Formulas for Curvature and Normal Vectors

Calculus 3 Section 11.9

• Position Vector:

r(t) = 〈x(t), y(t), z(t)〉

• Velocity Vector:

v(t) = 〈x′(t), y′(t), z′(t)〉 = r′(t)

• Acceleration Vector:

a(t) = 〈x′′(t), y′′(t), z′′(t)〉 = v′(t) = r

′′(t)

• Arc Length:

s(t) =

∫

t

a

√

(x′(t))2 + (y′(t))2 + (z′(t))2dt =

∫

t

a

|v(t)|dt

ds

dt= |v(t)|

• Unit Tangent Vector:

T(t) =v

|v|

write a short description of T:

• Curvature:

κ(t) =

∣

∣

∣

∣

dT

dt

∣

∣

∣

∣

=1

|v|

∣

∣

∣

∣

dT

dt

∣

∣

∣

∣

=|T′(t)|

|r′(t)|=

|a× v|

|v|3

write a short description of κ:

• Principal Unit Vector:

N(t) =dT/ds

|dT/ds|=

1

κ

dT

ds=

dT/dt

|dT/dt|

write a short description of N:

• Acceleration:

a = aNN+ aTT where an = κ|v|2a =|a× v|

|v|and aT =

d2s

dt2=

a · v

|v|

write a short description of a:

1