Mathematics of Physics and Engineering

28 Curves in Space

The equation of the tangent line at point P is

R(s) = r(to) + su{t 0 ). (1.3.9)

EXERCISE 1.3.2.c Let C be a planar curve defined by the vector function
r(t) = cosH + sintj, —n < t < n. Compute the tangent vector r'(t) and
the unit tangent vector u(t) as functions oft. Compute r'(0) and u(0).
Draw the curve C and the vectors r'(0), u'(0). Verify your results using
a computer algebra system, such as MAPLE, MATLAB, or MATHE-
MATICA.

EXERCISE 1.3.3.c Let C be a spatial curve defined by the vector function
r(t) = costi + sintj + tk. Compute the tangent vector r'(t), the unit
tangent vector u(t) and the vector u'(t). Compute r'{ix/2). Draw the
curve C for 0 < t < n/2 and draw U'(TV/2) at the point r(n/2). Verify
your results using your favorite computer algebra system.

Definition 1.6 A curve C, defined by a vector function r(t), a < t < b, is
called smooth if the unit tangent vector u = u(t) exists and is a continuous
function for all t € (a,b). If the curve is closed, then, additionally, we
must have r'(a) = r'(b). The curve is called piece-wise smooth if it is
continuous and consists of finitely many smooth pieces.

EXERCISE 1.3.4.A Give an example of a non-smooth curve C defined by a
vector function r(t), — 1 < t < 1, so that the derivative vector r'{t) exists
and is continuous for all t G (—1,1).

EXERCISE 1.3.5. c Explain how the graph of a function y = f(x) can be
interpreted as a curve in K^3. Show that this curve is smooth if and only if
the function f = f(x) has a continuous derivative, and show that, at the
point (xo,f(xo),0), formula (1.3.9) defines the same line as y = f(xo) +
f'(xo)(x-x 0 ),z = 0.

Given a curve C and two points with position vectors r(c),r(d), a <
c < d < b, on the curve, we define the distance between the two points
along the curve using a limiting process. The construction is similar to the
definition of the Riemann integral in ordinary calculus.
For each n > 2, choose points c — to < h < • • • < tn = d and form
n-l
the sums Ln = Y^, ll^rill> where AT-J = r{ti+) — r(U). Assume that

maxo<,<n_i(£i+i — U) —> 0 as n —> oo. If the limit linin-Kx, Ln exists for all
a < c < d < b, and does not depend on the particular choice of the points
tk, then the curve C is called rectifiable. By definition, the distance