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

By Cauchy-Schwarz Inequality,

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