Example

Section	You should be able to...	Examples	Review Exercises
15.1	1 Describe a vector field (p. 974)	1–6	1, 9
15.2	1 Define a line integral in the plane (p. 977)
	2 Find the value of a line integral along a smooth curve (p. 979)	1–4	11, 15
	3 Find line integrals of the form \(\int_{C}f(x,y) dx\) and \(\int_{C}f(x,y) dy\) (p. 981)	5–7	3, 4, 12(a),(c), 14, 23
	4 Find line integrals along a piecewise-smooth curve (p. 984)	8	12(b), 21
	5 Find the value of a line integral in space (p. 985)	9	7, 8
15.3	1 Identify a conservative vector field and a potential function (p. 990)	1	9(a)
	2 Use the Fundamental Theorem of Line Integrals (p. 991)	2, 3	6(b), 9(b),10
	3 Reconstruct a function from its gradient: Finding the potential function for a conservative vector field (p. 995)	4	6(a)
	4 Determine whether a vector field is conservative (p. 996)	5–8	5
15.4	1 Compute work (p. 1003)	1, 2	16–18
15.5	1 Use Green's Theorem to find a line integral (p. 1009)	1, 2	12(d), 20, 22, 24
	2 Use Green's Theorem to find area (p. 1010)	3	19
	3 Use Green's Theorem with multiply-connected regions (p. 1011)	4	25
15.6	1 Describe surfaces defined parametrically (p. 1017)	1, 2	26
	2 Find a parametric representation of a surface (p. 1018)	3–5	27
	3 Find equations for a tangent plane and a normal line (p. 1022)	6, 7	28
	4 Find the surface area of a parametrized surface (p. 1023)	8	29
15.7	1 Find a surface integral using a double integral (p. 1027)	1–3	13, 30, 31, 33–35
	2 Determine the orientation of a surface (p. 1030)	4–6	32
	3 Find the flux of a vector field across a surface (p. 1033)	7, 8	36, 46(a)
15.8	1 Find the divergence of a vector field (p. 1038)	1	37(a)–40(a)
	2 Use the Divergence Theorem (p. 1038)	2–4	41, 43, 44
	3 Interpret the divergence of \(\mathbf{F}\) (p. 1042)	5	46(a)
15.9	1 Find the curl of \(\mathbf{F}\) (p. 1046)	1	37(b)–40(b)
	2 Verify Stokes' Theorem (p. 1047)	2	37(c)–40(c)
	3 Use Stokes' Theorem to find an integral (p. 1048)	3, 4	42, 45
	4 Use Stokes' Theorem with conservative vector fields (p. 1049)	5	46(b), 47, 48
	5 Interpret the curl of \(\mathbf{F}\) (p. 1050)