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ch15 - Chapter 15 Topics in Vector Calculus Summary This...

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Chapter 15: Topics in Vector Calculus Summary: This chapter is the culmination of the discussion of multivariable scalar and vector-valued 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, so-called 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). 333
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334 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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