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Multiple integral

A multiple integral is a definite integral of a function of several real variables over a domain in n-dimensional Euclidean space, generalizing the single-variable definite integral to higher dimensions such as two or three variables.^[1] For instance, double integrals apply to functions of two variables over regions in the plane, while triple integrals extend to three variables over volumes in space.^[2] The concept of multiple integrals emerged in the 17th century with methods like indivisibles developed by Bonaventura Cavalieri for computing areas and volumes, evolving through contributions from mathematicians like Cauchy on double integrals in the 19th century.^[3] Multiple integrals are fundamental in multivariable calculus, where they enable the computation of accumulated quantities across multidimensional regions, such as the volume beneath a surface defined by z = f(x, y) or the mass of a thin plate with variable density.^[4] Defined as limits of Riemann sums over partitions of the domain, they reduce computationally to iterated single integrals via Fubini's theorem, which justifies integrating one variable at a time while treating others as constants.^[4] This structure allows evaluation over rectangular or general regions, often requiring adjustments for boundaries and coordinate systems.^[2] Beyond basic volumes, multiple integrals find broad applications in physics and engineering, including calculating centers of mass and moments of inertia for rigid bodies,^[5] and in probability for computing expectations and probabilities in multivariate distributions.^[6] They also support change-of-variables techniques, such as transformations to polar, cylindrical, or spherical coordinates, to simplify integration over symmetric domains.^[2] In vector calculus, multiple integrals underpin theorems like the divergence theorem and Stokes' theorem, linking surface and volume integrals to flux and circulation.^[4]

Introduction and Overview

Definition and Scope

A multiple integral generalizes the concept of a single integral to functions of several variables, allowing the computation of integrals over regions in higher-dimensional Euclidean spaces \mathbb{R}^n. In essence, it extends the idea of measuring signed areas under curves in one dimension to measuring signed volumes under hypersurfaces in multiple dimensions; for instance, when n=2, it corresponds to the area beneath a surface in the plane, while for n=3, it yields volumes within solid regions.^[4]^[2] This framework is fundamental in multivariable calculus for quantifying accumulated quantities, such as mass, probability, or total flux, over multi-dimensional domains.^[7] The basic notation for a multiple integral of a function f: \mathbb{R}^n \to \mathbb{R} over a domain D \subseteq \mathbb{R}^n is \int_D f(x_1, \dots, x_n) \, dV, where dV denotes the volume element in \mathbb{R}^n, often expressed as dx_1 \dots dx_n in Cartesian coordinates.^[7]^[1] This integral represents the limit of Riemann sums approximating the function's values multiplied by the volumes of small subregions partitioning D.^[4] Unlike line integrals, which accumulate quantities along one-dimensional curves in space, or surface integrals, which do so over two-dimensional surfaces, multiple integrals specifically evaluate volumes over n-dimensional regions, emphasizing the full interior rather than boundaries.^[8] The construction builds on the Riemann integral from one dimension by extending partitions to multi-dimensional grids, where each subregion's volume replaces the interval length in the sum.^[7]^[2] Multiple integrals inherit key properties from their single-variable counterparts, such as linearity, which will be explored further in subsequent sections.^[4]

Historical Development

The development of multiple integrals traces its roots to the foundational work on calculus by Isaac Newton and Gottfried Wilhelm Leibniz in the late 17th century, where they independently formulated the concepts of differentiation and single-variable integration as tools for analyzing motion and areas under curves.^[9] Newton's approach, outlined in his 1669 manuscript De analysi per aequationes numero terminorum infinitas, emphasized fluxions and fluents to compute integrals geometrically, while Leibniz's 1684 publication Nova methodus introduced the integral symbol ∫ and treated integration as summation of infinitesimals analytically.^[9] These single-variable techniques laid the groundwork for extensions to functions of multiple variables, though Newton and Leibniz themselves did not systematically explore multivariable cases.^[9] In the 18th century, Leonhard Euler advanced the theory by extending integral calculus to multiple dimensions, particularly through his comprehensive three-volume work Institutionum calculi integralis published between 1768 and 1770.^[10] There, Euler systematically investigated double integrals, deriving formulas for integrals over rectangular regions and connecting them to beta and gamma functions, which he had earlier introduced in 1729.^[10] A key contribution appears in his 1768 paper "De formulis integralibus duplicatis," later published in 1770, where he explored the evaluation and properties of double integrals, treating them as iterated single integrals while addressing challenges in changing variables.^[11] The 19th century brought rigorous foundations to multiple integrals amid growing concerns over convergence and discontinuities. Augustin-Louis Cauchy, in his 1823 Résumé des leçons sur le calcul infinitésimal, defined the definite integral for continuous functions using partitions and upper/lower sums, a framework that influenced multivariable extensions by emphasizing uniformity in limits.^[12] Cauchy also examined double integrals, noting in later works that interchanging integration order could yield inconsistencies unless absolute convergence holds, prompting early notions of absolute integrability.^[13] Bernhard Riemann, in his 1854 paper "Über die Darstellbarkeit einer Funktion durch eine trigonometrische Reihe" (published in 1868), generalized the integral to bounded functions with discontinuities via tagged partitions and oscillation criteria, providing a basis for multi-dimensional Riemann integrals that accommodated more complex domains.^[12] Toward the century's end, Italian mathematicians Giuseppe Peano and Vito Volterra formalized multiple integrals further; Peano's 1887 Calcolo vettoriale generalized measure concepts to one-, two-, and three-dimensional sets, defining internal and external measures for regions with boundaries and linking them to integrability.^[14] Volterra, in his 1881 papers, constructed examples of sets with positive outer content but nowhere dense interiors and discontinuous derivatives that failed Riemann integrability, highlighting limitations and paving the way for advanced theories around 1880–1890.^[14] The early 20th century saw Henri Lebesgue revolutionize multiple integration through his 1902 doctoral thesis Intégrale, longueur, aire, which introduced measure theory to handle non-continuous functions in higher dimensions.^[15] Lebesgue's approach defined the integral via measurable sets and simple functions, extending naturally to \mathbb{R}^n by product measures, thus resolving issues with Riemann's method for functions discontinuous on sets of positive measure and enabling integration over irregular domains.^[15] This framework, building on Riemann's foundations, became the standard for modern analysis of multiple integrals.^[15]

