VECTOR CALCULUS
Therefore, remembering thatu=x, from (10.12) the arc length betweenx=aandx=b
is given by
s=∫ba√
dr
du·
dr
dudu=∫ba√
1+
(
dy
dx) 2
dx.This result was derived using more elementary methods in chapter 2.
If a curveCis described byr(u) then, by considering figures 10.1 and 10.3, wesee that, at any given point on the curve,dr/duis a vector tangent toCat that
point, in the direction of increasingu. In the special case where the parameteru
is the arc lengthsalong the curve thendr/dsis aunittangent vector toCand is
denoted byˆt.
The rate at which the unit tangent ˆtchanges with respect tosis given bydˆt/ds, and its magnitude is defined as thecurvatureκof the curveCat a given
point,
κ=∣
∣
∣
∣dˆt
ds∣
∣
∣
∣=∣
∣
∣
∣d^2 ˆr
ds^2∣
∣
∣
∣.We can also define the quantityρ=1/κ, which is called theradius of curvature.
Sinceˆtis of constant (unit) magnitude, it follows from (10.8) that it is perpen-dicular todˆt/ds. The unit vector in the direction perpendicular toˆtis denoted
byˆnand is called theprincipal normalat the point. We therefore have
dˆt
ds=κnˆ. (10.13)The unit vectorˆb=ˆt׈n, which is perpendicular to the plane containingˆtandnˆ, is called thebinormaltoC. The vectorsˆt,nˆandbˆform a right-handed
rectangular cooordinate system (ortriad) at any given point onC(see figure 10.3).
Asschanges so that the point of interest moves alongC, the triad of vectors also
changes.
The rate at which bˆchanges with respect tosis given bydbˆ/dsand is ameasure of thetorsionτof the curve at any given point. Sincebˆis of constant
magnitude, from (10.8) it is perpendicular todbˆ/ds. We may further show that
dbˆ/dsis also perpendicular toˆt, as follows. By definitionbˆ·ˆt=0,whichon
differentiating yields
0=d
ds(
bˆ·ˆt)
=dˆb
ds·ˆt+ˆb·dˆt
ds=dˆb
ds·ˆt+ˆb·κˆn=dˆb
ds·ˆt,where we have used the fact thatbˆ·ˆn= 0. Hence, sincedˆb/dsis perpendicular
to bothˆbandˆt, we must havedˆb/ds∝ˆn. The constant of proportionality is−τ,