Chapter 2: Space Curves
Section 2.7: Frenet-Serret Formalism
Example 2.7.6
With in the prescription in Table 2.7.2, obtain and graph the plane curve whose curvature is .
Solution
Mathematical Solution
Figure 2.7.6(a) is a graph of the position vector , whose components are obtained by the following calculations.
=
Figure 2.7.6(a) Graph of
By way of corroboration, obtain
and
from which it follows that = = .
Maple Solution - Interactive
Initialize
Tools≻Load Package: Student Vector Calculus
Loading Student:-VectorCalculus
Set the display format for vectors by executing the BasisFormat command to the right, or use the task template.
Obtain
Calculus palette: Definite integral template
Context Panel: Assign Name
Obtain and display the position-vector form of the plane curve
Write R.
Context Panel: Evaluate and Display Inline
Verify that
Context Panel: Student Vector Calculus≻Differentiate≻With Respect To≻
Context Panel: Assign to a Name≻T
Write T.
Context Panel: Student Vector Calculus≻ Norm≻Euclidean
Context Panel: Simplify≻Assuming Real
Context Panel: Student Vector Calculus≻Norm≻Euclidean
Graph the plane curve
Context Panel: Student Vector Calculus≻Conversions≻To List
Context Panel: Plots≻Plot Builder Set
Maple Solution - Coded
Install the Student VectorCalculus package.
Invoke the BasisFormat command.
Apply the int command to obtain .
Verifications
Use the diff command to obtain T, and to it, apply the Norm command.
Use the diff command to obtain , and to it, apply the Norm and simplify commands.
Graph
Apply the ConvertVector command to R, making it a PositionVector.
Apply the PlotPositionVector command to obtain the graph.
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