Calculus: Analytic Geometry and Calculus, with Vectors

658 Iterated and multiple integrals

segments from (xo,yo,zo) to (x,yo,zo) and then to (x,y,zo) and then to (x,y,z) is

(6) Wa(x,y,z) = fzpP(«,Yo,zo) da +1 YOQ(x'a,zo) d,6 +L, R(x,y,y)dy

and that

(7)

ITS

az = R(x,y,z)

Remark: A force field is called conservative if the work done in moving a particle

around a closed curve is zero or (what amounts to the same thing) if the work

done in moving the particle from one point to another is the same for all paths

running from the first point to the second. If F is conservative, the functions

in (2), (4), and (6) are equal and we may set

(8) W(x,y,z) = Wi(x,Y,z) = W3(x,Ybz) = W3(x,Y,z)

Then (3), (5), and (7) give

(9)

so

(10)

aW aW aW

P= xa' Q ay' R az

Fawi.+awj+aWk=VW

and F is the gradient of W. If in addition, the particle being moved is a unit

mass in a gravitational field or a unit charge in an electric field, then W is called

a potential function, the potential at the point (x,y,z) being W(x,y,z) or W(x,y,z) +

C, where C is a constant. It is sometimes important to know that if P, Q, R are

the scalar components of a conservative vector function and if they have con-

tinuous partial derivatives of first order, then (9) implies that

aQ aP OR aP OR aQ

ax - ay' ax - az' ay - az'

More information about this matter appears in Problem 10 of Section 13.3.

(^16) Assuming that all of the integrals exist, tell why

f

b

dx fvz(s) dy ffi(s,L)Fi(x)Fs(x,Y)F3(x,Y,z) dz

a Ol(y) 13(z.Y)

b ,(xW) xz,Y)

= f F1(x) dx f F2(x,y) dy

1 (

a f Fs(x,y,z) dz.

"' / 1(x. Y)

17 Calculate

for the case in which

aaay faz ds fbyu(s,t) di

(a) p __>_ 0, q >= 0, u(s,t) = svt4

(b) u(s,t) = s + t

(c) u(s,t) = e sin t