}$$ but instead defined in terms of curvature and derivative of tangent?
 A: The answer is pretty straightforward. 
First, @Trunk, these are the standard notation for tangent, normal, and binormal in the Frenet frame in differential geometry. See, for instance, ONeill's Elementary Differential Geometry, or Google "Frenet Frame". 
Next: Since 
$$
T(s) = r'(s)
$$
if you compute 
$$
T'(s)
$$
you get 
$$
r''(s)
$$
If you normalize this, you get $N(s)$, according to your definition. And if you define $k(s) = 1/|r''(s)|$, you get that 
$$
T'(s) = k(s) N(s)
$$
In other words, both definitions arrive at the same place. One of them requires saying that $k$ is $1/r''$, the other makes that a consequence of the definition. 
There's one subtlety: the 2D case. In that case, it makes some sense to say that since $N$ is orthogonal to $T$ in general, you can just define $N$ to be $T$ rotated 90 degrees counterclockwise. When you do this, and write
$$
T'(s) = k(s) N(s)
$$
it turns out that $k$ isn't just $1/|r''(s)|$, but rather that it can be positive or negative, and is actually $1/r''(s)$; in the plane (and only in the plane) it makes sense to talk about "positive" and "negative" curvature when you make this choice. 
Frankly, I find this slightly confusing to students, and usually start in 3-space until they're comfortable with curvature and torsion, and then look at curves in the plane as a special case, and "discover" the possibility (in that special case) of adding a sign to the curvature. 
A: Your question seems confused.
If by 'normal', you mean the vector that is normal to a curve at each point along it, then I would see it being defined in relation to the first derivative of the curve function since the slope of the normal and tangent are negative reciprocal.
What is this $ r(s), N(s)$ and $T'(s) $ ?
Explain your nomenclature and maybe we can help you.
