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Volume element

In multivariable calculus, a volume element, denoted as dV, is an infinitesimally small portion of three-dimensional space that serves as the fundamental building block for evaluating triple integrals over a region. It represents the product of infinitesimal increments along each coordinate direction, allowing the summation of a scalar function—such as density or a constant—to compute totals like volume, mass, or charge. In Cartesian coordinates, the volume element is straightforwardly expressed as dV = [dx](/page/DX) \, [dy](/page/DY) \, [dz](/page/DZ), where x, y, and z are the orthogonal axes.^[1] When changing to curvilinear coordinate systems for regions with symmetry, the volume element transforms according to the absolute value of the Jacobian determinant of the coordinate transformation, ensuring the integral remains invariant. For cylindrical coordinates (r, \theta, z), this yields dV = r \, dr \, d\theta \, dz, accounting for the radial scaling. In spherical coordinates (\rho, \phi, \theta), it becomes dV = \rho^2 \sin \phi \, d\rho \, d\phi \, d\theta, incorporating both radial and angular factors. These adjusted forms facilitate computations in problems involving cylindrical or spherical geometries, such as calculating the volume of a cylinder or the mass of a uniform sphere.^[2]^[3] Volume elements underpin numerous applications across mathematics and physics, including the derivation of theorems like the divergence theorem, which relates surface fluxes to volume integrals of divergence. They are also critical in numerical methods, such as finite volume schemes for solving partial differential equations, where discrete approximations of dV conserve quantities like mass or momentum. By partitioning space into these elements, triple integrals provide a rigorous framework for modeling continuous distributions in three dimensions.^[4]^[3]

Volume Elements in Euclidean Space

In Cartesian Coordinates

In three-dimensional Euclidean space, the volume element in Cartesian coordinates provides the foundational measure for integrating over volumes. It is defined as the infinitesimal volume of a rectangular parallelepiped spanned by the orthogonal differentials dx, dy, and dz along the x-, y-, and z-axes, respectively. This small box-like element approximates the local volume contribution in a region, enabling the summation of such contributions to compute total volumes through limits of Riemann sums.^[5] The volume element is denoted mathematically as dV = dx \, dy \, dz, where the product directly reflects the orthogonality of the coordinate axes, ensuring that the basis vectors are mutually perpendicular and of unit length in the standard metric. This simplicity allows for straightforward additivity in Riemann sums, as the volume elements align perfectly with the grid formed by partitioning the space into rectangular cells without distortion or scaling. In differential form notation, it can also be expressed as dV = dx \wedge dy \wedge dz to emphasize its antisymmetric, oriented nature.^[6] The concept of such infinitesimal volume elements emerged in the late 17th century as part of the development of calculus by Isaac Newton and Gottfried Wilhelm Leibniz, who laid the groundwork for multiple integrals to handle areas, volumes, and physical quantities in higher dimensions.^[7] A representative example is the computation of the volume of a rectangular box with side lengths a, b, and c. The triple integral over the region $0 \leq x \leq a, $0 \leq y \leq b, $0 \leq z \leq c is \iiint dV = \int_0^c \int_0^b \int_0^a dx \, dy \, dz = abc, directly yielding the enclosed volume through iterated single integrals.

In Curvilinear Coordinate Systems

In Euclidean space, the volume element in curvilinear coordinates arises from a change of variables from the standard Cartesian coordinates (x, y, z) to new coordinates (u, v, w), where the position is given by \mathbf{r}(u, v, w) = x(u, v, w) \mathbf{i} + y(u, v, w) \mathbf{j} + z(u, v, w) \mathbf{k}. The infinitesimal volume element dV is the absolute value of the scalar triple product of the partial derivatives, dV = \left| \frac{\partial \mathbf{r}}{\partial u} \cdot \left( \frac{\partial \mathbf{r}}{\partial v} \times \frac{\partial \mathbf{r}}{\partial w} \right) \right| du \, dv \, dw, which equals \left| \det \left( \frac{\partial (x, y, z)}{\partial (u, v, w)} \right) \right| du \, dv \, dw.^[8]^[9] This scaling factor, often denoted as the Jacobian determinant J, accounts for the distortion of the coordinate grid relative to the uniform Cartesian grid, where J = 1.^[10] For common orthogonal curvilinear systems, the Jacobian can be computed explicitly from the transformation equations. In cylindrical coordinates (\rho, \theta, z), defined by x = \rho \cos \theta, y = \rho \sin \theta, z = z, the Jacobian matrix is

