Handbook of mathematics for engineers and scienteists part 52 potx

7.3. DOUBLE AND TRIPLE INTEGRALS 325
2. Additivity. If a domain U is split into two subdomains, U
1
and U
2
, that do not have
common internal points and if a function f(x, y, z) is integrable in either subdomain, then

U
f(x, y, z) dx dy dz =

U
1
f(x, y, z) dx dy dz +

U
2
f(x, y, z) dx dy dz.
3. Estimation theorem.Ifm ≤ f(x, y, z) ≤ M in a domain U,then
mV ≤

U
f(x, y, z) dx dy dz ≤ MV,
where V is the volume of U.
4. Mean value theorem.Iff(x, y, z) is continuous in U, then there exists at least one
internal point (¯x, ¯y, ¯z)
U such that

U
f(x, y, z) dx dy dz = f(¯x, ¯y, ¯z) V .
The number f(¯x, ¯y, ¯z) is called the mean value of the function f in the domain U.

5. Integration of inequalities.Ifϕ(x, y, z) ≤ f(x, y, z) ≤ g(x, y, z) in a domain U,then

U
ϕ(x, y, z) dx dy dz ≤

U
f(x, y, z) dx dy dz ≤

U
g(x, y, z) dx dy dz.
6. Absolute value theorem:





U
f(x, y, z) dx dy dz




≤

U


f(x, y, z)



dx dy dz.
7.3.5. Computation of the Triple Integral. Some Applications.
Iterated Integrals and Asymptotic Formulas
7.3.5-1. Use of iterated integrals.
1
◦
. Consider a three-dimensional body U bounded by a surface z = g(x, y) from above and
a surface z = h(x, y) from below, with a domain D being the projection of it onto the x, y
plane. In other words, the domain U is defined as {(x, y)
D : h(x, y) ≤ z ≤ g(x, y)}.Then

U
f(x, y, z) dx dy dz =

D
dx dy

g(x,y)
h(x,y)
f(x, y, z) dz.
2
◦
. If, under the same conditions as in Item 1
◦
, the domain D of the x, y plane is defined
as {a ≤ x ≤ b, y
1
(x) ≤ y ≤ y
2
(x)},then


U
f(x, y, z) dx dy dz =

b
a
dx

y
2
(x)
y
1
(x)
dy

g(x,y)
h(x,y)
f(x, y, z) dz.
326 INTEGRALS
7.3.5-2. Change of variables in the triple integral.
1
◦
.Letx = x(u, v, w), y = y(u, v, w), and z = z(u, v, w) be continuously differentiable
functions that map, one to one, a domain Ω of the u, v, w space into a domain U of the
x, y, z space, and let a function f(x, y, z) be continuous in U.Then

U
f(x, y, z) dx dy dz =


Ω
f

x(u, v, w), y(u, v, w), z(u, v, w)

|J(u, v, w)|du dv dw,
where J(u, v, w)istheJacobian of the mapping of Ω into U:
J(u, v, w)=
∂(x, y, z)
∂(u, v, w)
=







∂x
∂u
∂x
∂v
∂x
∂w
∂y
∂u
∂y
∂v
∂y
∂w

∂z
∂u
∂z
∂v
∂z
∂w







.
The expression in the middle is a very common notation for a Jacobian.
The absolute value of the Jacobian characterizes the expansion (or contraction) of an
infinitesimal volume element when passing from x, y, z to u, v, w.
2
◦
. The Jacobians of most common transformations in space are listed in Table 7.2.
TABLE 7.2
Common curvilinear coordinates in space and the respective Jacobians
Name of coordinates Transformation Jacobian, J
Cylindrical coordinates ρ, ϕ, z
x = ρ cos ϕ, y = ρ sin ϕ, z = z
ρ
Generalized cylindrical
coordinates ρ,ϕ, z
x = aρ cos ϕ, y = bρ sin ϕ, z = z abρ
Spherical coordinates r, ϕ, θ

x = r cos ϕ sin θ, y = r sin ϕ sin θ, z = r cos θ
r
2
sin θ
Generalized spherical
coordinates r, ϕ, θ
x = ar cos ϕ sin θ, y = br sinϕ sin θ,
z = cr cos θ
abcr
2
sin θ
Coordinates of prolate ellipsoid of
revolution σ, τ , ϕ (σ ≥ 1 ≥ τ ≥ –1)
x = a

(σ
2
– 1)(1 – τ
2
)cosϕ,
y = a

(σ
2
– 1)(1 – τ
2
)sinϕ,
z = aστ
a
3

(σ
2
– τ
2
)
Coordinates of oblate ellipsoid of
revolution σ, τ , ϕ (σ ≥ 0,–1 ≤ τ ≤ 1)
x = a

