Calculus III:

Vector functions
Curvature

Curvature

We will determine, in this section, the curvature of a smooth curve or helix , that is a curve with (t) continuous for all t's, and (t) ≠ (s).

At a given point, the curvature measures how fast a curve is changing direction .

The formal definition of curvature is:

κ = |d(t)/ds|

Where is the unit tangent and s is the arc length.

We can also use the following formulas:


or


In the case of a function of one variable, y = f(x), we have

(t) = x + f(x), and the curvature is

κ = |f'(x)|/[1 + (f'(x))²]^3/2

Example

Let's determine the curvature for the vector function:

(t) = (1, t, 2t)

(t) = 〈 0, 1, 2 〉

||(t)|| = √5

(t) = 〈 0, 1, 2 〉 /√5

'(t) = 〈 0, 0, 0 〉 /√5 =

|| '(t)|| = 0

The curvature κ = 0/√5 = 0