\begin{vmatrix} \cos \theta & -\rho \sin \theta & 0 \\ \sin \theta & \rho \cos \theta & 0 \\ 0 & 0 & 1 \end{vmatrix},

with determinant J = \rho (\cos^2 \theta + \sin^2 \theta) = \rho, yielding the volume element dV = \rho \, d\rho \, d\theta \, dz.^[11]^[12] Similarly, in spherical coordinates (\rho, \phi, \theta), where \rho is the radial distance, \phi is the polar angle, and \theta is the azimuthal angle, the transformation is x = \rho \sin \phi \cos \theta, y = \rho \sin \phi \sin \theta, z = \rho \cos \phi. The Jacobian matrix is

\begin{vmatrix} \sin \phi \cos \theta & \rho \cos \phi \cos \theta & -\rho \sin \phi \sin \theta \\ \sin \phi \sin \theta & \rho \cos \phi \sin \theta & \rho \sin \phi \cos \theta \\ \cos \phi & -\rho \sin \phi & 0 \end{vmatrix},

and its determinant is J = \rho^2 \sin \phi, so dV = \rho^2 \sin \phi \, d\rho \, d\phi \, d\theta.^[13]^[14] The \sin \phi factor emerges from the angular components of the partial derivatives, reflecting the varying "density" of the coordinate lines near the poles. The invariance of the volume element under coordinate transformations is ensured by the change of variables theorem for multiple integrals, which states that for a continuously differentiable transformation with non-vanishing Jacobian, the triple integral \iiint_R f(x, y, z) \, dV = \iiint_S f(g(u, v, w), h(u, v, w), k(u, v, w)) \left| \det \left( \frac{\partial (x, y, z)}{\partial (u, v, w)} \right) \right| du \, dv \, dw, where R and S are the regions in Cartesian and curvilinear coordinates, respectively.^[10]^[15] This theorem guarantees that the integral of any integrable function f over a fixed region remains the same regardless of the coordinate system, as the Jacobian compensates exactly for the local volume scaling.^[16] A representative example is the computation of the volume of a ball of radius R centered at the origin, which is rotationally symmetric and thus simplifies in spherical coordinates. The volume is \iiint dV = \int_0^{2\pi} d\theta \int_0^\pi \sin \phi \, d\phi \int_0^R \rho^2 \, d\rho = 2\pi \cdot 2 \cdot \frac{R^3}{3} = \frac{4}{3} \pi R^3, where the \sin \phi in the Jacobian arises from the transformation's angular scaling, ensuring the integral matches the known Cartesian result without direct computation of the latter.^[13]^[14]

Volume Elements on Linear Subspaces

Construction via Gram Determinant

In a k-dimensional linear subspace of Euclidean n-space (k \leq n), consider a basis consisting of vectors X_1, \dots, X_k \in \mathbb{R}^n. The volume element dV on this subspace, parameterized by coordinates (u_1, \dots, u_k) such that a point is \sum_{i=1}^k u_i X_i, is given by

