Calculus- Early Transcendentals, 2021a

538 Vector Calculus
Since f x = 2e x − y, f = 2e x − xy + g(y,z). Sincef y = −x, it must be that g y = 0, so g(y,z)=C + h(z).
Thus f = 2e x − xy +C + h(z) and
f z = h ′ (z)=e z ,
so h(z)=e z .Thisleavesf = 2e x − xy + e z +C.
♣
Exercises for 16.2
∫
Exercise 16.2.1 Let f = 〈xy,−xy〉 and let D be given by 0 ≤ x ≤ 1, 0 ≤ y ≤ 1. Compute
∫
f · Nds.
∂D
∫
Exercise 16.2.2 Let f = 〈ax 2 ,by 2 〉 and let D be given by 0 ≤ x ≤ 1, 0 ≤ y ≤ 1. Compute
∫
f · Nds.
∂D
∫
Exercise 16.2.3 Let f = 〈ay 2 ,bx 2 〉 and let D be given by 0 ≤ x ≤ 1, 0 ≤ y ≤ x. Compute
∫
f · Nds.
∂D
∂D
∂D
∂D
f · dr and
f · dr and
f · dr and
∫Exercise 16.2.4 ∫ Let f = 〈sinxcosy,cosxsiny〉 and let D be given by 0 ≤ x ≤ π/2, 0 ≤ y ≤ x. Compute
f · dr and f · Nds.
∂D
∂D
∫
Exercise 16.2.5 Let f = 〈y,−x〉 and let D be given by x 2 + y 2 ≤ 1. Compute
∫
Exercise 16.2.6 Let f = 〈x,y〉 and let D be given by x 2 + y 2 ≤ 1. Compute
∂D
∂D
∫
f · dr and
∫
f · dr and
∂D
∂D
f · Nds.
f · Nds.
Exercise 16.2.7 Prove Theorem 16.1.
Exercise 16.2.8 Prove Theorem 16.2.
Exercise 16.2.9 If ∇ · f = 0, f is said to be incompressible. Show that any vector field of the form
f(x,y,z)=〈 f (y,z),g(x,z),h(x,y)〉 is incompressible. Give a non-trivial example.
16.3 Line Integrals
We have so far integrated “over” intervals, areas, and volumes with single, double, and triple integrals. We
now investigate integration over or “along” a curve—“line integrals” are really “curve integrals”.