1. Quiz 13
Let ω = f (x)dx1 ∧ · · · ∧ dxn be a smooth n-form on an open subset U of Rn. Here xi : Rn→ R is the i-th coordinate function on Rn, i.e.
xi(a1, · · · , an) = ai
for any (a1, · · · , an) ∈ Rn. We define the integral of ω over a Jordan measurable set S ⊆ U by
Z
S
ω = Z
S
f (x)dµ
where dµ denotes the Jordan measure on Rn. When n = 1, the definition is obvious. If ω = f (x, y)dx ∧ dy is a smooth two form on R2 and S is a Jordan measurable set of R2, we define
Z
S
ω = Z Z
S
f (x, y)dA.
If ω = f (x, y, z)dx ∧ dy ∧ dz is a smooth three form on R3 and S is Jordan measurable, then Z
S
ω = Z Z Z
S
f (x, y, z)dV.
Remark. Since f (x, y)dy ∧ dx = −f (x, y)dx ∧ dy, we have Z
S
f (x, y)dy ∧ dx = − Z Z
S
f (x, y)dA.
If U and V are open subset of Rn and φ : U → V is a C1-diffeomorphism, then the change of variable formula reads : for any smooth n-form ω on V, one has
(1.1)
Z
φ(S)
ω = Z
S
φ∗ω where S is a Jordan measurable subset of U.
A singular 0 cube in Rn is a function c : {0} → Rn. A singular 0-cube in Rn is equivalent to a point in Rn. A zero form on Rn is a function f : Rn → R. The integral of a zero form f over a singular 0 cube c : {0} → Rn is defined to be
Z
c
f = f (c(0)).
Let k ≥ 1. A singular k cube in Rn is a function c : [0, 1]k → Rn such that there exists an open subset U containing [0, 1]k of Rn and a smooth function ψ : U → Rn such that ψ|[0,1]k = c. If ω is a k-form on Rn, we define the integral of ω over c by
Z
c
ω = Z
[0,1]k
ψ∗ω.
(Sometimes, we will use the notation c for ψ since ψ is an extension of c.) A singular 1-cube is usually called a singular curve and a singular 2-cube is usually called a singular surface.
The integral of a one form over a singular one cube is called a line integral and the integral of a two form over a singular two cube is called a surface integral.
1
2
(1) Evaluate the following integral of differential forms.
(a) Z
[0,1]×[0,1]
x
1 + xydx ∧ dy.
(b) Z
[0,1]×[0,1]
(5 − 3x + 2y)dy ∧ dx.
(c) Let R = {(x, y) ∈ R2 : 1 ≤ x2+ y2 ≤ 4, y ≥ 0}. Evaluate the integral Z
R
(3x + 4y2)dx ∧ dy
via φ : [1, 2] × [0, 2π] → R2, φ(r, θ) = (r cos θ, r sin θ) and by the change of variable formula (1.1).
(2) Evaluate the line integral (a)
Z
γ
xydx + 3y2dy where γ(t) = (11t4, t3) for 0 ≤ t ≤ 1.
(b) Evaluate the line integral Z
γ
xdx − zdy + ydz where γ(t) = (2t, 3t, −t2) for
−1 ≤ t ≤ 1.
(3) Evaluate the surface integral Z
X
xdy ∧ dz + ydz ∧ dx + zdx ∧ dy (x2+ y2+ z2)3/2
where X(θ, ψ) = (sin θ cos ψ, sin θ sin ψ, cos θ) for 0 ≤ θ ≤ π and 0 ≤ ψ ≤ 2π.
Let D be a compact Jordan measurable region on R2 whose boundary C = ∂D is a C1-oriented curve. If ω = P (x, y)dx + Q(x, y)dy is a smooth one-form on an open subset U of R2 so that D is contained in U. Green’s Theorem states that
I
C
ω = Z Z
D
dω.
It is easy to show that dω = (Qx− Py)dx ∧ dy. Hence the Green’s Theorem can be rewritten
as I
C
P (x, y)dx + Q(x, y)dy = Z Z
D
(Qx− Py)dx ∧ dy.
Use Green’s Theorem to evaluate the following integrals.
(1) Evaluate I
C
x4dx+xydy where C is the triangle curve consisting of the line segments from (0, 0) to (1, 0), from (1, 0) to (0, 1) and from (0, 1) to (0, 0).
(2) Evaluate I
C
(3y − esin x)dx + (7x +p
y4+ 1)dy where C is the circle x2+ y2 = 9 with positive orientation (counterclockwise).
(3) Let ω = − y
x2+ y2dx + x
x2+ y2dy be the smooth one form on R2 \ {(0, 0)} and C = {(x, y) ∈ R2: 4x2+ 16y2 = 16} with positive orientation. Evaluate
I
C
ω.
