VECTOR DIFFERENTIAL CALCULUS
The Frenet-Serret formulas
We will now round up our discussion on the three unit vectors - the unit tangent, the unit normal and the unit binormal by deriving certain formulas involving the first derivatives of each vector.
These formulas are called the Frenet-Serret formulas.
We will now round up our discussion on the three unit vectors - the unit tangent, the unit normal and the unit binormal by deriving certain formulas involving the first derivatives of each vector.
These formulas are called the Frenet-Serret formulas.
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VECTOR DIFFERENTIAL CALCULUS >
THE FRENET-SERRET FORMULAS
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INTERPRETATION OF TORSION
FUNDAMENTAL THEOREM OF SPACE CURVES.1
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VECTOR DIFFERENTIAL CALCULUS >
THE FRENET-SERRET FORMULAS
May want to check out:
INTERPRETATION OF TORSION
FUNDAMENTAL THEOREM OF SPACE CURVES.1
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VIDEO LECTURE
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CONCEPT
Recall our previous results, that is
It seems logical first that we find the first derivative of the unit normal
Together the three formulas are
And so I present to you The Frenet-Serret formulas which are of fundamental importance in the theory of curve in 3-space. These formulas can also be represented in matrix form as:
We shall see very soon how a curve can be solely described by the functions of curvature All information presentated, less questions and exercises, is original content of Donny, with slight references to various books.
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and 
with respect to arc length 
. To get that, we simply represent 
in its cross product and differentiate accordingly, and employing substitution with the previous equations.
and torsion 
leading up to the fundamental theorem of space curves. This will get quite exciting!
