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4 Multiple integrals; vectors
4.1 Multiple integrals
Let’s review this subject by doing various examples of integrating a function f (x, y)
over a region of 2-space:
Ex. 1:
I =
region
y
xdydx =
4
0
dx
x
√
x
0
y dy =
4
0
dx
x ·
x =
x
y
y=x
1/
2
4
Figure 1: Ex. 1
Q: how about changing the order of integration? We could do the x-integral
first obviously. But we need to be careful of the order of the limits.:
I =
2
y=
dy y
4
x=y
2
x dx =
Q: Why change the order? Sometimes it’s important:
Ex. 2:
I =
ln 16
x=
dx
4
y=e x/ 2
dy
ln y
Can’t do
dy/ ln y, so switch:
x
y
y=e
x/
4
Figure 2: Ex. 1
So let’s draw a new figure (always draw a figure if you switch!).
I =
4
y=
dy
ln y
2 ln y
0
dx =
4
y=
dy · 2 = 6 (5)
4.2 Change of variables: the Jacobian
First, let’s do a standard example where we don’t get into formalities:
x
y
dr
rd θ
1
1
dθ
y= (^) 1-x
2
Figure 3: Ex. 1
Ex. 3:
I =
1
x=
√
1 −x 2
y=
e
−(x
2 +y
2 ) dydx (6)
Figure 4: Ex. 5. Original and transformed regions.
is to calculate Jacobian:
J
u, v
x, y
−y/x
2 1 /x
x + y
x
2
J
x, y
u, v
x
2
x + y
So
I =
ve
v
x
2 (v)
x
2 (v)
v
dudv =
e
v dudv, (15)
nice and simple, but what are the transformed limits? Well, starting from the
original ones, we learn
y = 0 ⇒ u = 0 if x 6 = 0 ; y = x ⇒ u = 1 (16)
x = 1 ⇒ v = 1 + u ; y = x = 0 ⇒ v = 0. (17)
So
I =
1
0
du
1+u
0
dv e
v
1
0
du(e
1+u − 1) = e
2 − e − 1. (18)
measures for cylindrical and spherical coords.: (Derived in Boas)–check!
Spherical x = r sin θ cos φ ; y = r sin θ sin φ ; z = r cos θ:
J
x, y, z
r, θ, φ
= r
2 sin θ (19)
i.e. dxdydz → r
2 sin θdrdθdφ (20)
Cylindrical x = ρ cos θ ; y = ρ sin θ; z = z :
J
x, y, z
ρ, θ, z
= ρ (21)
i.e. dxdydz → ρdρdθdz (22)
(N.B. θ in spherical coords. is not the same θ as for cylindrical coords!
Might want to use φ for cylindrical coords instead.)
Ex. 6:
Calculate the moment of inertia of a cone of height equal to its base radius
h = R. Take the density of the material to be ρ 0
, assumed homogeneous. Moment
of inertia is then
I =
V
ρ 0 dV (x
2
2 ) = ρ 0
h
z=
dz
z
r=
rdr
2 π
0
dθ r
2 = ρ 0
πh
5
Note dimensions are correct, since [ρ 0
] = M/L
3 , so [ρ 0
h
5 ] = M L
2 .
4.3 Vectors
4.3.1 Properties of vectors
Vector: “set of components which transforms under rotation of a coordinate system
in the same way as the coordinates of a point ~r. ” (“A vector is something that
transforms like a vector”.) Huh? What does that mean? Take the coordinates
x 1 , x 2 of a point ~r in a Cartesian coordinate system.
Figure 5: Transformation of coordinates x 1 , x 2 → x
′
1
, x
′
2
.
If we rotate the coordinate axes to a new set x
′
1
, x
′
2
by an angle φ, geometry tells
A =
B ⇒ A
i
= B
i
A +
B =
C ⇒ A
i
+ B
i
= C
i
A = (aA 1
, aA 2
,... aA N
A = (−A
1
, −A
2 , · · · − aA N
A +
B =
B +
A
A +
B) = a
A + b
B
etc.
Magnitude of a vector:
A
2
N ∑
i=
A
2
i
N ∑
i=
A
′ 2
i
check! (39)
A| =
A
2 = A (40)
4.3.2 Products of vectors
- Dot or scalar product:
A ·
B =
i
A
i
B
i
- Cross product
A ×
B)
i
jk
ijk
A
j
B
k
where ≤ ijk is the so-called Levi-Civita symbol, sometimes called the completely
antisymmetric tensor:
ijk
0 if any two indices are the same
+1 if indices correspond to an even permutation of 123
− 1 if indices correspond to an odd permutation of 123
Very useful identity – worth memorizing!
k
ijk
lmk = δ iδ jm − δ im δ j
Ex 1:
A ×
B) · (
C ×
D) =? (45)
i
A ×
B)
i
C ×
D)
i
i
jk
ijk
A
j
B
k
`m
i`m
C
`
D
m
ijk`m
ijk
i`m
)A
j
B
k
C
`
D
m
jk`m
(δ jδ km − δ jm δ k
)A
j
B
k
C
`
D
m
`m
(A
`
B
m
C
`
D
m
− A
m
B
`
C
`
D
m
`
A
`
C
`
m
B
m
C
m
`
B
`
C
`
m
A
m
D
m
A ·
C)(
B ·
D) − (
B ·
C)(
A ·
D) (46)
Some more identities to check using these techniques:
A · (
B ×
C) =
B · (
C ×
A) = (
A ×
B) ·
C
A × (
B ×
C) =
B(
A ·
C) −
C(
A ·
B) (“ BAC -CAB rule”)
Remarks:
A ·
B = AB cos θ AB
. If
A ·
B = 0, two vectors are “orthogonal”,
θ AB
= π/2.
- unit vectors ˆe 1 , ˆe 2 ,.... We can choose mutually orthogonal, ˆe i · eˆ j = δ ij . Also
note ˆe i × eˆ j
ijk ˆe k
A in terms of D orthogonal unit vectors in D dimensions,
A =
i
A
i eˆ i
- The scalar or dot product of two vectors is invariant under coordinate trans-
formations:
A
′ ·
B
′
i
A
′
i
B
′
i
i
j
a ij
A
i
k
a ik
B
k
ijk
a ij a ik
A
j
B
k
jk
δ jk
A
j
B
k
k
A
k
B
k
A ·
B (47)
A ×
B| = AB sin θ AB
