Related: Vector-Valued Derivatives
Definition (Vector Valued Functions)
Let be real functions on , and let be the corresponding mapping of into . If increases monotonically on , to say that means that for . Then we define
Theorem (Vector FTC)
If and then
Proof is done by repeating Fundamental Theorem of Calculus repeatedly times.
Proposition (Absolute Vector Integral)
Proof: If are components of , then
By Proposition (Composition of Integrable is Integrable), each of the functions , and so does their sum. Since is a continuous function of , then by Theorem (Inverse Continuity), is continuous on . By composition integration theorem again, then .
With this lemma, let where . Then and
The "" is a dot product.
then by properties of the integral, and applying the integral to both, we have
If then we are done. If not, then by dividing this inequality with , we are done.
Curves
A curve in is a continuous function
where its domain is 1 dimensional.
- An arc is image of the map on some subset of the domain.
- The arc is closed if .
Definition (Arc Length Approximation)
Given a partition , define to be
Definition (Arc Length)
Let
If , we say is rectifiable and the length of is defined to be .
Related: Supremum
Theorem
If is a curve, exists, and is continuous, then is rectifiable and