Question

example for the conversion of line integral to surface integral and another example for conversion of surface integral to volume integral

Answer 1

Chapter 5

Line and surface integrals

5.1 Line integrals in two dimensions

Instead of integrating over an interval [a, b] we can integrate over a curve C. Such integrals are called line

integrals. They were invented in the early 19th century to solve problems involving forces, fluid flow and

magnetism.

Work Done We begin by recalling some basic ideas about work done. The work done W, by a variable

force f(x) in moving a particle from a point a to a point b along the x-axis is

W =

Z b

a

f(x)dx =

Xf(x)δx = Force × distance=Work.

We now generalise this idea to a particle moving a long a general curve C and this gives a line integral.

Suppose that the force is given by the vector F in the direction

*

P R pointing as shown in Figure 5.1. If

the force moves the object from P to Q, then the displacement vector is D =

*

P Q. The work done done by

this force is defined to be the product of the component of the force along D and the distance moved:

W = |D||F| cos θ = F · D ,

Figure 5.1: The force acting in the

*

P Q direction is |F| cos θ

1