Mathematical Definition

General Formulation

In the context of multivariable calculus, a multiple integral provides a means to compute the integral of a function over a multi-dimensional domain, extending the one-dimensional Riemann integral. For a function f: D \subset \mathbb{R}^n \to \mathbb{R}, where D is a bounded domain in n-dimensional Euclidean space, the multiple integral is formally defined as the limit of Riemann sums over partitions of D. Specifically, consider a partition P of D into finitely many subregions D_i with volumes \Delta V_i, and choose sample points \xi_i \in D_i. The Riemann sum is \sum_i f(\xi_i) \Delta V_i, and the integral is

\int_D f \, dV = \lim_{\|P\| \to 0} \sum_i f(\xi_i) \Delta V_i,

where \|P\| denotes the mesh of the partition, typically the maximum diameter of the subregions D_i, and the limit exists if the upper and lower Riemann sums converge to the same value.^[16]^[17] For the integral to be well-defined in the Riemann sense, f must be bounded on the compact set D, and the upper sum U(f, P) = \sum_i M_i \Delta V_i (where M_i = \sup_{D_i} f) and lower sum L(f, P) = \sum_i m_i \Delta V_i (where m_i = \inf_{D_i} f) must satisfy \inf_P U(f, P) = \sup_P L(f, P) as the partition is refined. A sufficient condition for Riemann integrability is that f is continuous on D, or more generally, discontinuous only on a set of n-dimensional Lebesgue measure zero. This ensures the limit is independent of the choice of sample points and partitions.^[16]^[17] In the n-dimensional case, the integral is commonly notated as \int_D f(x_1, \dots, x_n) \, dV, where dV represents the volume element, or more abstractly as \int_{\mathbb{R}^n} f \, d\mu with \mu denoting the Lebesgue measure on \mathbb{R}^n, which coincides with the Riemann integral for Riemann-integrable functions but extends to a broader class via measure-theoretic construction. The Lebesgue approach assigns a measure to subsets of \mathbb{R}^n based on coverings by rectangles, providing a rigorous foundation for integration over non-rectifiable domains, though the Riemann definition remains central for continuous functions on compact sets.^[16]^[18] For unbounded domains D \subset \mathbb{R}^n, the multiple integral is defined as an improper integral by taking the limit over an increasing sequence of compact exhaustion sets D_k \uparrow D, such that \int_D f \, dV = \lim_{k \to \infty} \int_{D_k} f \, dV, provided the limit exists and is finite. This extension allows integration over regions like \mathbb{R}^n itself, but requires additional convergence conditions on f to ensure the improper integral is well-behaved.^[17]

Properties of Multiple Integrals

Multiple integrals, defined as limits of Riemann sums over partitions of a bounded domain D \subset \mathbb{R}^n, inherit several fundamental algebraic and analytic properties analogous to those of the single-variable Riemann integral. These properties hold for Riemann-integrable functions, typically continuous functions on compact Jordan-measurable sets, and are derived directly from the definition via sums over subdomains.^[19] One key property is linearity. For Riemann-integrable functions f, g: D \to \mathbb{R} and constants a, b \in \mathbb{R}, the multiple integral satisfies

\int_D (a f + b g) \, dV = a \int_D f \, dV + b \int_D g \, dV,

where dV denotes the volume element. This follows from the linearity of the Riemann sum approximation: if \sigma is a partition of D into subdomains R_k with tags \mathbf{x}_k \in R_k and \Delta V_k = \mathrm{vol}(R_k), then the Riemann sum for a f + b g is \sum_k (a f(\mathbf{x}_k) + b g(\mathbf{x}_k)) \Delta V_k = a \sum_k f(\mathbf{x}_k) \Delta V_k + b \sum_k g(\mathbf{x}_k) \Delta V_k. Taking the limit as the mesh of the partition approaches zero preserves this relation, yielding the integral equality.^[19]^[20] Monotonicity is another essential property: if f \leq g on D, where f and g are Riemann-integrable, then \int_D f \, [dV](/page/DV) \leq \int_D g \, [dV](/page/DV). To see this, consider the non-negative function h = g - f \geq 0. By linearity, \int_D g \, dV = \int_D f \, dV + \int_D h \, dV, so it suffices to show \int_D h \, dV \geq 0 for non-negative integrable h. In the Riemann sum, each term h(\mathbf{x}_k) \Delta V_k \geq 0, so the sum is non-negative, and thus the limit (integral) is non-negative. This monotonicity extends to the case where $0 \leq f \leq g.^[19]^[20] Additivity over disjoint domains also holds: if D = D_1 \cup D_2 with D_1 \cap D_2 = \emptyset and f is Riemann-integrable on D, then \int_D f \, dV = \int_{D_1} f \, dV + \int_{D_2} f \, dV. This property arises from refining partitions of D to separately partition D_1 and D_2; the Riemann sums decompose additively over the subdomains, and the limit respects this decomposition since the integrals over each exist. For more general disjoint unions of Jordan-measurable sets, the property extends by finite additivity of the content (Jordan measure).^[19] For continuous functions on a compact domain D, the mean value theorem for multiple integrals states that if m \leq f(\mathbf{x}) \leq M for all \mathbf{x} \in D, then there exists some c \in [m, M] such that \int_D f \, dV = c \cdot \mathrm{vol}(D). More precisely, for continuous f on a connected compact Jordan-measurable D, there exists \mathbf{c} \in D with \int_D f \, dV = f(\mathbf{c}) \cdot \mathrm{vol}(D). This follows from the intermediate value theorem applied to the continuous function g(t) = \int_D (f - t) \, dV on [m, M], which changes sign or zero at some point, combined with monotonicity to ensure the integral scales with volume.^[19]^[20] Basic inequalities derive from these properties, such as the triangle inequality \left| \int_D f \, dV \right| \leq \int_D |f| \, dV for Riemann-integrable f. To prove this, note that |f| \geq f and |f| \geq -f, so by monotonicity, \int_D |f| \, dV \geq \int_D f \, dV and \int_D |f| \, dV \geq -\int_D f \, dV = \int_D (-f) \, dV. Adding these yields $2 \int_D |f| \, dV \geq \int_D f \, dV + \int_D (-f) \, dV = 0, but more directly, \left| \int_D f \, dV \right| = \int_D |f| \, dV if f \geq 0 or by linearity for general f = f^+ - f^- with f^\pm \geq 0, though the inequality holds via the above bounds.^[19]^[20]

