Arc Length Function

     Arc length is a measure of length along an arc. The following formula is based on the chain rule and the formula for the arc length of a function of one variable.  It is important to realize that arc length is independent of the vector function used to compute its value; that is to say, if another vector function is used to compute the arc length the values will be the same.

Proposition (Arc Length Function) Let be a piecewise smooth curve that is the graph of the vector function described parametrically by ,   and let be a particular point on (base point). Then the length of from the base point to the variable is given by the arc length function



    Proof. The arc length of the graph of on the interval is given by



Now if the graph is parametrized by equations and then by the chain rule,   so, for Thus



In , we have



But in fact the arc length of a graph is independent of the parametrization and thus,



where is the base point corresponding to

Example (Arc Length Function) Find the arc length of the curve



from to

    Solution.  We have, units because







and so the arc length is

Proposition (Speed as the Derivative of Arc Length) Suppose an object moves along a smooth curve that is the graph of the position function , where is continuous on the interval Then the object has speed for where  



    Proof. Given that is the position vector function for an object which moves along the graph of and given that is continuous on we can apply the Second Fundamental Theorem of Calculus to


to obtain

Example (Speed as the Derivative of Arc Length) If a moving object has a position vector function of



then find the speed of the object at time and the distance traveled by the object between times and

    Solution. The speed of the object at time is   because



  

and the distance traveled by the object between times and is because

   .
  

Example (Using Arc Length to Parametrize) Express the vector function in terms of arc length measured from the point corresponding to , in the direction of increasing :

    Solution. We have,
    


Solving for we have Thus

Proposition (Unit Tangent and Unit Normal Vectors) If has a piecewise smooth graph and is represented as in terms of the arc length parameter , then the unit tangent vector and the principal unit normal vector satisfies



where is a scalar function of

    Proof. Given a piecewise smooth graph represented by and in terms of arc length by , then by the chain rule,  



Also



and since and points in the same direction as and since is a unit vector



where

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