Week 13: Substitution in multiple integrals, Line integrals, Vector fields, Work, Circulation and Flux,

Thomas Calculus, 11th edition, pages: 1128 - 1137, 1143 - 1159

Sec 15.7 Substitution in Multiple Integrals

This section shows how to evaluate multiple integrals by substitution. As in single
integration, the goal of substitution is to replace complicated integrals by ones that are
easier to evaluate. Substitutions accomplish this by simplifying the integrand, the limits
of integration, or both.

Chapter 16 Integration in Vector Fields

This chapter treats integration in vector fields. It is the mathematics that engineers and physicists use to describe fluid flow, design underwater transmission cables,
explain the flow of heat in stars, and put satellites in orbit. In particular, we define line integrals, which are used to find the work done by a force field in moving an object along a path through the field. We also define surface integrals so we can find the rate that a fluid flows across a surface. Along the way we develop key concepts and results, such as conservative force fields and Green’s Theorem, to simplify our calculations of these new integrals by connecting them to the single, double, and triple integrals we have already studied.

Sec 16.1 Line Integrals

In Chapter 5 we defined the definite integral of a function over a finite closed interval [a, b] on the x-axis. We used definite integrals to find the mass of a thin straight rod, or the work done by a variable force directed along the x-axis. In this section we calculate the masses of thin rods or wires lying along a curve in the plane or space, or to find the work done by a variable force acting along such a curve. For these calculations we need a more general notion of a “line” integral than integrating over a line segment on the x-axis. Instead we need to integrate over a curve C in the plane or in space. These more general integrals are called line integrals, although “curve” integrals might be more descriptive. We make our definitions for space curves, remembering that curves in the xy-plane are just a special case with z-coordinate identically zero.

Sec 16.2 Vector Fields, Work, Circulation and Flux

When we study physical phenomena that are represented by vectors, we replace integrals over closed intervals by integrals over paths through vector fields. In this section, we use such integrals to find the work done in moving an object along a path against a variable force (such as a vehicle sent into space against Earth’s gravitational field) or to find the work done by a vector field in moving an object along a path through the field (such as the work done by an accelerator in raising the energy of a particle). We also use line integrals to find the rates at which fluids flow along and across curves.