(1 + σ
2
)(1 – τ
2
)cosϕ,
y = a

(1 + σ
2
)(1 – τ
2
)sinϕ,
z = aστ
a
3
(σ
2
+ τ
2
)
Parabolic coordinates σ, τ , ϕ

x = στ cos ϕ, y = στ sin ϕ, z =
1
2
(τ
2
– σ
2
)
στ(σ
2
+ τ
2
)
Parabolic cylinder
coordinates σ, τ , z
x = στ, y =
1
2
(τ
2
– σ
2
), z = z
σ
2
+ τ
2
Bicylindrical coordinates σ, τ , z
x =
a sinh τ

cosh τ –cosσ
, y =
a sin σ
cosh τ –cosσ
, z = z
a
2
(cosh τ –cosσ)
2
Toroidal coordinates σ, τ , ϕ
(–π ≤ σ ≤ π, 0 ≤ τ < ∞, 0 ≤ ϕ < 2π)
x =
a sinh τ cos ϕ
cosh τ –cosσ
, y =
a sinh τ sin ϕ
cosh τ –cosσ
,
z =
a sin σ
cosh τ –cosσ
a
3
sinh τ
(cosh τ –cosσ)
2
Bipolar coordinates σ, τ , ϕ
(σ is any, 0 ≤ τ < π, 0 ≤ ϕ < 2π)
x =
a sin τ cos ϕ

cosh σ –cosτ
, y =
a sin τ sin ϕ
cosh σ –cosτ
,
z =
a sinh σ
cosh σ –cosτ
a
3
sin τ
(cosh σ –cosτ)
2
7.3. DOUBLE AND TRIPLE INTEGRALS 327
7.3.5-3. Differentiation of the triple integral with respect to a parameter.
Let the integrand function and the integration domain of a triple integral depend on a
parameter, t. The derivative of this integral with respect to t is expressed as
d
dt

U(t)
f(x, y, z, t) dx dy dz
=

U(t)
∂
∂t
f(x, y, z, t) dx dy dz +

S(t)

(n ⋅ v)f(x, y, z, t) ds,
where S(t) is the boundary of the domain U(t), n is the unit normal to S(t), and v is the
velocity of motion of the points of S(t).
7.3.5-4. Some geometric and physical applications of the triple integral.
1. Volume of a domain U:
V =

U
dx dy dz.
2. Mass of a body of variable density γ = γ(x, y, z) occupying a domain U:
m =

U
γdxdydz.
3. Coordinates of the center of mass:
x
c
=
1
m

U
xγ dx dy dz, y
c
=
1
m

U
yγ dx dy dz, z

c
=
1
m

U
zγ dx dy dz.
4. Moments of inertia about the coordinate axes:
I
x
=

U
ρ
2
yz
γdxdydz, I
y
=

U
ρ
2
xz
γdxdydz, I
z
=

U
ρ

2
xy
γdxdydz,
where ρ
2
yz
= y
2
+ z
2
, ρ
2
xz
= x
2
+ z
2
,and ρ
2
xy
= x
2
+ y
2
.
If the body is homogeneous, then γ = const.
Example. Given a bounded homogeneous elliptic cylinder,
x
2
a

2
+
y
2
b
2
= 1, 0 ≤ z ≤ h,
find its moment of inertia about the z-axis.
Using the generalized cylindrical coordinates (see the second row in Table 7.2), we obtain
I
x
= γ

U
(x
2
+ y
2
) dx dy dz = γ

h
0

2π
0

1
0
ρ
2

(a
2
cos
2
ϕ + b
2
sin
2
ϕ)abρdρdϕdz
=
1
4
abγ

h
0

2π
0
(a
2
cos
2
ϕ + b
2
sin
2
ϕ) dϕ dz =
1
4

abγ

2π
0

h
0
(a
2
cos
2
ϕ + b
2
sin
2
ϕ) dz dϕ
=
1
4
abhγ

2π
0
(a
2
cos
2
ϕ + b
2
sin

2
ϕ) dϕ =
1
4
ab(a
2
+ b
2
)hγ.
328 INTEGRALS
5. Potential of the gravitational field of a body U at a point (x, y, z):
Φ(x, y, z)=

U
γ(ξ, η, ζ)
dξ dη dζ
r
, r =

(x – ξ)
2
+(y – η)
2
+(z – ζ)
2
,
where γ = γ(ξ, η, ζ) is the body density. A material point of mass m is pulled by the
gravitating body U with a force

