Chapter 2: Space Curves
Section 2.4: Curvature
Example 2.4.6
Obtain the curvature of the helix defined by Rp=cosp i+sinp j+p k.
Solution
Mathematical Solution
To compute the curvature via the formula κ=∥R′×R″∥/ρ3 , first obtain
R′=−sin(p)cos(p)1, R″=−cos(p)−sin(p)0, ρ=∥R′∥ = −sinp2+cos2p+1=2
then calculate R′×R″ = |ijk−sinpcosp1−cosp−sinp0| = sin(p)−cos(p)1 so that
∥R′×R″∥ρ3 = sin2p+cos2p+123 = 222 = 12
To compute the curvature via the definition κ=T′s, first apply the chain rule so that
T′s=dTdp dpds=dTdp 1ρ
Since Tp=R′p/ρ, it follows that T′s=12−cos(p)−sin(p)0 12, so that
T′s = 12 −cosp2+−sinp2 = 12⋅1=12
Maple Solution - Interactive
Initialize
Tools≻Load Package: Student Vector Calculus
Loading Student:-VectorCalculus
Execute the BasisFormat command at the right, or use the task template.
BasisFormatfalse:
Write R=… as per Table 1.1.1.
Context Panel: Assign Name
R=cosp,sinp,p→assign
Obtain the curvature
Write R.
Context Panel: Evaluate and Display Inline
Context Panel: Student Vector Calculus≻Frenet Formalism≻Curvature≻p
Context Panel: Simplify≻Simplify
R = →curvature142cosp2+2sinp22= simplify 12
Obtain the curvature as κ=∥R′×R″∥/ρ3
Make R′ an Atomic Identifier.
Calculus palette: Differentiation operator
R′=ⅆⅆ p R→assign
Make R″ an Atomic Identifier.
R″=ⅆ2ⅆp2 R→assign
Keyboard the norm bars.
ρ=∥R′∥→assign
Make R′ and R″ Atomic Identifiers.
Common Symbols palette: Cross product operator
R′×R″ρ3 = 12
Obtain the curvature as κ=T′s
T=R′/ρ→assign
Context Menu: Evaluate and Display Inline
ⅆⅆ p T/ρ = 12
Display computed quantities
R′,R″,ρ,T
Maple Solution - Coded
Install the Student VectorCalculus package.
withStudent:-VectorCalculus:
Define the helix as a position vector.
R≔cosp,sinp,p:
Apply the Curvature command.
simplifyCurvatureR,p = 12
Use the TangentVector command to obtain Tp.
T≔TangentVectorR,p,normalized:
Apply the diff command to obtain R′p.
Apply the Norm command to obtain ρ=∥R′∥.
ρ≔NormdiffR,p:
Apply the diff, CrossProduct, and Norm commands.
NormCrossProductdiffR,p,diffR,p,pρ3 = 12
Apply the diff and Norm commands.
NormdiffT,pρ = 12
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