The_Cambridge_Handbook_of_Physics_Formulas

2.6 Mensuration
39
Curve measure
Length of plane
curve l =
∫ b
Surface of
revolution A =2π
Volume of
revolution
a
V = π
[
1+
∫ b
a
∫ b
a
y
( ) ] 2 1/2
dy
dx (2.279)
dx
[
1+
a
b
y(x)
l
start point
end point
plane curve
length
( ) ] 2 1/2
dy
dx (2.280) A surface area
dx
y 2 dx (2.281) V volume
[ ( ) ]
Radius of
2 3/2 ( dy d 2 ) −1
y
curvature ρ = 1+
dx dx 2 (2.282)
ρ
radius of
curvature
2
Differential geometry a
Unit tangent
Unit principal normal
ˆτ = ṙ
|ṙ| = ṙ
v
ˆn =
¨r −˙v ˆτ
|¨r −˙v ˆτ|
τ tangent
(2.283) r curve parameterised by r(t)
v |ṙ(t)|
(2.284) n principal normal
Unit binormal ˆb = ˆτ ׈n (2.285) b binormal
Curvature
κ = |ṙרr|
|ṙ| 3 (2.286) κ curvature
Radius of curvature
ρ = 1 κ
(2.287) ρ radius of curvature
Torsion
λ = ṙ ·(¨r×... r )
|ṙרr| 2 (2.288) λ torsion
ˆn
˙ˆτ = κvˆn (2.289)
osculating
plane
Frenet’s formulas
˙ˆn = −κv ˆτ +λvˆb (2.290)
normal plane
ˆτ
˙ˆb = −λvˆn (2.291)
ˆb
r
rectifying
plane
origin
a For a continuous curve in three dimensions, traced by the position vector r(t).