Chapter 2: Space Curves
Section 2.8: Resolution of R″ along T and N
If C is the plane curve described by the position vector Rp=epcosp i+epsinp j, 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 2.8.2(a).
Table 2.8.2(a) Items from the Frenet formalism
Then R″=2 ep −sin(p)cos(p) and
=ddp2 ep ⁢cosp−sinp⁢cosp+sinp2+2ep2e−p2−⁢cos⁡p+sin⁡p⁢cosp−sinp2
Indeed, the scalar projections of R″ on T and N, respectively, are
R″·N=2ep2−sin(p)cos(p)·−⁢cos⁡p+sin⁡p⁢cosp−sinp = 2ep = e−p22ep2 = κ ρ2
Maple Solution - Interactive
Set R″ as an Atomic Identifier, and invoke it as an Atomic Identifier each time it is called.
Be sure to use Maple's exponential "e" for all references to the exponential function.
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=ⅇp cosp,ⅇp sinp→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 = 2⁢ⅇ2⁢p→assign to a nameρ
Set R″ as an Atomic Identifier.
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 = ⅇp⁢cos⁡pⅇp⁢sin⁡p→tangent vectorⅇp⁢cos⁡p−sin⁡pⅇp⁢cos⁡p+sin⁡p→Euclidean-normalize12⁢2⁢ⅇp⁢cos⁡p−sin⁡pⅇ2⁢p12⁢2⁢ⅇp⁢cos⁡p+sin⁡pⅇ2⁢p→assuming real12⁢2⁢cos⁡p−sin⁡p12⁢2⁢cos⁡p+sin⁡p→assign to a nameT
Context Panel: Student Multivariate Calculus≻Frenet Formalism≻Principal Normal≻p
Context Panel: Assign to a Name≻N
R = ⅇp⁢cos⁡pⅇp⁢sin⁡p→principal normal−12⁢2⁢ⅇp⁢cos⁡p+sin⁡pⅇ2⁢p12⁢2⁢ⅇp⁢cos⁡p−sin⁡pⅇ2⁢p→2-normalize−12⁢2⁢ⅇp⁢cos⁡p+sin⁡pⅇ2⁢p12⁢2⁢ⅇp⁢cos⁡p−sin⁡pⅇ2⁢p→assuming real−12⁢2⁢cos⁡p+sin⁡p12⁢2⁢cos⁡p−sin⁡p→assign to a nameN
Obtain the curvature κ
Context Panel: Student Multivariate Calculus≻Frenet Formalism≻Curvature≻p
Context Panel: Assign to a Name≻kappa
Rⅇp⁢cos⁡pⅇp⁢sin⁡p = →curvature12⁢−12⁢2⁢ⅇp⁢cos⁡p−ⅇp⁢sin⁡pⅇ2⁢p−2⁢ⅇp⁢sin⁡pⅇ2⁢p2+−12⁢2⁢ⅇp⁢sin⁡p+ⅇp⁢cos⁡pⅇ2⁢p+2⁢ⅇp⁢cos⁡pⅇ2⁢p2⁢2ⅇ2⁢p→assuming real12⁢2⁢ⅇ−p→assign to a nameκ
Compute ρ′T+κ ρ2N and compare with R″
ⅆⅆ p ρ T+κ ρ2 N = ⅇ2⁢p⁢cos⁡p−sin⁡p−ⅇ−p⁢ⅇ2⁢p⁢cos⁡p+sin⁡pⅇ2⁢p⁢cos⁡p+sin⁡p+ⅇ−p⁢ⅇ2⁢p⁢cos⁡p−sin⁡p→assuming real−2⁢ⅇp⁢sin⁡p2⁢ⅇp⁢cos⁡p
R″ = −2⁢ⅇp⁢sin⁡p2⁢ⅇp⁢cos⁡p
Compare the scalar projection of R″ on T with ρ′
Common Symbols palette: Dot product operator
R″·T = −ⅇp⁢sin⁡p⁢2⁢cos⁡p−sin⁡p+ⅇp⁢cos⁡p⁢2⁢cos⁡p+sin⁡p= simplify ⅇp⁢2
Calculus palette: Differentiation operator
ⅆⅆ p ρ = 2⁢ⅇ2⁢p→assuming realⅇp⁢2
Compare the scalar projection of R″ on N with κ ρ2
Common Symbols palette: Cross product operator
Context Panel: Simplify≻Simplify
R″·N = ⅇp⁢sin⁡p⁢2⁢cos⁡p+sin⁡p+ⅇp⁢cos⁡p⁢2⁢cos⁡p−sin⁡p= simplify ⅇp⁢2
κ ρ2 = 2⁢ⅇ−p⁢ⅇ2⁢p= simplify ⅇp⁢2
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, being sure to use Maple's exponential e.
R≔ⅇp cosp,ⅇp sinp:
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 assuming p∷real
Obtain the scalar projection of R″ on T and compare to ρ′
Apply the DotProduct and simplify commands.
Obtain ρ′ via the diff command.
diffρ,p = 2⁢ⅇp
Obtain the scalar projection of R″ on N and compare to κ ρ2
Apply the DotProduct
and simplify commands.
simplifyDotProductR″,N = 2⁢ⅇp
Apply the simplify command.
simplifyκ ρ2 = 2⁢ⅇp
<< Previous Example Section 2.8
Next Example >>
© Maplesoft, a division of Waterloo Maple Inc., 2021. All rights reserved. This product is protected by copyright and distributed under licenses restricting its use, copying, distribution, and decompilation.
For more information on Maplesoft products and services, visit www.maplesoft.com
Download Help Document
What kind of issue would you like to report? (Optional)