Particular Cases

Double Integrals

A double integral is a type of multiple integral that extends the concept of a single integral to functions of two variables over a region in the plane. It is denoted by \iint_D f(x,y) \, dA, where f(x,y) is the integrand, D is a bounded region in \mathbb{R}^2, and dA = dx \, dy represents the area element. This integral represents the signed volume under the surface z = f(x,y) over the domain D.^[21] The domain D for a double integral is typically a bounded subset of \mathbb{R}^2, often classified into Type I and Type II regions to facilitate computation via iterated integrals. A Type I region is described as D = \{(x,y) \mid a \leq x \leq b, \, g(x) \leq y \leq h(x)\}, where g(x) and h(x) are continuous functions with g(x) \leq h(x) for a \leq x \leq b, forming vertical strips bounded by the curves y = g(x) and y = h(x). Similarly, a Type II region is D = \{(x,y) \mid c \leq y \leq d, \, g(y) \leq x \leq h(y)\}, where g(y) and h(y) are continuous with g(y) \leq h(y) for c \leq y \leq d, consisting of horizontal strips. These classifications allow the double integral to be evaluated as an iterated integral in the appropriate order.^[22] The double integral is defined as the limit of Riemann sums over partitions of the region D. For a partition of D into subregions with areas \Delta A_{ij}, the Riemann sum is \sum_i \sum_j f(x_i^*, y_j^*) \Delta A_{ij}, where (x_i^*, y_j^*) is a sample point in the ij-th subregion; for rectangular partitions, \Delta A_{ij} = \Delta x_i \Delta y_j. As the mesh of the partition approaches zero, this sum converges to \iint_D f(x,y) \, dA. Double integrals inherit properties such as linearity from the general theory of multiple integrals.^[21] A function f(x,y) is Riemann integrable over a closed and bounded region D if it is bounded and continuous on D, except possibly on a set of measure zero; continuity on such a D ensures integrability. This condition guarantees that the limit of the Riemann sums exists and is independent of the choice of partitions and sample points.^[23]

Triple Integrals

A triple integral extends the concept of multiple integration to three dimensions, providing a means to compute the integral of a function over a bounded solid region E \subset \mathbb{R}^3. It is denoted by \iiint_E f(x,y,z) \, [dV](/page/DV), where f: E \to \mathbb{R} is the integrand and dV represents the volume element, typically expressed in Cartesian coordinates as dx \, dy \, dz. This integral quantifies quantities such as the total mass of a solid with variable density f(x,y,z), or the average value of f over E when normalized by the volume of E.^[24]^[25] The triple integral is rigorously defined as the limit of a triple Riemann sum over a partition of E. Consider a partition of E into sub-boxes with volumes \Delta V_{ijk} = \Delta x_i \Delta y_j \Delta z_k, where sample points (\xi_i, \eta_j, \zeta_k) are chosen within each sub-box. The Riemann sum is then \sum_i \sum_j \sum_k f(\xi_i, \eta_j, \zeta_k) \Delta V_{ijk}, and the triple integral is \iiint_E f(x,y,z) \, dV = \lim \sum_i \sum_j \sum_k f(\xi_i, \eta_j, \zeta_k) \Delta V_{ijk} as the mesh of the partition approaches zero. For the integral to exist, f must be bounded and continuous on the compact set E, ensuring uniform convergence of the Riemann sums.^[24]^[25]^[26] To evaluate triple integrals, the solid E is often described by its projections onto the coordinate planes, leading to iterated integrals with specific orders of integration. For a Type I region, E is defined such that a \leq x \leq b, g(x) \leq y \leq h(x), and u(x,y) \leq z \leq v(x,y), where g, h, u, and v are continuous functions; the integral becomes \int_a^b \int_{g(x)}^{h(x)} \int_{u(x,y)}^{v(x,y)} f(x,y,z) \, dz \, dy \, dx. Similar Type II and Type III descriptions project onto the yz- and xz-planes, respectively, by cycling the variables, allowing flexibility in setup based on the region's geometry. The volume element dV = dx \, dy \, dz facilitates this iteration, with integrability holding under the same continuity assumptions on f and the bounding surfaces of E. Building on double integrals over planar regions, these descriptions add a third layer of variable limits dependent on the prior coordinates.^[25]^[26]^[24]

Higher-Dimensional Integrals

The multiple integral in n dimensions, for n \geq 4, generalizes the formulation to integrals over subsets D \subset \mathbb{R}^n and is denoted as \int_D f(x_1, \dots, x_n) \, dx_1 \cdots dx_n, where f: \mathbb{R}^n \to \mathbb{R} is an integrable function and the measure is the standard Lebesgue measure on \mathbb{R}^n. This notation extends the product structure of lower-dimensional cases, treating the integral as an n-fold iteration over the coordinates.^[27] A primary challenge in higher-dimensional integration arises from the curse of dimensionality, where the volume of the unit cube in \mathbb{R}^n remains 1, but the complexity of partitioning the domain into manageable elements grows exponentially with n, rendering standard grid-based methods computationally intractable for n > 3. For instance, tensor product approximations require resources scaling as O(m^n) for grid size m, quickly exceeding practical limits even for moderate n around 10. This intractability extends to both analytical evaluation and numerical approximation, necessitating specialized techniques beyond simple iteration.^[28] To address limitations of Riemann integrals in higher dimensions, the Lebesgue integral provides a robust extension by defining it over the \sigma-algebra of Lebesgue measurable sets in \mathbb{R}^n, generated as the completion of the Borel \sigma-algebra with respect to the Lebesgue outer measure. A function f is measurable if the preimage of Borel sets under f belongs to this \sigma-algebra, enabling the integral to handle a broader class of functions, including those with singularities or on irregular domains, while preserving properties like countable additivity.^[29] An important abstract property of the higher-dimensional Lebesgue integral is its invariance under orthogonal transformations: for any orthogonal matrix Q \in O(n), the integral satisfies \int_{\mathbb{R}^n} f(Qx) \, dx = \int_{\mathbb{R}^n} f(x) \, dx for integrable f, reflecting the rotation invariance of the Lebesgue measure itself. This property ensures that the integral remains unchanged under rigid rotations of the domain, facilitating applications in symmetric problems across \mathbb{R}^n.^[29]

Computation Techniques

Iterated Integrals

Iterated integrals provide a practical method for computing multiple integrals by reducing them to successive single-variable integrations. For a function f(x,y) defined over a region D in the plane that can be described as a Type I region, where D = \{(x,y) \mid a \leq x \leq b, g(x) \leq y \leq h(x)\} with g and h continuous, the iterated integral is defined as

\int_a^b \left( \int_{g(x)}^{h(x)} f(x,y) \, dy \right) dx.

