Integration in Vector Fields
In this chapter we combine ideas from the preceding four chapters: space curves, vector functions, partial
derivatives, and multiple integrals.
The result is the development of
giving powerful mathematical tools for science and engineering.
to find the work done by a force in moving an object along a path, and to find the mass of a wire having a
Surface integrals are used to find the rate at which a fluid flows across a surface. Finally, we
conclude with three major theorems,
the Divergence Theorem,
These theorems connect the new tools (i.e., the new integrals) and offer insight into their mathematical
calculations and physical applications.
We start this chapter with the definition of a vector field as they will be a major component of this chapter.
Let us start with the formal definition of a vector field.