Chapter 2: Space Curves
Section 2.4: Curvature
Obtain the curvature of the helix defined by Rp=cosp i+sinp j+p k.
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
Tools≻Load Package: Student Vector Calculus
Execute the BasisFormat command at the right, or use the
Write R=… as per Table 1.1.1.
Context Panel: Assign Name
Obtain the curvature
Context Panel: Evaluate and Display Inline
Context Panel: Student Vector Calculus≻Frenet Formalism≻Curvature≻p
Context Panel: Simplify≻Simplify
R = cos⁡psin⁡pp→curvature14⁢2⁢cos⁡p2+2⁢sin⁡p2⁢2= 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.
Keyboard the norm bars.
Make R′ and R″ Atomic Identifiers.
Common Symbols palette: Cross product operator
R′×R″ρ3 = 12
Obtain the curvature as κ=T′s
Context Menu: Evaluate and Display Inline
ⅆⅆ p T/ρ = 12
Display computed quantities
Maple Solution - Coded
Install the Student VectorCalculus package.
Define the helix as a position vector.
Apply the Curvature command.
simplifyCurvatureR,p = 12
Use the TangentVector command to obtain Tp.
Apply the diff command to obtain R′p.
Apply the Norm command to obtain ρ=∥R′∥.
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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