Calculus: Analytic Geometry and Calculus, with Vectors

3.7 Rates, velocities 179

The scalar components of the velocity v or v(t) are sometimes denoted
by the symbols v., v, v, so that

(3.765)

and

(3.766)

vx

=

x/(t)

dt' vv =y'(t) dt' V. = z' (t) dt

V = vxi + vvj + vzk.

The acceleration a(t) is a vector which is defined in terms of velocities
in the same way that velocities are defined in terms of displacements.
Thus, provided the derivatives exist,

(3.77) a(t) = v'(t) = r"(t) = x"(t)i + y"(t)j + z"(t)k

or

(3.771)

t2

a(t) = k,

dt = dt2 = dt2 i +dt2j + dz

where the "double prime" in (3.77) and the number 2 appearing in "dee
squared x dee t squared" in (3.771) denote second derivatives, that is,
derivatives of derivatives. We still have to learn what is meant by the
speed (a scalar) of the bumblebee. It is defined by

(3.78) Speed = length of velocity vector,

so that, in our notation,

()2.

(3.781) Speed = 1V(t)1

()2

Perhaps it should be explained that the t appearing in the above equa-

tions is called a parameter, that (3.74) is a parametric equation of the path,

and that the path is the graph of the parametric equation. According to

this definition, a parameter is a number. It is an element of the domain

of the functions in (3.74), and we need not complicate our lives by harbor-

ing impressions that parameters are complicated things. Section 7.1

gives a careful explanation of circumstances in which the graph is called

a curve.

In Section 5.1 we shall give a rather detailed discussion of tangents to

graphs. Meanwhile, it can be remarked that if the vectors in (3.751)

and (3.761) are not 0, then the line through P(x,y,z) and P(x + Ax,

y + Ay, z + Az) is called a chord of the curve being considered, and the

line through P(x,y,z) having the direction of v(t) is called the tangent to

the curve at P(x,y,z). Therefore, we can find the direction of the tangent

to a sufficiently decent curve by finding the velocity of a particle which

moves along the curve with nonzero velocity. The tangent line and the

velocity vector have, by definition, the same direction. To bridge the

gap between our work and plebeian terminology used in the prosaic