Chapter 2: Space Curves
Section 2.8: Resolution of R″ along T and N
Example 2.8.3
If C is the circle given by Rt=3 cost i+3 sint j (where the parameter t is time), verify the validity of the decomposition R..=v. T+κ v2N, where the overdot denotes differentiation with respect to t.
Solution
Mathematical Solution
By the usual techniques of the Frenet formalism, obtain the results in Table 2.8.3(a).
T=−sin(t)cos(t)
N=−cos(t)−sin(t)
v=3
κ=13
Table 2.8.3(a) Items from the Frenet formalism
Then R..= −3cos(t)sin(t) and
v.T+κ v2N
=ddt3 (−sin(t)cos(t))+1332−cos(t)−sin(t)
=0+3−cos(t)−sin(t)
=−3cos(t)sin(t)
Indeed, the scalar projections of R.. on T and N, respectively, are
R..·T=−3cos(t)sin(t)·−sin(t)cos(t)=0=ρ.
and
R..·N=−3cos(t)sin(t)·−cos(t)−sin(t) = 3 = 1332 = 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=3 cost,3 sint→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 t R = 3→assign to a nameρ
Set R″ as an Atomic Identifier.
R″=ⅆ2ⅆt2 R→assign
Obtain T
Write R.
Context Panel: Student Multivariate Calculus≻Frenet Formalism≻Tangent Vector≻t
Context Panel: Student Multivariate Calculus≻Normalize≻Euclidean
Context Panel: Assign to a Name≻T
R = →tangent vector →2-normalize →assign to a nameT
Obtain N
Context Panel: Student Multivariate Calculus≻Frenet Formalism≻Principal Normal≻t
Context Panel: Assign to a Name≻N
R = →principal normal →assign to a nameN
Obtain the curvature κ
Context Panel: Student Multivariate Calculus≻Frenet Formalism≻Curvature≻t
Context Panel: Simplify≻Simplify
Context Panel: Assign to a Name≻kappa
R = →curvature13sint2+cost2= simplify 13→assign to a nameκ
Compute ρ′T+κ ρ2N and compare with R″
ⅆⅆ t 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 = 3cost2+3sint2= simplify 3
κ ρ2 = 3= simplify 3
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.
withStudent:-VectorCalculus:
Apply the BasisFormat command.
Define R and obtain R..,T,N,ρ,κ
Define C as a position vector.
R≔3 cost,3 sint:
Apply the diff command to obtain R.., setting the name R.. as an Atomic Identifier.
R..≔diffR,t,t:
Obtain T with the TangentVector command.
T≔TangentVectorR,normalized:
Obtain N with the PrincipalNormal command.
N≔PrincipalNormalR,normalized:
Obtain ρ with the diff command.
ρ≔NormdiffR,t:
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,t 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 = 3
κ ρ2 = 3
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