dV = \sqrt{\det G} \, du_1 \cdots du_k,

where G is the k \times k Gram matrix with entries G_{ij} = X_i \cdot X_j.^[17] To compute this, first form the Gram matrix by calculating the inner products G_{ij} for all i, j = 1, \dots, k, which capture the geometry of the basis vectors in the ambient space. Then, evaluate the determinant \det G, and take its positive square root to obtain the scaling factor \sqrt{\det G}, which measures the k-dimensional "density" or infinitesimal volume adjustment relative to the coordinate differentials.^[18]^[17] This factor \sqrt{\det G} equals the k-dimensional volume of the parallelepiped spanned by the basis vectors X_1, \dots, X_k, providing a direct geometric interpretation: it quantifies how the basis "stretches" the unit k-cube in the parameter space into the actual subspace volume.^[18] When k = n, this reduces to the standard Euclidean volume element using the absolute value of the determinant of the full matrix formed by the vectors.^[17] For example, consider a 2-dimensional subspace (plane) in \mathbb{R}^3 spanned by vectors X_1 = (1, 0, 0) and X_2 = (1, 1, 0). The Gram matrix is

G = \begin{pmatrix} 1 & 1 \\ 1 & 2 \end{pmatrix},

with \det G = 2 - 1 = 1, so \sqrt{\det G} = 1. Thus, the area element is dA = du_1 \, du_2, matching the unit area of the parallelogram formed by these vectors.^[18] The construction is invariant under orthogonal transformations of the ambient \mathbb{R}^n, as such transformations preserve inner products and thus the Gram matrix G itself (if Q is orthogonal, then the Gram matrix of Q X_i is G).^[19]

Geometric Interpretation and Properties

The volume element on a k-dimensional linear subspace V of Euclidean space \mathbb{R}^n provides a geometric measure of the k-dimensional content of infinitesimal parallelepipeds lying within V, capturing the intrinsic "volume" orthogonal to the (n-k)-dimensional complement subspace. This interpretation views the element as quantifying the scaling factor for k-volumes embedded in the ambient n-dimensional structure, ensuring that the measure is independent of the choice of embedding coordinates while respecting the Euclidean metric.^[20] Key properties of the volume element include positive homogeneity, where scaling all basis vectors of a parallelepiped by a positive factor \lambda multiplies the volume by \lambda^k; and additivity, allowing the total volume over disjoint unions of sets within the subspace to be the sum of individual volumes. These properties arise from the multilinearity inherent in the definition via the Gram determinant, making the volume element a positive density for the induced Lebesgue measure on the subspace, with orientation encoded separately through the choice of ordered basis for signed integrations. The volume itself is computed via the Gram determinant of an orthonormalized basis for V.^[20]^[17] The volume element on V exhibits compatibility with the ambient Euclidean volume through orthogonal complements: if \mathbb{R}^n = V \oplus V^\perp is the orthogonal direct sum decomposition, then the n-dimensional Lebesgue measure on sets of the form A \times B with A \subset V and B \subset V^\perp equals the product of the induced k-dimensional Lebesgue measure on A and the (n-k)-dimensional Lebesgue measure on B. This relation underscores how subspace volumes contribute to full-dimensional content via perpendicular slices.^[21] A concrete example occurs for a $1-dimensional [subspace](/page/Subspace) (a line) spanned by a direction vector \mathbf{X} \in \mathbb{R}^nwith [parameter](/page/Parameter)u, where the volume element reduces to the [arc length](/page/Arc_length) element ds = |\mathbf{X}| , du$, measuring infinitesimal lengths along the line in the Euclidean metric.^[22]

Volume Elements on Riemannian Manifolds

Definition Using the Metric Tensor

In an n-dimensional oriented Riemannian manifold (M, g), the volume form \omega_g is defined intrinsically as the unique nowhere-vanishing n-form such that its norm with respect to the metric induced on the space of differential forms satisfies \|\omega_g\| = 1. This provides a canonical measure on the manifold, generalizing the notion of volume from Euclidean space to curved geometries where the metric tensor g dictates the local structure. The volume form ensures that integrals over regions of M yield geometrically meaningful volumes, independent of coordinate choices up to orientation. The metric tensor g = (g_{ij}) at each point encodes the inner product on the tangent space, determining lengths of vectors, angles between them, and thus the geometry of infinitesimal parallelepipeds; its determinant \det g quantifies the local volume distortion relative to the coordinate frame, with \sqrt{|\det g|} serving as the scaling factor for the volume density. In local coordinates (x^1, \dots, x^n), the volume form takes the explicit expression

