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Pb. 7.24 Using the parametric equations for an ellipse given in Example
1.13, find the curvature of the ellipse as function of t.
a. At what points is the curvature a minimum, and at what points is
it a maximum?
b. What does the velocity do at the points of minimum and maximum
curvature?
c. On what dates of the year does the planet Earth pass through these
points on its trajectory around the sun?

7.7 Line Integral

As you may have already learned in an elementary physics course: if a force
r
r
F is applied to a particle that moves by an infinitesimal distance ∆l , then the
infinitesimal work done by the force on the particle is the scalar product of
the force by the displacement; that is,
r r
∆W = F ⋅ ∆l (7.58)

Now, to calculate the work done when the particle moves along a curve C,
located in a plane, we need to define the concept of a line integral.
Suppose that the curve is described parametrically [i.e., x(t) and y(t) are
given]. Furthermore, suppose that the vector field representing the force is
given by:
r
F = P xy ê +(, ) Q xy ê( , ) (7.59)
x y

The displacement element is given by:

∆l = ∆xê + ∆yê (7.60)
x y

The infinitesimal element of work, which is the dot product of the above two
quantities, can then be written as:

∆W = P ∆x Q+ ∆y (7.61)

This expression can be simplified if the curve is written in parametric form.
Assuming the parameter is t, then ∆W can be written as a function of the sin-
gle parameter t:

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