This inner integral treats x as fixed and integrates with respect to y, yielding a function of x that is then integrated over the x-interval. Similarly, for a Type II region D = \{(x,y) \mid c \leq y \leq d, p(y) \leq x \leq q(y)\}, the iterated integral takes the form

\int_c^d \left( \int_{p(y)}^{q(y)} f(x,y) \, dx \right) dy.

These definitions extend naturally to higher dimensions, such as triple integrals over regions described by nested bounds.^[30] For rectangular domains, such as R = [a,b] \times [c,d], the order of integration offers flexibility, allowing evaluation as either \int_a^b \int_c^d f(x,y) \, dy \, dx or \int_c^d \int_a^b f(x,y) \, dx \, dy. This interchange simplifies computation when one order yields an easier antiderivative. The equivalence of these iterated integrals to the multiple integral over the domain is justified by Fubini's theorem.^[31] Fubini's theorem states that if f(x,y) is continuous on a compact rectangular region R = [a,b] \times [c,d], then the double integral equals both iterated integrals:

\iint_R f(x,y) \, dA = \int_a^b \int_c^d f(x,y) \, dy \, dx = \int_c^d \int_a^b f(x,y) \, dx \, dy.

This holds more generally for continuous f on a compact bounded region D \subset \mathbb{R}^2, where the multiple integral \iint_D f \, dA equals the appropriate iterated integral over a description of D. The theorem extends to higher dimensions analogously.^[32]^[33] A sketch of the proof for the rectangular case relies on the uniform continuity of f on the compact set R, which follows from its continuity. Partition R into a grid of small rectangles with side lengths \Delta x = (b-a)/m and \Delta y = (d-c)/n. The Riemann sum for the double integral is \sum_{i=1}^m \sum_{j=1}^n f(x_i^*, y_j^*) \Delta x \Delta y, which can be grouped as a telescoping double sum: first summing over j for fixed i approximates the inner integral with respect to y, then over i for the outer with respect to x. Uniform continuity ensures that the choice of sample points (x_i^*, y_j^*) introduces an error bounded by \epsilon \cdot (b-a)(d-c) for sufficiently fine partitions, making the iterated sums converge to the same limit as the double integral. This argument generalizes to non-rectangular compact regions by extending f continuously to a containing rectangle.^[31]

Change of Variables

In multiple integrals, the change of variables theorem facilitates the substitution of new coordinates via a smooth, invertible transformation, accounting for the distortion of volumes through the Jacobian determinant. For an n-dimensional integral over a region D \subset \mathbb{R}^n and a diffeomorphism g: U \to D where U \subset \mathbb{R}^n, the formula states that

\int_D f(\mathbf{x}) \, d\mathbf{x} = \int_U f(g(\mathbf{u})) \left| \det J_g(\mathbf{u}) \right| \, d\mathbf{u},

with J_g denoting the Jacobian matrix of partial derivatives of g. This holds under conditions of continuity and invertibility, ensuring the transformation preserves the integral's value up to the scaling factor provided by the absolute determinant.^[34]^[35] The Jacobian matrix J_g(\mathbf{u}) is the n \times n matrix with entries (J_g)_{ij} = \partial g_i / \partial u_j, and its determinant measures the local volume scaling. In two dimensions, for coordinates x = x(u,v), y = y(u,v), the Jacobian determinant simplifies to

\left| \frac{\partial(x,y)}{\partial(u,v)} \right| = \left| \frac{\partial x}{\partial u} \frac{\partial y}{\partial v} - \frac{\partial x}{\partial v} \frac{\partial y}{\partial u} \right|,

which adjusts the area element dA = dx \, dy to \left| \frac{\partial(x,y)}{\partial(u,v)} \right| du \, dv in the transformed variables. This determinant arises from the linear approximation of the transformation near each point, analogous to the single-variable case where dx = |g'(u)| du.^[34]^[36] A standard application is the polar coordinate transformation in the plane, defined by x = r \cos \theta, y = r \sin \theta, with parameters r \geq 0 and \theta \in [0, 2\pi). To derive the Jacobian, compute the partial derivatives:

\frac{\partial x}{\partial r} = \cos \theta, \quad \frac{\partial x}{\partial \theta} = -r \sin \theta, \quad \frac{\partial y}{\partial r} = \sin \theta, \quad \frac{\partial y}{\partial \theta} = r \cos \theta.

The determinant is then

\frac{\partial(x,y)}{\partial(r,\theta)} = \begin{vmatrix} \cos \theta & -r \sin \theta \\ \sin \theta & r \cos \theta \end{vmatrix} = (\cos \theta)(r \cos \theta) - (-r \sin \theta)(\sin \theta) = r \cos^2 \theta + r \sin^2 \theta = r,

since \cos^2 \theta + \sin^2 \theta = 1 and r \geq 0. Thus, the area element becomes dA = r \, dr \, d\theta. For integration over a disk of radius R, the bounds are $0 \leq r \leq R and $0 \leq \theta \leq 2\pi, simplifying computations for circularly symmetric regions.^[34]^[37]^[38] In three dimensions, cylindrical coordinates extend polar coordinates by including height: x = r \cos \theta, y = r \sin \theta, z = z, with r \geq 0, \theta \in [0, 2\pi), and z varying according to the domain (e.g., a \leq z \leq b). The Jacobian matrix is

J = \begin{pmatrix} \cos \theta & -r \sin \theta & 0 \\ \sin \theta & r \cos \theta & 0 \\ 0 & 0 & 1 \end{pmatrix},

and its determinant is r, yielding the volume element dV = r \, dr \, d\theta \, dz. This transformation is useful for regions with cylindrical symmetry, such as infinite cylinders or finite heights.^[39]^[40]^[36] Spherical coordinates provide another common substitution: x = \rho \sin \phi \cos \theta, y = \rho \sin \phi \sin \theta, z = \rho \cos \phi, where \rho \geq 0, \theta \in [0, 2\pi), and \phi \in [0, \pi]. The Jacobian matrix is

J = \begin{pmatrix} \sin \phi \cos \theta & -\rho \sin \phi \sin \theta & \rho \cos \phi \cos \theta \\ \sin \phi \sin \theta & \rho \sin \phi \cos \theta & \rho \cos \phi \sin \theta \\ \cos \phi & 0 & -\rho \sin \phi \end{pmatrix},

with determinant \rho^2 \sin \phi (computed by cofactor expansion along the second row or verified via standard results). Since \sin \phi \geq 0 for \phi \in [0, \pi], the volume element is dV = \rho^2 [\sin \phi](/page/Sin) \, d\rho \, d\theta \, d\phi. For a ball of radius R, the bounds are $0 \leq \rho \leq R, $0 \leq \theta \leq 2\pi, $0 \leq \phi \leq \pi, aiding integrals over spherical domains.^[34]^[38]^[36]

