Calculus: Analytic Geometry and Calculus, with Vectors

7.2 Lengths and integrals 427

The right member of (2) is an example of a curve integral, and it is denoted by
the symbol in

(3) W = fC

F(r)is the vector function of the vector r defined by

(4) F(r) = F(x,y,z)

in which x, y, z are the numbers for which r = xi + yj + A. The right member

of (3) is read "the integral over C of F(r) dot dr," the fundamental idea being

that it is the curve C that is partitioned to produce the approximating sums.

To learn something about this curve integral, we can use the formulas

(5) F(x,Y,z) = fi(x,Y,z)i + f2(x,y,z)j + f3(x,y,z)k

(6) 6'k = 1'kl + Ayk1 + tlzkk

to put (2) in the form

(7) W - Ilm J [fl(x(tk), Y(tk), ZOO) AX! +f2(x(tk), Y(tk), z(tk)) Atk

+ f3(x(tk), Y(tk), z(tk))Dakj t.tk.

A proof very similar to the one centering around (7.255) enables us to show that

(8) W = fabfi(x(t), y(t), z(t))x'(t) dt + fabf2(x(t), y(t), z(t))y'(t) dt

• f b f3(x(t), y(t), z(t))z'(t) dt.f

This is a formula from which W can be calculated or approximated. Since we

are not electronic computers, we do not (at least at the present time) make cal-

culations, but we do point out that the integrals in (8) are abbreviated to those

in the formula

(9) W = JC[fi(x,Y,z) dx + f2(x,y,z) dy + fs(x,Y,z) dz].

The integrals in (9) are scalar curve integrals. Whenever we want to know

what these things mean and how they can be evaluated, we should have the wits

to check back to see what they abbreviate. While it may be possible to over-

emphasize the importance of the matter, we can observe that if we set

(10) F(r) = fi(x,Y,z)i + fs(x,y,z)j + f3(x,Y,z)k

and make the pretense that dr is a vector for which

(11) dr=dxi+dyj+dzk

and

(12) fj(x,y,z) dx + f2(x,y,z) dy + fs(x,y,z) dz,

then we can substitute (12) into (3) to obtain (9). Of course, it should be thor-

oughly understood that we have done nothing but explain symbols. This would