Chapter9
Applications of Vectors
The use of vectors in mathematics, physics, engineering and the sciences is
extensive. The applications presented within these pages have been selected mainly
from the study areas of physics and engineering.
Approximation of Vector Field
The Kriging^1 method is a numerical method to approximate a quantity using
a statistical weighting of known data values. The Kriging method can be used to
approximate many different kinds of quantities. The following illustrates an ap-
plication for the approximation of a vector field using interpolation. The weighted
average associated with a set of data values {Q 1 , Q 2 , Q 3 ,... , Q n}is defined
Q=w^1 Q^1 +w^2 Q^2 +w^3 Q^3 +···+wnQn
w 1 +w 2 +w 3 +···wn(9 .1)where w 1 ,... , w nare the assigned weighting factors. Note that if all the weights equal
unity, then equation (9.1) reduces down to a regular average of the given data values.
The following discussion illustrates how the Kriging method can be used to
approximate a vector field in the neighborhood of known points and known vectors
associated with these points. Note that the discussion presented can be generalized
and made applicable to any quantity Q=Q(x, y, z )which is a function of position
that one wants to approximate.
Given a finite number of known vectors
F 1 =F(x 1 , y 1 , z 1 ),F 2 =F(x 2 , y 2 , z 2 ), ···,Fn=F(xn, yn, zn)which are associated with the known points (x 1 , y 1 , z 1 ),.. .,(xn, y n, zn). It is assumed
that these known vectors are associated with a vector field F =F(x, y, z), but we
don’t know the form for F. In order to approximate the representation of the vector
field F =F(x, y, z )in some neighborhood of the known points (xi, yi, zi),i= 1,... , n
and known vectors at these points one can proceed as follows. In order to use the
known data values to estimate the value of F =F(x, y, z )at a general point (x, y, z)
one can define the distances
(^1) Danie Gerhardus Krige(1919- ) A South African geologist and mining engineer.