Symmetry and Reduction Methods

Symmetry methods provide powerful tools for simplifying the evaluation of multiple integrals by exploiting inherent properties of the integrand or the integration domain, often reducing the computational burden without requiring complete iteration or transformation. These techniques are particularly effective when the domain possesses reflectional, rotational, or other symmetries that align with the behavior of the function being integrated. In multiple variables, a function f(\mathbf{x}) is considered odd with respect to one variable, say x_i, if f(-x_i, x_1, \dots, x_{i-1}, x_{i+1}, \dots, x_n) = -f(x_1, \dots, x_n) for all points in the domain. If the domain D is symmetric with respect to the hyperplane where x_i = [0](/page/0) (i.e., if \mathbf{x} \in D implies that the point with x_i replaced by -x_i is also in D), then the multiple integral \int_D f(\mathbf{x}) \, d\mathbf{x} = [0](/page/0). This result follows from pairing points symmetric across the hyperplane, where the contributions cancel due to the oddness in x_i. The property extends analogously if the function is odd in multiple variables and the domain is symmetric in those directions. For instance, in double integrals over a region symmetric about the y-axis, if f(x,y) is odd in x, the integral vanishes. Radial symmetry arises when the integrand depends only on the radial distance r = \sqrt{x_1^2 + \dots + x_n^2} from the origin, and the domain is a ball or disk centered at the origin. For a double integral over the unit disk in the plane, if f(x,y) = g(\sqrt{x^2 + y^2}), the integral simplifies to

\iint_{x^2 + y^2 \leq 1} g(\sqrt{x^2 + y^2}) \, dx \, dy = 2\pi \int_0^1 g(r) r \, dr,

due to the rotational invariance around the origin, which distributes the area element uniformly in angular directions. This reduction leverages the full $2\pi rotational symmetry to convert the two-dimensional integral into a one-dimensional radial integral, weighted by the Jacobian factor r. Similar reductions apply in higher dimensions, such as for triple integrals over a ball, yielding factors involving the surface area of the unit sphere.^[41] Reduction formulas for multiple integrals often employ integration by parts or differentiation under the integral sign to express a given integral in terms of simpler ones, particularly for parametric families or power functions. For example, consider a parametric double integral I(a) = \iint_D f(x,y,a) \, dx \, dy; differentiating with respect to the parameter a yields I'(a) = \iint_D \frac{\partial}{\partial a} f(x,y,a) \, dx \, dy, assuming suitable continuity and differentiability conditions to justify interchanging derivative and integral. Integrating back and applying boundary conditions can reduce I(a) to lower-order terms. Integration by parts in one variable, treating others as fixed, similarly reduces powers: for \iint_D x^m y^n \, dx \, dy over a rectangular domain, parts on x^m yields a recurrence relating it to integrals with m-2. These methods are especially useful for evaluating moments or polynomial integrals without full expansion.^[42] Domain decomposition exploits symmetry by partitioning an irregular or complex domain into congruent symmetric subregions, computing the integral over one subregion, and multiplying by the number of subregions. For a domain invariant under a group of symmetries (e.g., reflections across coordinate planes), the integral over the full domain equals the number of symmetric copies times the integral over a fundamental domain. In triple integrals over a sphere of radius R, the domain decomposes into 8 octants symmetric under sign changes in x, y, z; thus, the volume integral is $8 times the integral over the positive octant x \geq 0, y \geq 0, z \geq 0. This approach minimizes setup and computation while preserving accuracy, provided the integrand respects the symmetries or averages appropriately across subregions.

Examples and Illustrations

Rectangular Domains

In rectangular domains, multiple integrals are evaluated over regions that are Cartesian products of intervals, such as a rectangle R = [a, b] \times [c, d] in the plane or a box in three dimensions. For a double integral \iint_R f(x,y) \, [dA](/page/DA), where f is continuous on R, the integral is computed using iterated integrals in the form \int_a^b \int_c^d f(x,y) \, dy \, dx.^[21] This approach treats the inner integral with respect to y as a function of the fixed outer variable x, followed by integration with respect to x.^[30] A concrete example is the double integral \iint_{[0,1] \times [0,1]} (x^2 + y^2) \, dA. Compute it as the iterated integral \int_0^1 \int_0^1 (x^2 + y^2) \, dy \, dx:

\begin{align*} \int_0^1 (x^2 + y^2) \, dy &= \left[ x^2 y + \frac{y^3}{3} \right]_{y=0}^{y=1} = x^2 \cdot 1 + \frac{1}{3} - 0 = x^2 + \frac{1}{3}, \\ \int_0^1 \left( x^2 + \frac{1}{3} \right) \, dx &= \left[ \frac{x^3}{3} + \frac{x}{3} \right]_{x=0}^{x=1} = \frac{1}{3} + \frac{1}{3} = \frac{2}{3}. \end{align*}

Thus, \iint_{[0,1] \times [0,1]} (x^2 + y^2) \, dA = \frac{2}{3}. This step-by-step iteration leverages the fixed limits of the rectangular domain, simplifying the evaluation.^[30] For triple integrals over a rectangular box B = [a, b] \times [c, d] \times [e, f], the integral \iiint_B f(x,y,z) \, dV is expressed as the iterated integral \int_a^b \int_c^d \int_e^f f(x,y,z) \, dz \, dy \, dx, assuming f is continuous on B.^[43] The innermost integral treats x and y as constants, followed by successive integrations over the remaining variables with their fixed limits. Over rectangular domains, the order of integration is independent, meaning \int_a^b \int_c^d f(x,y) \, dy \, dx = \int_c^d \int_a^b f(x,y) \, dx \, dy for continuous f, allowing flexibility in choosing the sequence that simplifies computation.^[30] This property extends to higher dimensions for boxes, where any of the six possible orders yields the same result.^[43]

Non-Rectangular Domains

In non-rectangular domains, the boundaries of the integration region are described using inequalities that depend on the variables, often requiring the region to be classified as Type I (vertical strips) or Type II (horizontal strips) for double integrals, or analogous descriptions for higher dimensions.^[44] This allows the multiple integral to be expressed as an iterated integral with variable limits, ensuring the integration covers exactly the desired area or volume without overlap or omission. Accurate boundary description is crucial, as it determines the limits of integration and directly affects the result.^[44] Consider the double integral \iint_D (x + y) \, dA over the triangular region D defined by $0 \leq x \leq 1 and $0 \leq y \leq x, which lies below the line y = x in the first quadrant.^[44] As a Type I region, the iterated integral is set up as

\int_0^1 \int_0^x (x + y) \, dy \, dx.

