Begin2.DVI

so that

dη

dξ

=−ξ−h

η−k

=f′(x) and d

(^2) η
dξ^2
=−
1 +
(dη
dξ
) 2
η−k
=f′′(x)

This shows that the first and second derivatives at the common point of intersection

of the curve and circle are the same and so this intersection is called a contact of

order two.

Scalar and Vector Fields

Of extreme importance in science and engineering are the concepts of a scalar

field and a vector field.

Scalar and vector fields

Let Rdenote a region of space in a cartesian coordinate sys-

tem. If corresponding to each point (x, y, z )of the region Rthere

corresponds a scalar function φ=φ(x, y, z ), then a scalar field is

said to exist over the region R. If to each point (x, y, z )of a region

Rthere corresponds a vector function

F
 =F
(x, y, z ) = F 1 (x, y, z )ˆe 1 +F 2 (x, y, z )ˆe 2 +F 3 (x, y, z )ˆe 3 ,

then a vector field is said to exist in the region R.

That is, a scalar field is a one-to-one correspondence between points in space and

scalar quantities and a vector field is a one-to-one correspondence between points in

space and vector quantities. The functions which occur in the representation of a

vector or scalar fields are assumed to be single valued, continuous, and differentiable

everywhere within their region of definition.

Example 6-23.

An example of a vector field is the velocity

of a fluid. In such a velocity field, at each point

in some specified region a velocity vector exists

which describes the fluid velocity. The velocity

vector is a function of position within the speci-

fied region. Consider water flowing in a channel