Chapter 2: Space Curves
Section 2.8: Resolution of R″ along T and N
If C is the curve given by Rp=3 p−p3 i+3 p2 j+3 p+p3 k in Example 2.6.4, verify the validity of the decomposition R″p=ρ′T+κ ρ2N.
By the usual techniques of the Frenet formalism, obtain the results in Table 2.8.8(a).
Table 2.8.8(a) Items from the Frenet formalism
Then R″=−6⁢p66⁢p and
=ddp3⁢2⁢p2+1 T+13⁢p2+12 3⁢2⁢p2+12 N
=6⁢2⁢p ⁢1− p22(p2+1)2⁢pp2+11/2 +6−2⁢pp2+1−p2−1p2+10
Indeed, the scalar projections of R″ on T and N, respectively, are
=−6 p66 p·⁢1− p22(p2+1)2⁢pp2+11/2
=−6 p66 p·−2⁢pp2+1−p2−1p2+10
Maple Solution - Interactive
Set R″ as an Atomic Identifier, and invoke it as an Atomic Identifier each time it is called.
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
R=3 p−p3,3 p2,3 p+p3→assign
Obtain ρ=R′p and R″p
Keyboard the norm bars.
Calculus palette: Differentiation operator
Context Panel: Evaluate and Display Inline
Context Panel: Simplify≻Assuming Positive
Context Panel: Assign to a Name≻rho
ⅆⅆ p R = 3⁢2⁢p2+12→assuming positive3⁢2⁢p2+1→assign to a nameρ
Context Panel: Simplify≻Simplify
Context Panel: Assign to a Name≻Temp
Set R″ as an Atomic Identifier and equate to Temp.
ⅆ2ⅆp2 R = −6⁢p66⁢p= simplify −6⁢p66⁢p→assign to a nameTemp
Context Panel: Student Multivariate Calculus≻Frenet Formalism≻Tangent Vector≻p
Context Panel: Student Multivariate Calculus≻Normalize≻Euclidean
Context Panel: Simplify≻Assuming Real
Context Panel: Assign to a Name≻T
R = −p3+3⁢p3⁢p2p3+3⁢p→tangent vector−3⁢p2+36⁢p3⁢p2+3→Euclidean-normalize16⁢2⁢−3⁢p2+3p2+122⁢pp2+1216⁢2⁢3⁢p2+3p2+12→assuming real−12⁢2⁢p2−1p2+12⁢pp2+112⁢2→assign to a nameT
Context Panel: Student Multivariate Calculus≻Frenet Formalism≻Principal Normal≻p
Context Panel: Assign to a Name≻N
R−p3+3⁢p3⁢p2p3+3⁢p = →principal normal12⁢2⁢csgn⁡1,p2+1⁢p4−4⁢p⁢csgn⁡p2+1−csgn⁡1,p2+1p2+12−2⁢csgn⁡1,p2+1⁢p3+p2⁢csgn⁡p2+1+p⁢csgn⁡1,p2+1−csgn⁡p2+1p2+12−12⁢2⁢csgn⁡1,p2+1→2-normalize12⁢2⁢csgn⁡1,p2+1⁢p4−4⁢p⁢csgn⁡p2+1−csgn⁡1,p2+1csgn⁡1,p2+12⁢p4+2⁢csgn⁡1,p2+12⁢p2+csgn⁡1,p2+12+2p2+12⁢p2+12−2⁢csgn⁡1,p2+1⁢p3+p2⁢csgn⁡p2+1+p⁢csgn⁡1,p2+1−csgn⁡p2+1csgn⁡1,p2+12⁢p4+2⁢csgn⁡1,p2+12⁢p2+csgn⁡1,p2+12+2p2+12⁢p2+12−12⁢2⁢csgn⁡1,p2+1csgn⁡1,p2+12⁢p4+2⁢csgn⁡1,p2+12⁢p2+csgn⁡1,p2+12+2p2+12→assuming real−2⁢pp2+1−p2−1p2+10→assign to a nameN
Obtain the curvature κ
Context Panel: Student Multivariate Calculus≻Frenet Formalism≻Curvature≻p
Context Panel: Assign to a Name≻kappa
R−p3+3⁢p3⁢p2p3+3⁢p = →curvature16⁢−16⁢2⁢−3⁢p2+3⁢csgn⁡1,p2+1csgn⁡p2+12⁢p2+1−13⁢2⁢−3⁢p2+3⁢pcsgn⁡p2+1⁢p2+12−2⁢pcsgn⁡p2+1⁢p2+12+−2⁢p⁢csgn⁡1,p2+1csgn⁡p2+12⁢p2+1−2⁢2⁢p2csgn⁡p2+1⁢p2+12+2csgn⁡p2+1⁢p2+12+−16⁢2⁢3⁢p2+3⁢csgn⁡1,p2+1csgn⁡p2+12⁢p2+1−13⁢2⁢3⁢p2+3⁢pcsgn⁡p2+1⁢p2+12+2⁢pcsgn⁡p2+1⁢p2+12⁢2csgn⁡p2+1⁢p2+1→assuming real13⁢p2+12→assign to a nameκ
Compute ρ′T+κ ρ2N and compare with R″
ⅆⅆ p ρ T+κ ρ2 N = −6⁢p⁢p2−1p2+1−12⁢pp2+112⁢p2p2+1−6⁢p2−1p2+16⁢p→assuming real−6⁢p66⁢p
R″ = −6⁢p66⁢p
Compare the scalar projection of R″ on T with ρ′
Common Symbols palette: Dot product operator
R″·T = 3⁢p⁢2⁢p2−1p2+1+6⁢2⁢pp2+1+3⁢2⁢p= simplify 6⁢2⁢p
ⅆⅆ p ρ = 6⁢2⁢p
Compare the scalar projection of R″ on N with κ ρ2
Common Symbols palette: Cross product operator
R″·N = 12⁢p2p2+1−6⁢p2−1p2+1= simplify 6
κ ρ2 = 6
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 written as an Atomic Identifier.
Install the Student VectorCalculus package.
Apply the BasisFormat command.
Define R and obtain R″,T,N,ρ,κ
Define C as a position vector.
R≔3 p−p3,3 p2,3 p+p3:
Apply the diff command to obtain R″, setting the name R″ as an Atomic Identifier.
Obtain T with the TangentVector and simplify commands.
Temp≔TangentVectorR,normalized:T≔simplifyTemp assuming p∷real:
Obtain N with the PrincipalNormal and simplify commands.
Temp≔PrincipalNormalR,normalized:N≔simplifyTemp assuming p∷real:
Obtain ρ with the diff and simplify commands.
ρ≔simplifyNormdiffR,p assuming p∷real:
Obtain κ with the Curvature and simplify commands.
κ≔simplifyCurvatureR assuming p∷real:
Obtain the right-hand side of the decomposition formula
Apply the diff and simplify commands to construct the right-hand side of the decomposition formula.
simplifydiffρ,p T+κ ρ2 N
Obtain the scalar projection of R″ on T and compare to ρ′
Apply the DotProduct and simplify commands.
simplifyDotProductR″,T = 6⁢2⁢p
Obtain ρ′ via the diff
diffρ,p = 6⁢2⁢p
Obtain the scalar projection of R″ on N and compare to κ ρ2
Apply the DotProduct
and simplify commands.
simplifyDotProductR″,N = 6
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