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dx dy  dx dy
+
P t
∆W = () ∆tQ t () ∆t =  () + Qt()  ∆t (7.62)
P t
dt dt  dt dt 
and the total work can be written as an integral over the single variable t:
t  ∫ 0 P t() dt Qt () dt  dt (7.63)
W = t 1   dx + dy 

Homework Problems

Pb. 7.25 How much work is done in moving the particle from the point
r
(0, 0) to the point (3, 9) in the presence of the force F along the following two
different paths?
2
a. The parabola y = x .
b. The line y = 3x.
The force is given by:
r
F = xyê + ( x + y ê )
2
2
x y
r
Pb. 7.26 Let F = yê + xê . Calculate the work moving from (0, 0) to (1, 1)
x
y
along each of the following curves:
a. The straight line y = x.
2
b. The parabola y = x .
c. The curve C described by the parametric equations:
xt() = t 32 and y t() = t 5
/

A vector field such as the present one, whose line integral is independent
of the path chosen between fixed initial and final points, is said to be conser-
vative. In your vector calculus course, you will establish the necessary and
sufficient conditions for a vector field to be conservative. The importance of
conservative fields lies in the ability of their derivation from a scalar poten-
tial. More about this topic will be discussed in electromagnetic courses.

7.8 Infinite Dimensional Vector Spaces*

This chapter section introduces some preliminary ideas on infinite-dimen-
sional vector spaces. We assume that the components of this vector space are

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