\omega_g = \sqrt{|\det g|} \, dx^1 \wedge \cdots \wedge dx^n,

where the absolute value accounts for orientation conventions, ensuring positivity for the standard orientation. This coordinate expression arises from the metric's action on the coordinate basis vectors \partial_i, yielding the Gram determinant as the volume of the parallelepiped they span. When restricted to Euclidean space with the standard metric g_{ij} = \delta_{ij}, \det g = 1, so \omega_g = dx^1 \wedge \cdots \wedge dx^n, recovering the familiar Lebesgue volume form without scaling. As a top-degree form, \omega_g is alternating and transforms as a pseudotensor under coordinate changes: it picks up a sign under orientation-reversing diffeomorphisms but remains consistent globally on oriented manifolds. This structure allows the volume form to integrate functions over M to define total volumes, such as \int_M f \, \omega_g for a function f. The intrinsic definition of the volume form via the metric tensor emerged in the 19th and early 20th centuries as part of the development of differential geometry. Bernhard Riemann introduced the foundational concept of a metric in his 1854 habilitation lecture, enabling intrinsic descriptions of geometry including volumes on manifolds. Tullio Levi-Civita later advanced this framework in the 1910s through his work on the covariant derivative and parallel transport, which preserve the metric and underpin volume computations in curved spaces; contributions from the Italian school, including Eugenio Beltrami and Gregorio Ricci-Curbastro, further solidified these ideas.^[23]^[24]

Expression in Local Coordinates

In a local coordinate chart (x^1, \dots, x^n) on a Riemannian manifold (M, g), the volume element is expressed as the volume form dV_g = \sqrt{|\det g_{ij}(x)|} \, dx^1 \wedge \cdots \wedge dx^n, where g_{ij}(x) are the components of the metric tensor in these coordinates.^[25] For a manifold embedded in Euclidean space via a map r: U \to \mathbb{R}^N, the metric components are given by g_{ij} = \frac{\partial r}{\partial x^i} \cdot \frac{\partial r}{\partial x^j}, using the standard dot product in \mathbb{R}^N.^[26] Under a change of coordinates from (x^i) to (y^k), the metric components transform as a covariant tensor: g'_{kl}(y) = \frac{\partial x^i}{\partial y^k} \frac{\partial x^j}{\partial y^l} g_{ij}(x).^[27] This induces a transformation on the determinant such that \det g' = (\det J)^{2} \det g, where J is the Jacobian matrix \partial x / \partial y; combined with the coordinate differentials transforming as dx^1 \wedge \cdots \wedge dx^n = (\det J) \, dy^1 \wedge \cdots \wedge dy^n, the volume form \omega_g = \sqrt{|\det g|} \, dx^1 \wedge \cdots \wedge dx^n becomes \sqrt{|\det g'|} \, dy^1 \wedge \cdots \wedge dy^n = \sqrt{|(\det J)^2 \det g|} \, dy^1 \wedge \cdots \wedge dy^n = |\det J| \sqrt{|\det g|} \, dy^1 \wedge \cdots \wedge dy^n, matching \sqrt{|\det g|} \, |dx^1 \wedge \cdots \wedge dx^n| (taking absolute value for the unsigned volume measure dV_g), ensuring it defines a consistent measure independent of the chart.^[27] When the coordinate basis is orthogonal, the metric tensor g_{ij} is diagonal with entries g_{ii} = h_i^2 (no sum), where the h_i are the scale factors h_i = \sqrt{g_{ii}}.^[28] In this case, \sqrt{|\det g_{ij}|} = h_1 h_2 \cdots h_n, so the volume element simplifies to dV_g = h_1 h_2 \cdots h_n \, du^1 \wedge \cdots \wedge du^n.^[28] The position-dependent nature of g_{ij}(x) in general coordinates implies that the Christoffel symbols, which involve partial derivatives of the metric, are nonzero, signaling departure from flat geometry.^[27] As an example on a flat manifold, consider cylindrical coordinates (r, \theta, z) on \mathbb{R}^3 with the Euclidean metric; here the induced metric is diagonal with g_{rr} = 1, g_{\theta\theta} = r^2, g_{zz} = 1, yielding scale factors h_r = 1, h_\theta = r, h_z = 1 and dV_g = r \, dr \wedge d\theta \wedge dz, matching the standard Euclidean volume element in these coordinates.^[28]