F . The projections of


F onto the x-, y-, and z-axes are
given by
F
x
= km
∂Φ
∂x
= km

U
γ(ξ, η, ζ)
ξ – x
r
3
dξ dη dζ,
F
y
= km
∂Φ
∂y
= km

U
γ(ξ, η, ζ)
η – y
r
3
dξ dη dζ,
F

z
= km
∂Φ
∂z
= km

U
γ(ξ, η, ζ)
ζ – z
r
3
dξ dη dζ,
where k is the gravitational constant.
7.3.5-5. Multiple integrals. Asymptotic formulas.
Multiple integrals in n variables of integration are an obvious generalization of double and
triple integrals.
1
◦
. Consider the Laplace-type multiple integral
I(λ)=

Ω
f(x)exp[λg(x)] dx,
where x = {x
1
, ,x
n
}, dx = dx
1
dx

n
, Ω is a bounded domain in R
n
, f(x)andg(x)
are real-valued functions of n variable, and λ is a real or complex parameter.
Denote by
S
ε
=

λ :arg|λ| ≤
π
2
– ε

, 0 < ε <
π
2
,
a sector in the complex plane of λ.
T
HEOREM 1.
Let the following conditions hold:
(1) the functions
f(x)
and
g(x)
are continuous in
Ω
,

(2) the maximum of
g(x)
is attained at only one point
x
0
Ω
(
x
0
is a nondegenerate
maximum point), and
(3) the function
g(x)
has continuous third derivatives in a neighborhood of
x
0
.
Then the following asymptotic formula holds as
λ →∞
,
λ S
ε
:
I(λ)=(2π)
n/2
exp[λg(x
0
)]
f(x
0

)+O(λ
–1
)
√
λ
n
det[g
x
i
x
j
(x
0
)]
,
where the
g
x
i
x
j
(x)
are entries of the matrix of the second derivatives of
g(x)
.
2
◦
. Consider the power Laplace multiple integral
I(λ)=


Ω
f(x)[g(x)]
λ
dx.
T
HEOREM 2.
Let
g(x)>0
and let the conditions of Theorem 1 hold. Then the following
asymptotic formula holds as
λ →∞
,
λ S
ε
:
I(λ)=(2π)
n/2
[g(x
0
)]
(2λ+n)/2
f(x
0
)+o(1)
√
λ
n
det[g
x
i

x
j
(x
0
)]
.
7.4. LINE AND SURFACE INTEGRALS 329
7.4. Line and Surface Integrals
7.4.1. Line Integral of the First Kind
7.4.1-1. Definition of the line integral of the first kind.
Let a function f(x, y, z)bedefi ned ona piecewise smooth curve

AB inthe three-dimensional
space R
3
.Letthecurve

AB be divided into n subcurves by points A = M
0
, M
1
, M
2
, ,
M
n
= B, thus defi ning a partition L
n
. The longest of the chords M
0

M
1
, M
1
M
2
, ,
M
n–1
M
n
is called the diameter of the partition L
n
and is denoted λ = λ(L
n
). Let us select
on each arc

M
i–1
M
i
an arbitrary point (x
i
, y
i
, z
i
), i = 1, 2, , n, and make up an integral
sum

s
n
=
n

i=1
f(x
i
, y
i
, z
i
) Δl
i
,
where Δl
i
is the length of

M
i–1
M
i
.
If there exists a finite limit of the sums s
n
as λ(L
n
) → 0 that depends on neither the
partition L

n
nor the selection of the points (x
i
, y
i
, z
i
), then it is called the line integral of
the first kind of the function f(x, y, z) over the curve

AB and is denoted

AB
f(x, y, z) dl = lim
λ→0
s
n
.
A line integral is also called a curvilinear integral or a path integral.
If the function f (x, y, z) is continuous, then the line integral exists. The line integral
of the first kind does not depend of the direction the path

AB is traced; its properties are
similar to those of the definite integral.
7.4.1-2. Computation of the line integral of the first kind.
1. If a plane curve is definedintheformy = y(x), with x [a, b], then

AB
f(x, y) dl =


b
a
f

x, y(x)


1 +(y

x
)
2
dx.
2. If a curve

AB is defined in parametric form by equations x = x(t), y = y(t), and
z = z(t), with t
[α, β], then

AB
f(x, y, z) dl =

β
α
f

x(t), y(t), z(t)


(x


t
)
2
+(y

t
)
2
+(z

t
)
2
dt.(7.4.1.1)
If a function f(x, y)isdefined on a plane curve x = x(t), y = y(t), with t
[α, β], one
should set z

t
= 0 in (7.4.1.1).
Example. Evaluate the integral

AB
xy dl,where

AB is a quarter of an ellipse with semiaxes a and b.
Let us write out the equations of the ellipse for the first quadrant in parametric form:
x = acos t, y = bsint (0 ≤ t ≤ π/2).
330 INTEGRALS

