critical points. Whenever the determinant of the Hessian matrix is zero at a critical
point, then the critical point ( ̄x 0 , ̄λ 0 )is said to be degenerate and one must seek an
alternative method to test for an extremum.
Vector Field and Field Lines
A vector field is a vector-valued function representing a mapping from Rn to
a vector V. Any vector which varies as a function of position in space is said to
represent a vector field. The vector field V =V(x, y, z )is a one-to-one correspondence
between points in space (x, y, z ) and a vector quantity V. This correspondence is
assumed to be continuous and differentiable within some region R. Examples of
vector fields are velocity, electric force, mechanical force, etc. Vector fields can be
represented graphically by plotting vectors at selected points within a region. These
kind of graphical representations are called vector field plots. Alternative to plotting
many vectors at selected points to visualize a vector field, it is sometimes easier to
use the concept of field lines associated with a vector field. A field line is a curve
where at each point (x, y, z )of the curve, the tangent vector to the curve has the
same direction as the vector field at that point. If r =x(t)ˆe 1 +y(t)ˆe 2 +z(t)ˆe 3 is the
position vector describing a field line, then by definition of a field line the tangent
vector dtdr =dxdt eˆ 1 +dydt eˆ 2 +dzdt ˆe 3 evaluated at a point t 0 must be in the same direction
as the vector V 0 =V(x(t 0 ), y (t 0 ), z(t 0 ). If this relation is true for all values of the
parameter t, then one can state that the vectors drdt and V must be colinear at each
point on the curve representing the field line. This requires
dr
dt
=dx
dt
ˆe 1 +dy
dt
ˆe 2 +dz
dt
ˆe 3 =k[V 1 (x, y, z)ˆe 1 +V 2 (x, y, z)ˆe 2 +V 3 (x, y, z)ˆe 3 ]
where kis some proportionality constant. Equating like components in the above
equation one obtains the system of differential equations
dx
dt =kV^1 (x, y, z ),
dy
dt =kV^2 (x, y, z),
dz
dt =kV^3 (x, y, z )
which must be solved to obtain the equations of the field lines.
Surface Integrals
In this section various types of surface integrals are introduced. In particular,
surface integrals of the form
∫∫
R
f(x, y, z)dS,
∫∫
R
F·dS ,
∫∫∫
R
F×dS ,
