ben green
(Ben Green)
#1
Chapter8
Vector Calculus II
In this chapter we examine in more detail the operations of gradient, divergence
and curl as well as introducing other mathematical operators involving vectors.
There are several important theorems dealing with the operations of divergence and
curl which are extremely useful in modeling and representing physical problems.
These theorems are developed along with some examples to illustrate how powerful
these results are. Also considered is the representation of the many vector operations
and their use when dealing with a general orthogonal coordinate system.
Vector Fields
Let F(x, y, z ) = F 1 (x, y, z )ˆe 1 +F 2 (x, y, z)ˆe 3 +F 3 (x, y, z)ˆe 3 denote a continuous vector
field with continuous partial derivatives in some region R of space. A vector field
is a one-to-one correspondence between points in space and vector quantities so by
selecting a discrete set of points {(xi, yi, zi)}| i= 1 ,... , n, (xi, y i, zi)∈ R} one could
sketch in tiny vectors each proportional to the given vector evaluated at the selected
points. This would be one way of visualizing the vector field. Imagine a surface
being placed in this vector field, then at each point (x, y, z)on the surface there is
associated a vector F(x, y, z). This is another way of visualizing a vector field. One
can think of the surface as being punctured by arrows of different lengths. These
arrows then represent the direction and magnitude of the vectors in the vector field.
The situation is illustrated in figure 8-1.
Figure 8-1. Representation of vector field F(x, y, z )at a select set of points.