Lower-Dimensional Analogues and Examples

Surface Area Elements

In an n-dimensional Riemannian manifold (M, g), a hypersurface \Sigma is an embedded (n-1)-dimensional submanifold, and its surface area element dA provides the measure for integration over \Sigma. The induced metric g_\Sigma on \Sigma is the pullback g_\Sigma = i^* g via the inclusion map i: \Sigma \to M, restricting the ambient metric to the tangent spaces of \Sigma. In local coordinates u^1, \dots, u^{n-1} on \Sigma, the surface area element is the volume form of g_\Sigma:

dA = \sqrt{|\det g_\Sigma|} \, du^1 \wedge \dots \wedge du^{n-1},

where g_\Sigma = (g_{ij}) with g_{ij} = g(\partial/\partial u^i, \partial/\partial u^j). This formula generalizes the arc length element on curves to higher codimension-1 submanifolds, enabling the computation of areas as \int_\Sigma f \, dA for scalar functions f. The surface area element relates to the ambient volume element \mathrm{vol}_g on M through the unit normal vector field N to \Sigma, which is orthogonal to T_p\Sigma for each p \in \Sigma and chosen to orient \Sigma. Specifically,

dA = i^* (N \lrcorner \mathrm{vol}_g),

where \lrcorner denotes interior product (contraction), ensuring dA inherits the orientation from N and \mathrm{vol}_g. In Euclidean \mathbb{R}^n, this manifests as the magnitude of the cross product of tangent vectors for orientation; for a parametrization X: U \subset \mathbb{R}^{n-1} \to \mathbb{R}^n with partial derivatives \partial X / \partial u^i, the induced metric components are g_{ij} = \langle \partial X / \partial u^i, \partial X / \partial u^j \rangle, yielding dA = \sqrt{|\det(g_{ij})|} \, du^1 \dots du^{n-1}. In three dimensions, this simplifies to dA = \|\partial X / \partial u \times \partial X / \partial v\| \, du \, dv, viewing dA as the infinitesimal "slice" volume perpendicular to N.^[29] For hypersurfaces expressed as graphs, such as \Sigma given by x^n = f(x^1, \dots, x^{n-1}) over an open set in \mathbb{R}^{n-1}, the parametrization is X(x^1, \dots, x^{n-1}) = (x^1, \dots, x^{n-1}, f(x^1, \dots, x^{n-1})). The induced metric becomes g_{ij} = \delta_{ij} + \frac{\partial f}{\partial x^i} \frac{\partial f}{\partial x^j}, so

dA = \sqrt{\det(\delta_{ij} + \partial_i f \, \partial_j f)} \, dx^1 \dots dx^{n-1}.

In the common case of a surface z = f(x,y) in \mathbb{R}^3,

dA = \sqrt{1 + \left( \frac{\partial f}{\partial x} \right)^2 + \left( \frac{\partial f}{\partial y} \right)^2} \, dx \, dy,

which approximates the Euclidean area dx \, dy for flat graphs (f_x = f_y = 0) and accounts for tilting via the normal tilt factors.^[30]^[29] The surface area element is invariant under reparametrizations of \Sigma, as \sqrt{|\det g_\Sigma|} transforms as a density under coordinate changes, preserving the intrinsic geometry defined by the first fundamental form (the matrix of g_\Sigma). This invariance ties indirectly to the second fundamental form II, which measures extrinsic bending via the normal's variation II(X,Y) = \langle \nabla_X Y, N \rangle for tangent vectors X,Y, but dA depends solely on the induced metric without altering under isometries of M.^[29] Surface area elements appear in the divergence theorem on oriented Riemannian manifolds, stating that for a compact domain \Omega \subset M with boundary \partial \Omega = \Sigma,

