Chapter 2: Space Curves
Section 2.7: Frenet-Serret Formalism
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Example 2.7.15
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If is the curve given by in Example 2.6.4,
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Obtain its Darboux vector .
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c)
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Show that , where primes denote differentiation with respect to arc length .
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Solution
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Mathematical Solution
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Part (a)
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By the usual techniques, obtain the items in Table 2.7.15(a).
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Table 2.7.15(a) Frenet formalism
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The Darboux vector is then
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Part (b)
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The derivatives of the vectors in the TNB-frame must be taken with respect to the arc length . Since R is given in terms of the parameter , the chain rule must be invoked. Hence, .
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= =
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= =
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= =
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Part (c)
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The derivatives of the vectors in the TNB-frame must be taken with respect to the arc length . Since R is given in terms of the parameter , the chain rule must be invoked. Hence, .
= =
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Maple Solution - Interactive
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Part (a)
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Initialize
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Tools≻Load Package: Student Vector Calculus
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Loading Student:-VectorCalculus
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Execute the BasisFormat command at the right, or use the
task template to set the display of vectors as columns.
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Frenet formalism:
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Context Panel: Assign Name
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Keyboard the norm bars.
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Calculus palette: Differentiation operator
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Context Panel: Evaluate and Display Inline
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Context Panel: Simplify≻Assuming Positive
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Context Panel: Assign to a Name≻rho
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Write R.
Context Panel: Evaluate and Display Inline
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Context Panel: Student Vector Calculus≻Frenet Formalism≻Curvature≻
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Context Panel: Simplify≻Assuming Positive
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Context Panel: Assign to a Name≻kappa
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Write R.
Context Panel: Evaluate and Display Inline
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Context Panel: Student Vector Calculus≻Frenet Formalism≻Torsion≻
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Context Panel: Simplify≻Assuming Positive
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Context Panel: Assign to a Name≻tau
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Write R.
Context Panel: Evaluate and Display Inline
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Context Panel: Student Vector Calculus≻Frenet Formalism≻TNB Frame≻
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Context Panel: Simplify≻Assuming Positive
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Context Panel: Assign to a Name≻
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Context Panel: Assign Name
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Context Panel: Assign Name
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Context Panel: Assign Name
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Obtain and display the Darboux vector
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Context Panel: Assign Name
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Write d.
Context Panel: Evaluate and Display Inline
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Part (b)
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Calculus palette: Differentiation operator or cross-product operator
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Context Panel: Evaluate and Display Inline
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Context Panel: Simplify≻Assuming Positive (where needed)
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Part (c)
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Calculus palette: Differentiation operators
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Common Symbols palette: Cross product operator
Press the Enter key.
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Context Panel: Simplify≻Simplify
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Context Panel: Evaluate and Display Inline
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Maple Solution - Coded
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Part (a)
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Initialize
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Install the Student VectorCalculus package.
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Define the position vector R and obtain
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Obtain and display the vectors of the TNB-frame
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Apply the TNBFrame command to R, assigning the result to a temporary name.
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Use the map command to simplify the three vectors of the TNB-frame.(This two-step process for obtaining the TNB-frame overcomes a deep-seated problem with simplifying a rooted vector carrying assumptions.)
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Extract and display the individual vectors of the TNB-frame.
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Obtain the Darboux vector
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Implement the definition of the Darboux vector, assigning it to a temporary name. To this temporary vector, apply the simplify command and call the result the Darboux vector.
(This two-step process for obtaining the Darboux vector overcomes a deep-seated problem with simplifying a rooted vector carrying assumptions.)
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Part (b)
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The derivatives of the vectors in the TNB-frame must be taken with respect to the arc length . Since R is given in terms of the parameter , the chain rule must be invoked. Hence, .
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Part (c)
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The derivatives of the vectors in the TNB-frame must be taken with respect to the arc length . Since R is given in terms of the parameter , the chain rule must be invoked. Hence, .
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