Evaluating the inner integral first gives \int_0^x (x + y) \, dy = \left[ x y + \frac{y^2}{2} \right]_0^x = x^2 + \frac{x^2}{2} = \frac{3}{2} x^2. The outer integral then yields \int_0^1 \frac{3}{2} x^2 \, dx = \frac{3}{2} \cdot \frac{1}{3} = \frac{1}{2}.^[44] The order of integration can be switched for the same region D, now viewed as a Type II region with $0 \leq y \leq 1 and y \leq x \leq 1.^[45] The iterated integral becomes

\int_0^1 \int_y^1 (x + y) \, dx \, dy,

which evaluates to the same value of \frac{1}{2}, confirming consistency via Fubini's theorem for continuous functions over bounded regions.^[45] For a triple integral example, consider the volume of the solid under the paraboloid surface z = x^2 + y^2 and above the unit disk D: x^2 + y^2 \leq 1 in the xy-plane. This volume is given by the triple integral \iiint_E 1 \, dV, where E is the region (x,y,z) with (x,y) \in D and $0 \leq z \leq x^2 + y^2 .^[46] As an iterated integral in Cartesian coordinates, it would require describing D with variable limits, such as integrating over y from -\sqrt{1-x^2} to \sqrt{1-x^2} and x from -1 to $1, then z from 0 to x^2 + y^2. However, the curved boundary of the disk suggests a setup in polar coordinates for the base, yielding

\int_0^{2\pi} \int_0^1 \int_0^{r^2} r \, dz \, dr \, d\theta = \int_0^{2\pi} \int_0^1 r^3 \, dr \, d\theta,

focusing on the boundary description where the radial limit r \leq 1 and angular sweep $0 \leq \theta \leq 2\pi capture the disk precisely.^[46]

Volume and Surface Computations

Multiple integrals provide a powerful method for computing geometric quantities such as volumes and surface areas in higher dimensions. The volume beneath a surface defined by z = f(x,y) \geq 0 over a region D in the xy-plane is given by the double integral V = \iint_D f(x,y) \, dA./03%3A_Multiple_Integrals/3.01%3A_Double_Integrals) A representative example is the volume under the upper hemisphere z = \sqrt{4 - x^2 - y^2} over the disk D: x^2 + y^2 \leq 4. Using polar coordinates, where x = r \cos [\theta](/page/Theta), y = r \sin [\theta](/page/Theta), and dA = r \, dr \, d\theta, the integral becomes

V = \int_0^{2\pi} \int_0^2 \sqrt{4 - r^2} \, r \, dr \, d\theta.

The inner integral evaluates to \int_0^2 \sqrt{4 - r^2} \, r \, dr = \frac{8}{3} via the substitution u = 4 - r^2, yielding V = 2\pi \cdot \frac{8}{3} = \frac{16\pi}{3}. For the volume of a solid region E in three dimensions, the triple integral \iiint_E 1 \, dV computes the enclosed volume. Consider the tetrahedron with vertices at (0,0,0), (1,0,0), (0,1,0), and (0,0,1), bounded by the planes x=0, y=0, z=0, and x + y + z = 1. Using iterated integrals over the projection D in the xy-plane (the triangle $0 \leq x \leq 1, $0 \leq y \leq 1 - x), the volume is

V = \int_0^1 \int_0^{1-x} \int_0^{1-x-y} dz \, dy \, dx = \int_0^1 \int_0^{1-x} (1 - x - y) \, dy \, dx = \frac{1}{6}.

This result matches the known formula for the volume of a tetrahedron with those vertices./15%3A_Multiple_Integration/15.02%3A_Double_Integrals_over_General_Regions) Multiple integrals also determine centroids and moments of solids. For a solid E with volume V, the centroid coordinates are \bar{x} = \frac{1}{V} \iiint_E x \, dV, \bar{y} = \frac{1}{V} \iiint_E y \, dV, and \bar{z} = \frac{1}{V} \iiint_E z \, dV. For the tetrahedron above, symmetry implies \bar{x} = \bar{y} = \bar{z}, and evaluating \bar{x} = \frac{1}{V} \int_0^1 \int_0^{1-x} \int_0^{1-x-y} x \, dz \, dy \, dx = \frac{1}{4}, so the centroid is at (\left( \frac{1}{4}, \frac{1}{4}, \frac{1}{4} \right)./15%3A_Multiple_Integration/15.06%3A_Calculating_Centers_of_Mass_and_Moments_of_Inertia)/15%3A_Multiple_Integration/15.02%3A_Double_Integrals_over_General_Regions) Surface area computations use double integrals for the graph of z = f(x,y) over D, given by \iint_D \sqrt{1 + f_x^2 + f_y^2} \, dA, which accounts for the metric induced by the surface parameterization. This formula arises from the arc length analogy extended to surfaces./13%3A_Multiple_Integration/13.05%3A_Surface_Area)

Advanced Concepts

Improper Multiple Integrals

Improper multiple integrals arise when the domain of integration is unbounded or the integrand has singularities within the domain, extending the concept of proper multiple integrals by taking limits of integrals over bounded approximating regions. Formally, for an unbounded domain D \subset \mathbb{R}^n and a function f: D \to \mathbb{R}, the improper integral is defined as \int_D f \, dV = \lim_{k \to \infty} \int_{D_k} f \, dV, where \{D_k\} is an increasing sequence of bounded domains such that D_k \uparrow D. Similarly, for singularities, the integral excludes a neighborhood of the discontinuity and takes the limit as the neighborhood shrinks to the point. The integral converges if this limit exists and is finite; otherwise, it diverges.^[47] A key property is absolute convergence: the improper multiple integral \int_D f \, dV converges absolutely if \int_D |f| \, dV < \infty. Absolute convergence implies ordinary convergence and ensures the value is independent of the choice of approximating sequence \{D_k\}, provided the sequence exhausts D properly. This criterion is particularly useful for testing convergence, as it reduces the problem to the positive integrand |f|, where comparison tests or other methods from single-variable calculus can often be applied after suitable transformations.^[47] A classic example is the Gaussian integral over the plane, \iint_{\mathbb{R}^2} e^{-(x^2 + y^2)} \, dA = \pi, which converges due to the rapid decay of the integrand at infinity. This improper double integral can be evaluated using iterated limits: \left( \int_{-\infty}^{\infty} e^{-x^2} \, dx \right)^2 = \pi, where each one-dimensional integral equals \sqrt{\pi}. The rotation invariance of the integrand suggests using polar coordinates, transforming the integral to \int_0^{2\pi} \int_0^{\infty} e^{-r^2} r \, dr \, d\theta = 2\pi \cdot \frac{1}{2} = \pi, confirming the result and illustrating how symmetry aids computation over unbounded domains.^[48] In higher dimensions, improper multiple integrals exhibit a crucial difference from their one-dimensional counterparts: conditional convergence does not occur. If the integral converges, it must converge absolutely; there are no cases where \int_D f \, dV exists finitely while \int_D |f| \, dV = \infty. This stems from the nature of the exhaustion process in multiple variables, where the positive and negative contributions cannot cancel in a way that allows conditional convergence without absolute integrability, avoiding pathologies like those in alternating series or one-dimensional oscillatory integrals.^[49]