\int_\Omega (\mathrm{div}_g X) \, \mathrm{vol}_g = \int_\Sigma \langle X, N \rangle_g \, dA,

relating volume integrals of divergence to oriented surface fluxes, with dA providing the boundary measure.

Volume Element on the Sphere

The 2-sphere S^2 of radius r, embedded in \mathbb{R}^3, serves as a fundamental example of a compact Riemannian manifold where the volume element, or area form, can be explicitly computed using the induced metric from the ambient Euclidean space. The standard parametrization employs spherical coordinates: \theta \in [0, \pi] as the colatitude (polar angle from the positive z-axis) and \phi \in [0, 2\pi) as the longitude (azimuthal angle). The embedding map is given by

\mathbf{x}(\theta, \phi) = (r \sin \theta \cos \phi, r \sin \theta \sin \phi, r \cos \theta).

This parametrization covers the entire sphere except for a set of measure zero (the poles and a meridian).^[31] The induced metric on S^2 arises from the pullback of the Euclidean metric ds^2 = dx^2 + dy^2 + dz^2 in \mathbb{R}^3 via the embedding map \iota: S^2 \to \mathbb{R}^3. To derive it, compute the partial derivatives of the position vector:

\frac{\partial \mathbf{x}}{\partial \theta} = r (\cos \theta \cos \phi, \cos \theta \sin \phi, -\sin \theta), \quad \frac{\partial \mathbf{x}}{\partial \phi} = r (-\sin \theta \sin \phi, \sin \theta \cos \phi, 0).

The metric tensor components are the inner products of these tangent vectors:

g_{\theta\theta} = \left\langle \frac{\partial \mathbf{x}}{\partial \theta}, \frac{\partial \mathbf{x}}{\partial \theta} \right\rangle = r^2, \quad g_{\phi\phi} = \left\langle \frac{\partial \mathbf{x}}{\partial \phi}, \frac{\partial \mathbf{x}}{\partial \phi} \right\rangle = r^2 \sin^2 \theta, \quad g_{\theta\phi} = g_{\phi\theta} = 0.

Thus, the line element is

ds^2 = r^2 \, d\theta^2 + r^2 \sin^2 \theta \, d\phi^2.

This diagonal metric tensor g = \operatorname{diag}(r^2, r^2 \sin^2 \theta) describes the geometry of the sphere.^[31]^[32] The volume element, or area form dA, on the Riemannian manifold S^2 is \sqrt{\det g} \, d\theta \wedge d\phi. The determinant is \det g = r^4 \sin^2 \theta, so \sqrt{\det g} = r^2 |\sin \theta|. Since \sin \theta \geq 0 for \theta \in [0, \pi], the area element simplifies to

dA = r^2 \sin \theta \, d\theta \, d\phi.

This form measures infinitesimal areas on the sphere's surface. To verify its consistency, integrate over the entire manifold:

\int_{S^2} dA = \int_0^{2\pi} \int_0^\pi r^2 \sin \theta \, d\theta \, d\phi = r^2 \cdot 2\pi \cdot \left[ -\cos \theta \right]_0^\pi = 4\pi r^2,

recovering the well-known surface area of the sphere.^[31]^[33] This construction generalizes to the n-sphere S^n of radius r embedded in \mathbb{R}^{n+1}, where hyperspherical coordinates involve angles \theta_1, \dots, \theta_n with \theta_1 \in [0, \pi] (colatitude), \theta_2, \dots, \theta_{n-1} \in [0, \pi], and \phi \in [0, 2\pi). The induced volume element is

dV = r^n \sin^{n-1} \theta_1 \sin^{n-2} \theta_2 \cdots \sin \theta_{n-1} \, d\theta_1 \, d\theta_2 \cdots d\theta_{n-1} \, d\phi,

reflecting the successive angular dependencies from the embedding. Integrating this yields the total surface area S_n(r) = \frac{2 \pi^{(n+1)/2} r^n}{\Gamma((n+1)/2)}.^[34]^[35]

