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 ⋅ dr,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 ⋅ dr =– BA a ⋅ dr. 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 ⋅ 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 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 ⋅ dr = 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 ⋅ dr = 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,ora = ∂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 ⋅ dr = 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 curla = ∂R ∂y – ∂Q ∂z i + ∂P ∂z – ∂R ∂x j + ∂Q ∂x – ∂P ∂y k = i j k ∂ ∂x ∂ ∂y ∂ ∂z PQR . The vector curla characterizes the rate of rotation of a and can also be described as the circulation density of a. Alternative notations: curl a ≡∇× a ≡ rota. . 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 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. 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