Fubini's Theorem and Iterated vs. Multiple Integrals

Fubini's theorem establishes the equivalence between multiple integrals and iterated integrals under suitable conditions in measure theory. The multiple integral of a measurable function f: X \times Y \to \mathbb{R} is defined intrinsically as the Lebesgue integral with respect to the product measure \mu \times \nu on the product \sigma-algebra of \sigma-finite measure spaces (X, \mathcal{A}, \mu) and (Y, \mathcal{B}, \nu). In contrast, the iterated integral computes f by successive integration: first with respect to one variable, then the other, serving as a practical computational tool rather than a fundamental definition.^[50] Fubini's theorem asserts that if f is \mu \times \nu-integrable, meaning \int_{X \times Y} |f| \, d(\mu \times \nu) < \infty, then the iterated integrals exist, are finite and equal to each other, and coincide with the multiple integral:

\int_{X \times Y} f \, d(\mu \times \nu) = \int_X \left( \int_Y f(x,y) \, d\nu(y) \right) d\mu(x) = \int_Y \left( \int_X f(x,y) \, d\mu(x) \right) d\nu(y).

This holds for \sigma-finite measures, ensuring the product measure is well-defined and the slicing functions are measurable. The absolute integrability condition is crucial, as it implies the inner integrals are well-behaved.^[50]^[51] For non-negative measurable functions f \geq 0, Tonelli's theorem relaxes the integrability requirement: the iterated integrals always exist (possibly infinite) and equal the multiple integral, regardless of whether \int |f| < \infty. This follows from the monotone convergence theorem applied to simple functions approximating f, allowing interchange without absolute convergence. Tonelli's result is particularly useful for positive integrands in applications like probability, where expectations of non-negative random variables can be computed iteratively.^[50] When the absolute integrability condition fails, the iterated integrals may differ, one may diverge while the other converges, or both may fail to match the multiple integral (which may not exist). A standard counterexample is f(x,y) = \frac{x - y}{(x + y)^3} over the unit square [0,1] \times [0,1] with Lebesgue measure, where \int_0^1 \left( \int_0^1 f(x,y) \, dy \right) dx = 0 but \int_0^1 \left( \int_0^1 f(x,y) \, dx \right) dy = +\infty, showing f is not Lebesgue integrable since \int \int |f| \, dy \, dx = \infty. This discontinuity along x = y causes the pathology, violating Fubini's hypothesis. In the context of Riemann integrals, Fubini's theorem requires stricter conditions, such as continuity of f on a compact rectangle, to ensure equivalence; otherwise, counterexamples exist where Riemann iterated integrals differ despite the function being bounded. The Lebesgue framework succeeds more broadly for integrable functions, as measurability and absolute integrability suffice, encompassing cases where Riemann integration fails due to discontinuities of positive measure. For instance, Lebesgue integration allows Fubini to apply to characteristic functions of non-rectifiable sets, where Riemann definitions break down.^[52]

Applications

In Geometry and Physics

In geometry, multiple integrals are essential for computing moments of inertia of solid bodies, which quantify resistance to rotational acceleration about a specified axis. For a solid body with density function \rho(x, y, z), the moment of inertia I_z about the z-axis is given by the triple integral

I_z = \iiint_V (x^2 + y^2) \rho(x, y, z) \, dV,

where V is the volume of the body; this formula arises from integrating the squared perpendicular distance from the axis over the mass distribution.^[24] A classic example is the moment of inertia of a uniform disk of radius R and mass M about its central axis perpendicular to the plane. Using polar coordinates for the double integral over the disk's area, with constant density \rho = M / (\pi R^2), the computation yields

I_z = \iint_D r^2 \rho \, r \, dr \, d\theta = \rho \int_0^{2\pi} d\theta \int_0^R r^3 \, dr = 2\pi \rho \cdot \frac{R^4}{4} = \frac{1}{2} M R^2,

demonstrating how multiple integrals simplify such geometric calculations.^[53] In physics, multiple integrals determine key quantities like the center of mass for extended objects under gravitational or other fields. The position vector of the center of mass \bar{\mathbf{r}} for a body of total mass M = \iiint_V \rho \, dV is

\bar{\mathbf{r}} = \frac{1}{M} \iiint_V \mathbf{r} \, \rho(\mathbf{r}) \, dV,

where \mathbf{r} = (x, y, z) is the position vector; the components follow similarly as \bar{x} = (1/M) \iiint_V x \rho \, dV, and so on.^[54] Another fundamental application is the electrostatic potential \phi(\mathbf{r}) at a point \mathbf{r} due to a volume charge distribution \rho(\mathbf{r}'), expressed as the triple integral

\phi(\mathbf{r}) = \frac{1}{4\pi \epsilon_0} \iiint_{V'} \frac{\rho(\mathbf{r}')}{|\mathbf{r} - \mathbf{r}'|} \, dV',

which integrates the contributions from infinitesimal charge elements across the volume V'; this formula underpins the superposition principle in electrostatics.^[55] Multiple integrals also play a central role in fluid dynamics, particularly for flux calculations like mass flow rates through surfaces. The mass flow rate through a surface S is the surface integral \iint_S \rho \mathbf{v} \cdot \mathbf{n} \, dS, where \rho is fluid density, \mathbf{v} is velocity, and \mathbf{n} is the outward unit normal; this represents the net rate of mass transport across S.^[56] By the divergence theorem, this surface integral equals the volume integral \iiint_V \nabla \cdot (\rho \mathbf{v}) \, dV over the enclosed volume V, linking local divergence of the mass flux to global flow behavior and deriving the continuity equation for mass conservation in fluids.^[57] In the context of work done by vector fields, multiple integrals exploit linearity properties to compute total work via superposition of contributions over regions. For conservative fields, such as gravitational or electrostatic forces, the work along a path decomposes into volume integrals of field potentials, allowing efficient evaluation without path tracing; this linearity ensures that the total work is the sum of integrals over disjoint subregions or charge distributions.^[58]