Other Examples: Torus and Hyperboloid

The torus provides another example of a surface embedded in Euclidean space where the volume element can be computed from the induced metric. Consider the standard parametrization of a torus of major radius R and minor radius r (with R > r > 0) given by

\mathbf{r}(u, v) = \big( (R + r \cos v) \cos u, \, (R + r \cos v) \sin u, \, r \sin v \big),

where u, v \in [0, 2\pi). The induced metric tensor from \mathbb{R}^3 yields the line element

ds^2 = (R + r \cos v)^2 \, du^2 + r^2 \, dv^2. $$ [](https://mathworld.wolfram.com/Torus.html) The corresponding volume element (surface area element) is $\sqrt{\det g} \, du \, dv = r (R + r \cos v) \, du \, dv$, where $g$ is the metric tensor with diagonal entries $g_{uu} = (R + r \cos v)^2$ and $g_{vv} = r^2$. This element reflects the varying geometry along the toroidal direction, with the factor $(R + r \cos v)$ accounting for the distance from the axis of revolution. Integrating over the parameter domain gives the total surface area

A = \int_0^{2\pi} \int_0^{2\pi} r (R + r \cos v) , du , dv = 4 \pi^2 R r, $$ ^[36] which arises because the inner integral over u yields $2\pi, and the outer over v averages the cosine term to zero. In contrast, the hyperboloid model illustrates a volume element on a surface of constant negative curvature embedded in Minkowski space. The two-dimensional hyperbolic plane H^2 is realized as the upper sheet of the hyperboloid \{ (x, y, z) \in \mathbb{R}^{2,1} \mid x^2 + y^2 - z^2 = -1, \, z > 0 \}, parametrized by

\mathbf{r}(u, v) = \big( \sinh u \cos v, \, \sinh u \sin v, \, \cosh u \big),

with u \geq 0 and v \in [0, 2\pi). The induced metric from the Lorentzian form ds^2 = dx^2 + dy^2 - dz^2 is

ds^2 = du^2 + \sinh^2 u \, dv^2. $$ [](http://math.uchicago.edu/~may/REU2012/REUPapers/Dunn-Weiss.pdf) The volume element is then $\sqrt{\det g} \, du \, dv = \sinh u \, du \, dv$, where the metric tensor has diagonal entries $g_{uu} = 1$ and $g_{vv} = \sinh^2 u$. This form highlights the exponential growth in the circumferential direction as $u$ increases, characteristic of hyperbolic geometry. Unlike the torus, where the Gaussian curvature varies (positive in outer regions and negative in inner ones), the hyperboloid has constant Gaussian curvature $K = -1$, providing a benchmark for negative curvature spaces.[](http://math.uchicago.edu/~may/REU2012/REUPapers/Dunn-Weiss.pdf) These examples extend the positive curvature case of [the sphere](/page/The_Sphere) by demonstrating how volume elements adapt to zero-mean curvature variations on the [torus](/page/Torus) and uniform negative curvature on the [hyperboloid](/page/Hyperboloid), influencing applications in [geometry](/page/Geometry) and physics such as flux computations or [orbital mechanics](/page/Orbital_mechanics).[](https://mathworld.wolfram.com/Torus.html)[](http://math.uchicago.edu/~may/REU2012/REUPapers/Dunn-Weiss.pdf)

