Figure 6-17. Contour plots of selected two-dimensional scalar functions.
A vector field is a one–to–one correspondence between points in space and vector
quantities, whereas a scalar field is a one–to–one correspondence between points
in space and scalar quantities. The concept of scalar and vector fields has many
generalizations. A scalar field assigns a single number φ(x, y, z)to each point of space.
A two-dimensional vector field would assign two numbers (F 1 (x, y, z ), F 2 (x, y, z )) to
each point of space, and a three-dimensional vector field would assign three numbers
(F 1 (x, y, z), F 2 (x, y, z ), F 3 (x, y, z )) to each point of space. An immediate generalization
would be that an n-dimensional vector field would assign an n-tuple of numbers
(F 1 , F 2 ,... , F n) to each point of space. Here each component Fi is a function of
position, and one can write
Fi=Fi(x, y, z ), i = 1 ,... , n.
Other immediate ideas that come to mind are the concepts of assigning n^2 numbers
to each point in space or n^3 numbers to each point in space. These higher dimensional
correspondences lead to the study of matrices and tensor fields which are functions
of position. In science and engineering, there is great interest in how such scalar
and vector fields change with position and time.
Partial Derivatives
If a vector field 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 is referenced
with respect to a fixed set of cartesian axes, then the partial derivatives of this vector
field are given by:
