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.. Multiple integral
Trang 12 Additivity If a domain U is split into two subdomains, U1and U2, 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 =
U1
f (x, y, z) dx dy dz +
U2
f (x, y, z) dx dy dz.
3 Estimation theorem If m≤f (x, y, z)≤M in a domain U , then
mV ≤
U f (x, y, z) dx dy dz≤M V,
where V is the volume of U
4 Mean value theorem If f (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 If ϕ(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 Item1◦ , the domain D of the x, y plane is defined
as{a≤x≤b, y1(x)≤y ≤y2(x)}, then
U f (x, y, z) dx dy dz =
b
y2x)
g(x,y)
h(x,y) f (x, y, z) dz.
Trang 2326 INTEGRALS
7.3.5-2 Change of variables in the triple integral
1◦ Let x = 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 =
Ωf x(u, v, w), y(u, v, w), z(u, v, w)
|J(u, v, w)| du dv dw, where J(u, v, w) is the Jacobian of the mapping of Ω into U:
J (u, v, w) = ∂(x, y, z)
∂(u, v, w) =
∂x
∂y
∂z
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
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 r2sin θ
Generalized spherical
coordinates r, ϕ, θ
x = ar cos ϕ sin θ, y = br sin ϕ sin θ,
z = cr cos θ abcr2sin θ
Coordinates of prolate ellipsoid of
revolution σ, τ , ϕ (σ≥ 1 ≥τ ≥ – 1 )
x = a
(σ2– 1 )( 1– τ2) cos ϕ,
y = a
(σ2– 1 )( 1– τ2) sin ϕ,
z = aστ
a3(σ2– τ2)
Coordinates of oblate ellipsoid of
revolution σ, τ , ϕ (σ≥ 0 , – 1 ≤τ≤ 1 )
x = a
( 1+ σ2)( 1– τ2) cos ϕ,
y = a
( 1+ σ2)( 1– τ2) sin ϕ,
z = aστ
a3(σ2+ τ2)
Parabolic coordinates σ, τ , ϕ x = στ cos ϕ, y = στ sin ϕ, z = 12(τ2– σ2) στ (σ2+ τ2) Parabolic cylinder
coordinates σ, τ , z x = στ , y = 12(τ2– σ2), z = z σ2+ τ2 Bicylindrical coordinates σ, τ , z x= a sinh τ
cosh τ – cos σ , y =
a sin σ cosh τ – cos σ , z = z
a2
(cosh τ – cos σ)2
Toroidal coordinates σ, τ , ϕ
(–π≤σ≤π, 0 ≤τ<∞, 0 ≤ϕ< 2π)
x= a sinh τ cos ϕ
cosh τ – cos σ , y =
a sinh τ sin ϕ cosh τ – cos σ,
z= a sin σ
cosh τ – cos σ
a3sinh τ (cosh τ – cos σ)2
Bipolar coordinates σ, τ , ϕ
(σ is any,0 ≤τ < π, 0 ≤ϕ< 2π)
x= a sin τ cos ϕ
cosh σ – cos τ , y =
a sin τ sin ϕ cosh σ – cos τ,
z= a sinh σ
cosh σ – cos τ
a3sin τ (cosh σ – cos τ )2
Trang 37.3.5-3 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 γ dx dy dz.
3 Coordinates of the center of mass:
xc = 1
m
U xγ dx dy dz, yc=
1
m
U yγ dx dy dz, zc =
1
m
U zγ dx dy dz.
4 Moments of inertia about the coordinate axes:
I x=
2
yz γ dx dy dz, I y =
2
xz γ dx dy dz, I z=
2
xy γ dx dy dz,
where ρ2yz = y2+ z2, ρ2xz = x2+ z2, and ρ2xy = x2+ y2
If the body is homogeneous, then γ = const.
Example Given a bounded homogeneous elliptic cylinder,
x2
a2 + y
2
b2 = 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 = γ
U (x2+ y2) dx dy dz = γ
h 0
2π 0
1 0
ρ2(a2cos2ϕ + b2sin2ϕ )abρ dρ dϕ dz
= 1
4abγ
h 0
2π
0 (a2cos2ϕ + b2sin2ϕ ) dϕ dz = 1
4abγ
2π 0
h
0 (a2cos2ϕ + b2sin2ϕ ) dz dϕ
= 1
4abhγ
2π
0 (a2cos2ϕ + b2sin2ϕ ) dϕ = 1
4ab (a2+ b2)hγ.
Trang 4328 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
(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
r3 dξ dη dζ,
F y = km ∂Φ
∂y = km
U γ (ξ, η, ζ)
η – y
r3 dξ dη dζ,
F z = km ∂Φ
∂z = km
U γ (ξ, η, ζ)
ζ – z
r3 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) =
Ωf (x) exp[λg(x)] dx,
where x ={x1, , x n}, dx = dx1 dx n,Ω is a bounded domain in Rn , f (x) and g(x)
are real-valued functions of n variable, and λ is a real or complex parameter.
Denote by
S ε=
λ: arg|λ| ≤ π2 – ε
4
, 0< ε < π2,
a sector in the complex plane of λ.
THEOREM1 Let the following conditions hold:
(1) the functions f (x) and g(x) are continuous inΩ,
(2) the maximum of g(x) is attained at only one point x0 Ω (x0 is a nondegenerate maximum point), and
(3) the function g(x) has continuous third derivatives in a neighborhood of x0
Then the following asymptotic formula holds as λ → ∞, λS ε:
I(λ) = (2π) n/2exp[λg(x0)] f(x0) + O(λ
– 1)
√
λ n det[g x
i x j(x0)],
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) =
Ωf (x)[g(x)]
λ dx.
THEOREM2 Let g(x) >0and let the conditions of Theorem 1 hold Then the following
asymptotic formula holds as λ → ∞, λS ε:
I (λ) = (2π) n/2[g(x0)](2λ+n)/2 f(x0) + o(1)
√
λ n det[g x
i x j(x0)].
Trang 57.4 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) be defined on a piecewise smooth curve AB in the three-dimensional spaceR3 Let the curveAB be divided into n subcurves by points A = M0, M1, M2, ,
M n = B, thus defining a partition L n The longest of the chords M0M1, M1M2, ,
M n–1M n is called the diameter of the partition L n and is denoted λ = λ( L n) Let us select
on each arcM i– 1M 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
whereΔl iis the length ofM i– 1M i
If there exists a finite limit of the sums s n as λ( L n) → 0that depends on neither the partitionL 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→0s 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 pathAB 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 defined in the form y = y(x), with x[a, b], then
AB f(x, y) dl =
b
a f x, y(x)
1+ (y x )2dx.
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 t2+ (y t 2+ (z t 2dt. (7.4.1.1)
If a function f (x, y) is defined on a plane curve x = x(t), y = y(t), with t [α, β], one should set z t =0in (7.4.1.1)
Example Evaluate the integral
AB
xy dl, whereAB 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 = a cos t, y = b sin t ( 0 ≤t≤π/2 ).
Trang 6330 INTEGRALS
We have
(x t)2+ (y t )2 =
a2sin2t + b2cos2t 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)
a2sin2t + b2cos2t dt
= ab
2
π/2 0
sin 2t a
2
2(1– cos2t) +
b2
2(1+ cos2t ) dt =
ab
4
1
– 1
a2+ b2
2 +
b2– a2
2 z dz
= ab
4
2
b2– a2
2 3
a2+ b2
2 +
b2– a2
2 z
3/2
1
– 1
= ab
3
a2+ ab + b2
a + b .
7.4.1-3 Applications of the line integral of the first kind
1 Length of a curve AB:
L=
2 Mass of a material curve AB with a given line density γ = γ(x, y, z):
m=
AB γ dl.
3 Coordinates of the center of mass of a material curve AB:
xc= 1
m
AB xγ dl, yc =
1
m
AB yγ dl, zc=
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 curveAB be defined in some domain inR3 By dividing the curve
by points A = M0, M1, M2, , M n = B into n subcurves, we obtain a partition L n Let
us select on each arcM i– 1M 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–1M i
called an integral sum.
If there exists a finite limit of the sums s n as λ( L n) → 0that depends on neither the partitionL 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 P dx + Q dy + R dz.
Trang 7The line integral of the second kind depends on the direction the path is traced, so that
AB a⋅dr= –
BA a⋅dr.
A line integral over a closed contourC is called a closed path integral (or a circulation)
of a vector field a around C and is denoted
8
C a⋅dr.
Physical meaning of the line integral of the second kind:
AB a⋅dr determines the
work done by the vector field a(x, y, z) on a particle of unit mass when it travels along the
arcAB
7.4.2-2 Computation of the line integral of the second kind
1◦ For a plane curveAB defined as y = y(x), with x
[a, b], and a plane vector field a,
AB a⋅dr=
b
a
P x, y(x)
+ Q x, y(x)
y
x (x)
dx.
2◦ LetAB 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
=
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 curveAB and a plane vector field a, one should set z (t) =0in (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 functionΦ(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 pathAB is equal to the increment of the potential along the path:
AB a⋅dr=Φ
B–Φ
A.
2◦ The curl of a vector field a(x, y, z) = Pi + Qj + Rk 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 ∂
P Q R
.
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.
... 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... 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... 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