Calculus- Early Transcendentals, 2021a

550 Vector Calculus
We can now rewrite Green’s Theorem using the concepts of divergence and curl; these rewritten versions
in turn are closer to some later theorems we will see.
Suppose we write a two dimensional vector field in the form f = 〈 f 1 , f 2 ,0〉, where f 1 and f 2 are functions
of x and y. Then
i j k
∇ × f =
∂ ∂ ∂
∂x ∂y ∂z
∣ f 1 f 2 0 ∣ = 〈0,0, ∂ f 2
∂x − ∂ f 1
∂y 〉,
and so (∇ × f) · k = 〈0,0, ∂ f 2
∂x − ∂ f 1
∂y 〉·〈0,0,1〉 = ∂ f 2
∫
∂D
∫
f · dr =
∂D
∫∫
f 1 dx+ f 2 dy =
∂x − ∂ f 1
∂y
D
. So Green’s Theorem says
∂ f 2
∂x − ∂ f 1
∂y
∫∫D
dA = (∇ × f) · kdA. (16.2)
Roughly speaking, the right-most integral adds up the curl (tendency to swirl) at each point in the region;
the left-most integral adds up the tangential components of the vector field around the entire boundary.
Green’s Theorem says these are equal, or roughly, that the sum of the “microscopic” swirls over the region
is the same as the “macroscopic” swirl around the boundary.
Next, suppose that the boundary ∂D has a vector form r(t), sothatr ′ (t) is tangent to the boundary,
and T = r ′ (t)/|r ′ (t)| is the usual unit tangent vector. Writing r = 〈x(t),y(t)〉 we get
T = 〈x′ ,y ′ 〉
|r ′ (t)|
and then
N = 〈y′ ,−x ′ 〉
|r ′ (t)|
is a unit vector perpendicular to T, that is, a unit normal to the boundary. Now
∫
∂D
∫
f · Nds =
∫
=
∂D
∂D
〈 f 1 , f 〉·〈y′ ,−x ′ ∫
〉
2
|r ′ |r ′ (t)|dt = f 1 y ′ dt − f 2 x ′ dt
(t)|
∂D
∫
f 1 dy− f 2 dx = − f 2 dx+ f 1 dy.
So far, we’ve just rewritten the original integral using alternate notation. The last integral looks just
like the left side of Green’s Theorem (16.12) except that f 1 and f 2 have traded places and f 2 has acquired
a negative sign. Then applying Green’s Theorem we get
∫
∂D
∫∫
− f 2 dx+ f 1 dy =
Summarizing the long string of equalities,
∫
∂D
D
∂D
∫∫
f · Nds =
∂ f 1
∂x + ∂ f 2
∂y
∫∫D
dA = ∇ · fdA.
D
∇ · fdA. (16.3)
Roughly speaking, the first integral adds up the flow across the boundary of the region, from inside to out,
and the second sums the divergence (tendency to spread) at each point in the interior. The theorem roughly
says that the sum of the “microscopic” spreads is the same as the total spread across the boundary and out
of the region.