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Previous editions of this book have been adopted as a textbook at over 375 universities and colleges and have been translated into three languages. Axler has ...
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Determinants and Volumes
The absolute value of a matrix's determinant equals the volume of the parallelepiped it defines. The determinant can also be a signed volume.Missing: element | Show results with:element
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275A, Notes 2: Product measures and independence - Terry Tao
Oct 12, 2015 · Product measures —. It is intuitively obvious that Lebesgue measure {m^2} on {{\bf R}^2} ought to be related to Lebesgue measure {m} on {{\bf ...
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Arc Length - Calculus II - Pauls Online Math Notes
Nov 16, 2022 · In this section we are going to look at computing the arc length of a function. Because it's easy enough to derive the formulas that we'll use in this section ...Arc Length with Vector Functions · Paul's Online Notes · Surface Area
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[PDF] Introduction to RIEMANNIAN GEOMETRY - Mathematics
The ideas of Riemann were taken up by the Italian geometers Enrico Betti and Francesco Brioschi, and subsequently by Eugenio. Beltrami (student of Brioschi), ...
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Schouten, Levi-Civita and the notion of parallelism in Riemannian ...
As is well known, Levi-Civita introduced his generalization of parallelism for contravariant vectors by considering the (local) embedding of a Riemannian ...
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[PDF] Geometry Primer
Recall that the metric on an oriented Riemannian manifold (M,g) determines a volume form given in local coordinates by dvg = pdet gij dx1 ∧ dx2 ∧···∧ dxn.
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[PDF] Chapter 4. The First Fundamental Form (Induced Metric)
Just as the Euclidean dot product contains all geometric information about IR3, the induced metric contains all geometric information about M, as we shall see.Missing: embedding partial
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[PDF] Lectures on Differential Geometry Math 240C
Jun 6, 2011 · This connection is called the Levi-Civita connection of the Riemannian manifold. (M, h·, ·i). To prove the theorem we express the two ...
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Orthogonal Curvilinear Coordinates - Particle in Cell Consulting, LLC
Jan 16, 2012 · The h1=|∂r/∂u1| h 1 = | ∂ r / ∂ u 1 | term is called a scale factor. It relates the actual displacement in a given coordinate direction to the ...Missing: riemannian element
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None
Below is a merged summary of the sections on Surface Area Element, Induced Metric, Parametrized Surfaces, and Graph Surfaces from *Differential Geometry of Curves and Surfaces* by Do Carmo. To retain all information in a dense and organized manner, I will use a combination of narrative text and a table in CSV format for detailed references and key formulas. The narrative provides an overview, while the table captures specific details such as definitions, formulas, page references, and examples across all provided segments.
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[PDF] Math 396. Metric tensor on hypersurfaces 1. Motivation Let U ⊆ R n ...
This is the most general formula for the induced metric tensor on a surface of revolution about the x-axis when the surface admits a parameterization by polar ...Missing: dot | Show results with:dot
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[PDF] lecture 2: the riemannian metric
Let M = S2 be the unit 2-sphere in R3. The induced Riemannian metric g (from the canonical Riemannian metric g0 on R3) is known as the round metric. To ...
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### Summary of Induced Metric on Embedded Surface in R³ (Sphere)
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Spherical Coordinates -- from Wolfram MathWorld
A system of curvilinear coordinates that are natural for describing positions on a sphere or spheroid.
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[PDF] Volumes of n-balls in R 1. Folland & Jones 10:29 Use “double p
Apr 12, 2021 · Then compute the Gram matrix) Show that the 3-D volume of S3 is 2π2 (and if our 3-sphere has radius R then it has 3-D volume 2π2R3). 4. Find a ...
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Hypersphere -- from Wolfram MathWorld
The n-hypersphere (often simply called the n-sphere) is a generalization of the circle (called by geometers the 2-sphere) and usual sphere (called by geometers ...
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Torus -- from Wolfram MathWorld
1/4pi^2(R+r)(R-r)^2. (14). The volume can also be found by integrating the Jacobian computed from the parametric equations ...
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[PDF] introduction to geometry - UChicago Math
Aug 24, 2012 · We will introduce all of the models, and compute the metric on the hyperboloid model. The relationship between these three models, as well as ...