Chapter 2: Space Curves
Section 2.4: Curvature
Example 2.4.6
Obtain the curvature of the helix defined by .
Solution
Mathematical Solution
To compute the curvature via the formula , first obtain
, , =
then calculate = = so that
= = =
To compute the curvature via the definition , first apply the chain rule so that
Since , it follows that , so that
= =
Maple Solution - Interactive
Initialize
Tools≻Load Package: Student Vector Calculus
Loading Student:-VectorCalculus
Execute the BasisFormat command at the right, or use the task template.
Write as per Table 1.1.1.
Context Panel: Assign Name
Obtain the curvature
Write R.
Context Panel: Evaluate and Display Inline
Context Panel: Student Vector Calculus≻Frenet Formalism≻Curvature≻
Context Panel: Simplify≻Simplify
=
Obtain the curvature as
Make an Atomic Identifier.
Calculus palette: Differentiation operator
Keyboard the norm bars.
Make and Atomic Identifiers.
Common Symbols palette: Cross product operator
Context Menu: Evaluate and Display Inline
Display computed quantities
Maple Solution - Coded
Install the Student VectorCalculus package.
Define the helix as a position vector.
Apply the Curvature command.
Use the TangentVector command to obtain .
Apply the diff command to obtain .
Apply the Norm command to obtain .
Apply the diff, CrossProduct, and Norm commands.
Apply the diff and Norm commands.
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