1 Introduction
These exercises serve as homework for our lesson on the two dimensional diver-
gence theorem, which says∫∫
Ω
div(F⃗ )dA =
∫
∂Ω
F⃗ · n⃗ds,
where F⃗ is some differentiable vector field in Ω and n⃗ is the outward pointing
unit normal vector at each point on the boundary ∂Ω of Ω.
If the components of F are F = ⟨P,Q⟩ and the boundary is positively
oriented, meaning you follow the boundary curve in such a way that the interio
of the region is on your left, then n⃗ds = ⟨dy,−dx⟩ and so this can be written∫∫
Ω
∂P
∂x
+
∂Q
∂y
dA =
∫
∂Ω
P (x, y)dy −Q(x, y)dx.
Alternatively, we can replace P and Q respectively with Q and −P in that
formula to get∫
∂Ω
F⃗ · dx⃗ =
∫
∂Ω
P (x, y)dx+Q(x, y)dy =
∫∫
Ω
∂Q
∂x
− ∂P
∂y
dA,
which when written this way is called Green’s Theorem. Ultimately the 2D-
Divergence and Green’s Theorems are the same theorem but phrased in slightly
different notation.
A couple comments:
ˆ I can’t stress enough just how powerful this theorem is. Clever choices of
F and Ω lead to so many useful results in math.
ˆ If the boundary has multiple components, you add up the integral ove
each component.
ˆ Careful with orientation! Especially if the region has an inner and oute
oundary component. Always remember that Ω should be on your left
when you walk along the boundary curve.
ˆ Since we have to work with tangent and normal vectors to curves, I’ll
emind you that if you rotate a vector ⟨a, b⟩ clockwise by 90 degrees you
get the vector ⟨b,−a⟩. Keep this in mind when setting up your formulas.
In particular, if a positively oriented curve is given, we would have
Tds = ⟨dx, dy⟩
and so the outward pointing unit normal (times ds) is the clockwise 90
degree rotation of that, giving us
nds = ⟨dy,−dx⟩.
1
2 Problems
Exercise 1 Consider the vector field
F⃗ =
−y⃗i+ x⃗j
x2 + y2
1. Show that
∂
∂x
(
x
x2 + y2
)
=
∂
∂y
(
−y
x2 + y2
)
.
2. Show that, as long as x > 0, ∇(arctan(y/x)) = F .
Because of this, we can (formally) think of F as the gradient of the angle
function: F = ∇θ, where x = r cos θ and y = r sin θ are polar coordinates.
Note that θ is not actually a well defined function in the entire xy-plane,
ut is well defined if we restrict its domain to things like x > 0 or to a
simply connected region away from the origin.
3. Suppose a curve γ in the xy-plane starts at the point with polar coordi-
nates (r1, θ1) and ends at a nea
y point with polar coordinates (r2, θ2).
Keeping the previous comment in mind, use the fundamental theorem of
line integrals to show ∫
γ
F⃗ · dx⃗ = θ2 − θ1
4. If Ω is all of R2 except for the origin, show that F is NOT the gradient of
a function on Ω by directly computing∫
γ
F · Tds
where γ = ⟨cos(t), sin(t)⟩, 0 ≤ t ≤ 2π. Remember if F was a gradient in
the region this would have to be zero, but it isn’t.
5. What is
∫
γ
F · Tds for the curves γ = ⟨R cos(t), R sin(t)⟩, 0 ≤ t ≤ 2π?
Does your answer depend on R?
6. What is
∫
γ
F ·Tds for the curves γ = ⟨cos(nt), sin(nt)⟩, 0 ≤ t ≤ 2π? Does
your answer depend on n?
7. Now apply Green’s theorem with P = −y(x2+y2)−1 and Q = x(x2+y2)−1,
where Ω has two nested boundary curves γ1 and γ2. What does this say
about
∫
γ1
F · Tds and
∫
γ2
F · Tds?
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Exercise 2 If we use Green’s theorem with P = 0 and Q = x we get∫
∂Ω
xdy = Area(Ω)
Use this to find the area of the region enclosed by the curve
x = t(1− t)
y = t3(1− t)
where 0 ≤ t ≤ 1. Again, remember in Green’s theorem that the boundary has to
e positively oriented.
Exercise 3 Use the divergence theorem to compute the value of
∫
γ
x2ds, where
γ is the circle x2 + y2 = 1. As a hint, try to find a vector field F where
x2 = F · n
Notice that the outward unit normal at the point (x, y) on this circle is simply
n = ⟨x, y⟩.
Exercise 4 Use Green’s theorem to show that the x-coordinate of the center of
mass of a uniformly dense region Ω is
x̄ =
1
Area(Ω)
∫
∂Ω
x2
2
dy
while the y-coordinate is
ȳ =
1
Area(Ω)
∫
∂Ω
−y2
2
dx
Yet again, remember in Green’s theorem that the boundary has to be positively
oriented.
Exercise 5 If f(x, y) is a twice continuously differentiable function its Lapla-
cian is defined to be
∆f = div(∇f) = ∂
2f
∂x2
+
∂2f
∂y2
Laplacians are very important because they serve as diffusion terms in a variety
of phenomena, from electromagnetism to stock prices. For example, heat flow
through an object is governed by the equation ∂tf = ∆f .
1. If f(x, y) = x2 + y2 what is ∆f?
2. If f(x, y) = x3 − 3xy2 what is ∆f?
3. If f(x, y) = ex sin(y) what is ∆f?
3
4. If f(x, y) = ex sin(2y) what is ∆f?
5. Use the Divergence theorem to prove the formula∫∫
Ω
∆fdA =
∫
∂Ω
∇f · n⃗ds
6. Let’s apply this to Ω = {x⃗|f(x⃗) ≤ c} and assume that ∂Ω = {x⃗|f(x⃗) = c}.
Note this assumption isn’t true for all functions.
In this case, show ∫∫
{f≤c}
∆fdA =
∫
{f=c}
|∇f |ds
As a hint, what do you recall about level sets and gradients?
Exercise 6 Suppose that f and g are smooth functions. Show∫∫
Ω
(f∆g +∇f · ∇g) dA =
∫
∂Ω
f∇g · n⃗ds
This is called Green’s first identity. Hint: Consider F = f∇g and use the
Divergence theorem and product rule.
Exercise 7 Use Green’s first identity to show that, if f and g are both zero on
∂Ω, then ∫∫
Ω
f∆gdA =
∫∫
Ω
g∆fdA
You just proved that ∆ is what is called a self adjoint operator on these sort of
functions, a concept which is very important in linear alge
a.
4
Introduction
Problems