Mathematics of Physics and Engineering

26 Curves in Space

Given a fixed frame O, the formulas of differential calculus for vector

functions in this frame are easily obtained by following the corresponding

derivations for scalar functions in ordinary calculus. As in ordinary calculus,

there are several rules for computing derivatives of vector-valued functions.

All these rules follow directly from the definition (1.3.2).

The derivative of a sum:

ft(u(t)+v(t))=u'(t)+v'(t). (1.3.3)

Product rule for multiplication by a scalar: if X(t) is a scalar function,

then

ft(X(t)r (t)) = X'(t)r(t) + X(t)r '(t). (1.3.4)

Product rules for scalar and cross products:

d ,.. ... du dv

_(„(t).wW) = -.« + «•-, (1.3.5)

and

d ,.. ... du dv

s(«(i)x„(i)) = -x„ + «x_. (1.3.6)

The chain rule: If t — (j>(s) and ri(s) = r(<p(s)), then

dr\ dr d(f>

ds dt ds (1.3.7)

From the two rules (1.3.3) and (1.3.4), it follows that if (£, j, k) are

constant vectors in the frame O so that r(i) = x(t) l + y{t)j+ z(t)k, then

r'(t)=x'(t)i + y'{t)j + z'(t)k.

Remark 1.5 The underlying assumption in the above rules for differen-
tiation of vector functions is that all the functions are defined in the same
frame. We will see later that these rules for computing derivatives can fail
if the vectors are defined in different frames and the frames are moving
relative to each other.

Lemma 1.1 If r is differentiable on (a,b) and \r(t)\ does not depend
on t for t G (a,b), then r(t) _L r'(t) for all t G (a,b). In other words, the
derivative of a constant-length vector is perpendicular to the vector itself.

Proof. By assumption, r(t) • r(t) is constant for all t. By the product rule
(1.3.5), 2r '(t) • r{t) = 0 and the result follows. •