In Probability and Statistics

In probability theory, multiple integrals play a central role in defining and computing probabilities for continuous multivariate random variables. For two continuous random variables X and Y with joint probability density function (PDF) f_{X,Y}(x,y), the probability that (X,Y) lies in a region A \subseteq \mathbb{R}^2 is given by the double integral P((X,Y) \in A) = \iint_A f_{X,Y}(x,y) \, dx \, dy.^[59] This formulation extends the univariate case, where probabilities are areas under the density curve, to joint distributions where they represent volumes under the surface defined by the joint PDF. The joint PDF must satisfy f_{X,Y}(x,y) \geq 0 for all x,y and normalize to total probability 1, i.e., \iint_{\mathbb{R}^2} f_{X,Y}(x,y) \, dx \, dy = 1.^[60] Marginal and conditional densities are derived from the joint PDF using multiple integrals. The marginal PDF of X is obtained by integrating out Y: f_X(x) = \int_{-\infty}^{\infty} f_{X,Y}(x,y) \, dy, and similarly for f_Y(y).^[60] This process, often facilitated by Fubini's theorem for iterated integrals, yields the univariate distribution of one variable regardless of the other. A prominent example is the bivariate normal distribution, where the joint PDF is f_{X,Y}(x,y) = \frac{1}{2\pi \sigma_X \sigma_Y \sqrt{1-\rho^2}} \exp\left( -\frac{1}{2(1-\rho^2)} \left[ \frac{(x-\mu_X)^2}{\sigma_X^2} + \frac{(y-\mu_Y)^2}{\sigma_Y^2} - 2\rho \frac{(x-\mu_X)(y-\mu_Y)}{\sigma_X \sigma_Y} \right] \right); integrating over y produces the marginal f_X(x), which is normal with mean \mu_X and variance \sigma_X^2, independent of the correlation \rho.^[61] Multiple integrals also compute expectations and related quantities for functions of joint random variables. For a measurable function g: \mathbb{R}^2 \to \mathbb{R}, the expectation is E[g(X,Y)] = \iint_{\mathbb{R}^2} g(x,y) f_{X,Y}(x,y) \, dx \, dy, provided the integral exists.^[60] This includes moments like the covariance, which measures linear dependence: \text{Cov}(X,Y) = E[(X - \mu_X)(Y - \mu_Y)] = \iint_{\mathbb{R}^2} (x - \mu_X)(y - \mu_Y) f_{X,Y}(x,y) \, dx \, dy = E[XY] - \mu_X \mu_Y.^[59] Transformations of random variables often require change of variables in multiple integrals, incorporating the Jacobian determinant to preserve probabilities. If U = g_1(X,Y) and V = g_2(X,Y) with differentiable inverses X = h_1(U,V), Y = h_2(U,V), the joint PDF of (U,V) is f_{U,V}(u,v) = f_{X,Y}(h_1(u,v), h_2(u,v)) \cdot |J|, where J = \det \begin{pmatrix} \frac{\partial h_1}{\partial u} & \frac{\partial h_1}{\partial v} \\ \frac{\partial h_2}{\partial u} & \frac{\partial h_2}{\partial v} \end{pmatrix} is the Jacobian of the inverse transformation.^[62] This technique is essential for deriving distributions of functions of joint variables, such as sums or ratios in statistical models.

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1. Absolute convergence of the integrals defined by thefirst method. Before attacking the main problem we must prove some preliminary theorems. When the proof ...
[48]
[PDF] An Introduction to Measure Theory - Terry Tao
tant example of a measure constructed by Carathéodory's theorem is given ... Theorem 1.7.21 (Fubini's theorem). Let (X, BX,µX) and (Y, BY ,µY ) be ...
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[PDF] Product Measure and Fubini's Theorem - MIT OpenCourseWare
Fubini's theorem holds under two different sets of conditions: (a) nonnega- tive functions g (compare with the MCT); (b) functions g whose absolute value has a ...
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[PDF] 8 # » » » » ¹ - LSU Math
We remark that a Fubini theorem for the Riemann integral is much less general and more cumbersome in its statement than is the case for the Lebesgue integral.
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15.4 Applications of Double Integrals - Courses
Compute the moment about the x-axis Ix, the moment about the y-axis Iy, and the moment about the origin I0 for a uniform density disk ρ(x,y)=ρ of radius a ...
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Triple integrals in physics
The location of the center-of-mass is as (ˉx,ˉy,ˉz), where ˉx=1M∭Sxρ(x,y,z)dV,ˉy=1M∭Syρ(x,y,z)dV,ˉz=1M∭Szρ(x,y,z)dV.
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[PDF] Chapter 4 The Electric Potential
In the last expression we are using the charge density ρ(r) of the distribution to get the element of charge dq for the volume element dτ. 4.1.7 Potential ...
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[PDF] 11–Applications of the Divergence Theorem - UC Davis Math
Conclude: The continuity equation implies that the total rate of change of mass inside any volume is equal to minus the flux of mass out through the boundary, a ...
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[PDF] Lecture 7 Gauss' and Stokes' Theorems
“The net rate of mass flow of fluid out of the control volume must equal the rate of decrease of the mass of fluid within the control volume" q. S d. Control ...
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Calculus III - Conservative Vector Fields - Pauls Online Math Notes
Nov 16, 2022 · In this section we will take a more detailed look at conservative vector fields than we've done in previous sections.
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Double integrals in probability
Example: Suppose that two numbers are chosen uniformly from the interval [0,1]. We call the larger of the two numbers X and the smaller one Y. Their joint pdf ...
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[PDF] Multiple Random Variables
Definition 4.1.3 A function f(x, y) from R2 into R is called a joint probability density function or joint pdf of the continuous bivariate random vector (X ...
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[PDF] INTRODUCTION TO STOCHASTIC PROCESSES. BIVARIATE ...
If u = g(x) and u + ∆u = g(x + ∆x), the shaded areas in this figure represent the same probability. Thus, in the limit ∆x → 0, we must have fU (u)∆u ...
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23.1 - Change-of-Variables Technique | STAT 414
This pdf is known as the double exponential or Laplace pdf. « PreviousLesson 23: Transformations of Two Random Variables · Next23.2 - Beta Distribution ...<|control11|><|separator|>