540 Polar, cylindrical, and spherical coordinates
at a point P on it and, in particular, to have information about the angle
¢ (psi) between this tangent and the vector running from the origin to
P. The most informative way to attack these and related questions is
by use of vectors; in fact it is not improbable that, in the long run, experi-
ence gained by working with vectors may be more valuablethan informa-
tion about ¢. We may start with a curve having the polar equation
p = f(o), where f is supposed to have a continuous derivative. We
may suppose that a particle P moves along C insuch a way that its
polar coordinates at time t are f(o(t)) and q5(t). When this is so, we
can set p(t) = f(¢(t)) and say that P has polarcoordinates p(t) and 0(t)
at time t. We now free ourselves from the supposition that p was a
function of 4, in the first place, and we consider the general situation in
which a particle P has polar coordinates p(t) and 0(t) at time t. When-
ever we wish to do so, we can reduce our work to thespecial case simply
by setting 4(t) = t, but it is very much worthwhile to handle the more
general situation. Moreover, we can still further increase the appli-
cability of our work by studying a still more general situation.
One who needs the medicine can free himself from the notion that
matters have become mysterious by supposing that P is a bumblebee or
electron that is buzzing around in E3 in such a way that its cylindrical
coordinates at time t are p(t), q5(t), z(t). One who wishes to consider
only polar coordinates can put z(t) = 0 at all times. The rectangular
coordinates x(t), y(t), z(t) are determined in terms of the cylindrical
coordinates by the formulas(10.25) x(t) = p(t) cos o(t), y(t) = p(t) sin 0(t), z(t) = z(t).
Thus, in terms of the standard unit vectors i, j, k, the vector r(t) run-
ning from the origin to P at time t is(10.26) r(t) = p(t)[cos ¢(t)i + sin ¢(t)j] + z(t)k.
In all of the following work, we suppose that t is confined to an interval
over which p, -0, and z are functions having continuous derivatives.
Let C be the curve (or arc) traversed by P as t increases over this interval.
Differentiating (10.26) gives the formula(10.261) v(t) = p'(t)[cos O(t)i + sin -0(t)j]
+ p(t)0'(t)[- sin ¢(t)i + cos c5(t)j] + z'(t)kfor the vector v(t) which is the velocity of P at time t and is also the
forward tangent to C at P. This can be put in the form(10.262) v(t) = p'(t)ur(t) + p(t)0'(t)u2(t) + z'(t)k
where(10.263) uI(t) = cos 0(t)i + sin O(t)j
Iu2(t) = - sin -O(t)i + cos 0(t)j.