Chapter 2: Space Curves
Section 2.8: Resolution of R″ along T and N
Example 2.8.5
If C is the helix Rp=cosp i+sinp j+p k in Example 2.6.1, verify the validity of the decomposition R″p=ρ′T+κ ρ2N.
Solution
Mathematical Solution
By the usual techniques of the Frenet formalism, obtain the results in Table 2.8.5(a).
T=12−sinpcosp1
N=− cosp−sinp0
ρ=2
κ=1/2
Table 2.8.5(a) Items from the Frenet formalism
Then R″=−cos(p)−sin(p)0 and
ρ′T+κ ρ2N
=ddp2 T+2212 N
=0+1⋅N
=− cosp−sinp0
Indeed, the scalar projections of R″ on T and N, respectively, are
R″·T=N·T=0=ρ′
and
R″·N=N·N = 1 = 1222 = κ ρ2
Maple Solution - Interactive
Set R″ as an Atomic Identifier, and invoke it as an Atomic Identifier each time it is called.
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 ρ=R′p and R″p
Keyboard the norm bars.
Calculus palette: Differentiation operator
Context Panel: Evaluate and Display Inline
Context Panel: Assign to a Name≻rho
ⅆⅆ p R→assign to a nameρ =
Set R″ as an Atomic Identifier.
Apply the simplify command.
R″=ⅆ2ⅆp2 R→assign
Obtain T
Write R.
Context Panel: Student Multivariate Calculus≻Frenet Formalism≻Tangent Vector≻p
Context Panel: Student Multivariate Calculus≻Normalize≻Euclidean
Context Panel: Assign to a Name≻T
R = →tangent vector →Euclidean-normalize →assign to a nameT
Obtain N
Context Panel: Student Multivariate Calculus≻Frenet Formalism≻Principal Normal≻p
Context Panel: Assign to a Name≻N
R = →principal normal →2-normalize →assign to a nameN
Obtain the curvature κ
Context Panel: Student Multivariate Calculus≻Frenet Formalism≻Curvature≻p
Context Panel: Simplify≻Simplify
Context Panel: Assign to a Name≻kappa
R = →curvature142cosp2+2sinp22= simplify 12→assign to a nameκ
Compute ρ′T+κ ρ2N and compare with R″
ⅆⅆ p ρ T+κ ρ2 N =
Write R″ (as an Atomic Identifier).
R″ =
Compare the scalar projection of R″ on T with ρ′
Common Symbols palette: Dot product operator
R″·T = 0
ⅆⅆ p ρ = 0
Compare the scalar projection of R″ on N with κ ρ2
Common Symbols palette: Cross product operator
R″·N = sinp2+cosp2= simplify 1
κ ρ2 = 1= simplify 1
Maple Solution - Coded
To assign to the symbol R″, it must be converted to an Atomic Identifier. Any reference to it thereafter must also be write as an Atomic Identifier.
Install the Student VectorCalculus package.
withStudent:-VectorCalculus:
Apply the BasisFormat command.
Define R and obtain R″,T,N,ρ,κ
Define C as a position vector.
R≔cosp,sinp,p:
Apply the diff command to obtain R″, setting the name R″ as an Atomic Identifier.
R″≔simplifydiffR,p,p:
Obtain T with the TangentVector and simplify commands.
T≔TangentVectorR,normalized:
Obtain N with the PrincipalNormal and simplify commands.
N≔PrincipalNormalR,normalized:
Obtain ρ with the diff and simplify commands.
ρ≔NormdiffR,p:
Obtain κ with the Curvature and simplify commands.
κ≔simplifyCurvatureR :
Display R″,T,N,ρ,κ
R″,T,N,ρ,κ
Obtain the right-hand side of the decomposition formula
Apply the diff command to construct the right-hand side of the decomposition formula.
diffρ,p T+κ ρ2 N
Obtain the scalar projection of R″ on T and compare to ρ′
Apply the DotProduct command.
DotProductR″,T = 0
Obtain ρ′ via the diff command.
diffρ,p = 0
Obtain the scalar projection of R″ on N and compare to κ ρ2
Apply the DotProduct and simplify commands.
simplifyDotProductR″,N = 1
κ ρ2 = 1
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