We have

(x

t
)
2
+(y

t
)
2
=

a
2
sin
2
t + b
2
cos
2
t. To evaluate the integral, we use formula (7.4.1.1) with
z

t
= 0:

AB
xy dl =


π/2
0
(a cos t)(b sin t)

a
2
sin
2
t + b
2
cos
2
tdt
=
ab
2

π/2
0
sin 2t

a
2
2
(1 –cos2t)+
b
2
2
(1 +cos2t) dt =

ab
4

1
–1

a
2
+ b
2
2
+
b
2
– a
2
2
zdz
=
ab
4
2
b
2
– a
2
2
3

a

2
+ b
2
2
+
b
2
– a
2
2
z

3/2




1
–1
=
ab
3
a
2
+ ab + b
2
a + b
.
7.4.1-3. Applications of the line integral of the first kind.
1. Length of a curve


AB:
L =

AB
dl.
2. Mass of a material curve

AB with a given line density γ = γ(x, y, z):
m =

AB
γdl.
3. Coordinates of the center of mass of a material curve

AB:
x
c
=
1
m

AB
xγ dl, y
c
=
1
m

AB

yγ dl, z
c
=
1
m

AB
zγ dl.
To a material line with uniform density there corresponds γ = const.
7.4.2. Line Integral of the Second Kind
7.4.2-1. Definition of the line integral of the second kind.
Let a vector field
a(x, y, z)=P (x, y, z)

i + Q(x, y, z)

j + R(x, y, z)

k
and a piecewise smooth curve

AB be defined in some domain in R
3
. By dividing the curve
by points A = M
0
, M
1
, M
2

, , M
n
= B into n subcurves, we obtain a partition L
n
.Let
us select on each arc

M
i–1
M
i
an arbitrary point (x
i
, y
i
, z
i
), i = 1, 2, , n, and make up a
sum of dot products
s
n
=
n

i=1
a(x
i
, y
i
, z

i
) ⋅
−−−−− →
M
i–1
M
i
,
called an integral sum.
If there exists a finite limit of the sums s
n
as λ(L
n
) → 0 that depends on neither the
partition L
n
nor the selection of the points (x
i
, y
i
, z
i
), then it is called the line integral of
the second kind of the vector field a(x, y, z) along the curve

AB and is denoted

AB
a ⋅ dr,or


AB
Pdx+ Qdy+ Rdz.
7.4. LINE AND SURFACE INTEGRALS 331
The line integral of the second kind depends on the direction the path is traced, so that

AB
a ⋅ dr =–

BA
a ⋅ dr.
A line integral over a closed contour C is called a closed path integral (or a circulation)
of a vector field a around C and is denoted

C
a ⋅ dr.
Physical meaning of the line integral of the second kind:

AB
a ⋅ dr determines the
work done by the vector field a(x, y, z) on a particle of unit mass when it travels along the
arc

AB.
7.4.2-2. Computation of the line integral of the second kind.
1
◦
. For a plane curve

AB defined as y = y(x), with x [a, b], and a plane vector field a,
we have


AB
a ⋅ dr =

b
a

P

x, y(x)

+ Q

x, y(x)

y

x
(x)

dx.
2
◦
.Let

AB be defined by a vector equation r =r(t)=x(t)

i + y(t)

j + z(t)


k, with t [α, β].
Then

AB
a ⋅ dr =

AB
Pdx+ Qdy+ Rdz
=

β
α

P

x(t), y(t), z(t)

x

t
(t)+Q

x(t), y(t), z(t)

y

t
(t)+R


x(t), y(t), z(t)

z

t
(t)

dt.(7.4.2.1)
For a plane curve

AB and a plane vector field a, one should set z

(t)=0 in (7.4.2.1).
7.4.2-3. Potential and curl of a vector field.
1
◦
. A vector field a = a(x, y, z) is called potential if there exists a function Φ(x, y, z)such
that
a =gradΦ,ora =
∂Φ
∂x

i +
∂Φ
∂y

j +
∂Φ
∂z


k.
The function Φ(x, y, z) is called a potential of the vector field a. The line integral of the
second kind of a potential vector field along a path

AB is equal to the increment of the
potential along the path:

AB
a ⋅ dr = Φ


B
– Φ


A
.
2
◦
.Thecurl of a vector field a(x, y, z)=P

i + Q

j + R

k is the vector defined as
curla =

∂R
∂y

–
∂Q
∂z


i +

∂P
∂z
–
∂R
∂x


j +

∂Q
∂x
–
∂P
∂y


k =








i

j

k
∂
∂x
∂
∂y
∂
∂z
PQR






.
The vector curla characterizes the rate of rotation of a and can also be described as the
circulation density of a. Alternative notations: curl a ≡∇× a ≡ rota.