Chapter 15: Topics in Vector Calculus
Summary:
This chapter is the culmination of the discussion of multivariable
scalar and vectorvalued functions and the applications of double and triple inte
grals. The main focus in this chapter is to extend integration by parts to double
and triple integrals.
The basic idea of integration by parts was to simplify the
integration of some single integrals by rewriting the integral as some terms evalu
ated at the boundary of the interval added together with a new integral. In double
and triple integrals, the boundaries are usually either curves in a plane or surfaces.
Line integrals and surface integrals are introduced to discuss boundary terms sim
ilar to those that show up when integration by parts is used.
The chapter begins by discussing vector fields. In particular, the divergence (a
scalar value) and curl (a vector) of a vector field are introduced. After the intro
duction of vector fields, the concept of line integrals and their relationship to such
quantities as work is described. In particular, socalled conservative vector fields
are found to obey the Fundamental Theorem of Line Integrals. Green’s Theorem
uses line integrals as a means to simplify the evaluation of some double inte
grals. For example, some work and area problems may be reduced using Green’s
Theorem to a problem of evaluating a line integral. Analogous to line integrals
along a curve, a surface integral may be evaluated when a surface is parameter
ized. This has applications relating to flow fields and flux of a vector field. The
chapter concludes by discussing the Divergence Theorem and Stokes’ Theorem.
The Divergence Theorem has some important applications involving flux and in
electromagnetic problems. Stokes’ Theorem is essentially an extension of Green’s
Theorem to higher dimensions. Both Green’s Theorem and Stokes’ Theorem are
closely related to integration by parts in one dimension.
OBJECTIVES: After reading and working through this chapter
you should be able to do the following:
1. Define a vector field (§15
.
1).
2. Find the divergence and curl of a vector field (§15
.
1).
3. Find the gradient field of a multivariable scalar function (§15
.
1).
4. Calculate line integrals on a smooth or piecewise smooth curve (§15
.
2).
5. Calculate the line integral of a vector field along a curve (§15
.
2).
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6. Find work using line integrals (§15
.
2).
7. Determine if a vector field is conservative (§15
.
3).
8. Evaluate line integrals of conservative fields using the Fundamental Theo
rem of Line Integrals (§15
.
3).
9. Use Green’s Theorem to evaluate double integrals by evaluating line inte
grals (§15
.
4).
10. Define and evaluate surface integrals (§15
.
5).
11. Use surface integrals to determine the flux across a surface (§15
.
6).
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 Spring '10
 lee
 Vector Calculus, Vector field, Stokes' theorem

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