332 CHAPTER 16. SPACE CURVES

The function of h on the left is called the difference quotient just as it was for a scalarvalued function. If f (t) = ( f1 (t) , · · · , fp (t)) and

∫ ba fi (t) dt exists for each i = 1, · · · , p,

then∫ b

a f (t) dt is defined as the vector(∫ b

af1 (t) dt, · · · ,

∫ b

afp (t) dt

).

This is what is meant by saying f is Riemann integrable.

Here is a simple proposition which is useful to have.

Proposition 16.2.2 Let a ≤ b, f = ( f1, · · · , fn) is vector valued and each fi is contin-uous, then ∣∣∣∣∫ b

af (t)dt

∣∣∣∣≤√n∫ b

a|f (t)|dt.

Proof: This follows from the following computation.∣∣∣∣∫ b

af (t)dt

∣∣∣∣≡ ∣∣∣∣(∫ b

af1 (t)dt, · · · ,

∫ b

afn (t)dt

)∣∣∣∣=

(n

∑i=1

∣∣∣∣∫ b

afi (t)dt

∣∣∣∣2)1/2

≤

(n

∑i=1

(∫ b

a| fi (t)|dt

)2)1/2

≤

(nmax

i

(∫ b

a| fi (t)|dt

)2)1/2

=√

nmaxi

(∫ b

a| fi (t)|dt

)≤

√n∫ b

a|f (t)|dt

As in the case of a scalar valued function, differentiability implies continuity but notthe other way around.

Theorem 16.2.3 If f ′ (t) exists, then f is continuous at t.

Proof: Suppose ε > 0 is given and choose δ 1 > 0 such that if |h|< δ 1,∣∣∣∣f (t +h)−f (t)h

−f ′ (t)∣∣∣∣< 1.

then for such h, the triangle inequality implies |f (t +h)−f (t)| < |h|+∣∣f ′ (t)

∣∣ |h| . Now

letting δ < min(

δ 1,ε

1+|f ′(x)|

)it follows if |h| < δ , then |f (t +h)−f (t)| < ε. Letting

y = h+ t, this shows that if |y− t|< δ , |f (y)−f (t)|< ε which proves f is continuous att.

As in the scalar case, there is a fundamental theorem of calculus.

Theorem 16.2.4 If f ∈ R([a,b]) and if f is continuous at t ∈ (a,b), then

ddt

(∫ t

af (s) ds

)= f (t) .