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Generalized Stokes theorem

The Generalized Stokes' theorem, also known as Stokes' theorem on manifolds, is a central result in differential geometry and topology that equates the integral of the exterior derivative of a differential (k-1)-form over an oriented k-dimensional manifold with boundary to the integral of the form itself over the oriented boundary of the manifold.^[1]^[2] In precise terms, for a compact oriented smooth k-dimensional manifold with boundary M (in \mathbb{R}^n or more generally on an oriented smooth manifold) and a smooth (k-1)-form \omega, the theorem states \int_M d\omega = \int_{\partial M} \omega, where d denotes the exterior derivative and \partial M inherits the induced orientation from M.^[1]^[3] This formulation relies on the theory of differential forms, where a k-form is a multilinear, alternating object that can be integrated over k-dimensional oriented submanifolds, and the exterior derivative d generalizes the gradient, curl, and divergence operators while satisfying d^2 = 0.^[2]^[1] The theorem unifies several classical results in vector calculus as special cases: for k=1, it recovers the fundamental theorem of calculus; for k=2, it includes Green's theorem in the plane and the classical Stokes' theorem in \mathbb{R}^3 relating the surface integral of the curl to the line integral around the boundary; and for k=3, it gives the divergence theorem (or Gauss's theorem).^[1]^[2] This generalization extends these identities from Euclidean spaces to arbitrary smooth oriented manifolds, enabling applications in physics (e.g., electromagnetism via Maxwell's equations), topology (e.g., de Rham cohomology), and geometry.^[3]^[2] Historically, the classical Stokes' theorem emerged in the mid-19th century, with George Gabriel Stokes posing it in a 1854 Cambridge examination question based on ideas from a 1850 letter by William Thomson (Lord Kelvin), though proofs appeared earlier in works by Mikhail Ostrogradsky (1826) and others for related forms like the divergence theorem.^[4] The modern generalized version using differential forms was developed by Élie Cartan, who formalized it in his 1945 book Les systèmes différentiels extérieurs et leurs applications géométriques, building on his earlier introduction of exterior calculus in 1899.^[4]^[3] Earlier precursors include Vito Volterra's 1889 unification of vector calculus theorems and Ostrogradsky's 1836 multivariable extensions.^[4]

Overview

Introduction

The generalized Stokes' theorem is a cornerstone of differential geometry, establishing a profound relationship between the integration of differential forms over oriented manifolds and their boundaries. For an oriented manifold M with boundary \partial M and a smooth (k-1)-form \omega on M, the theorem asserts that

\int_M d\omega = \int_{\partial M} \omega,

where d denotes the exterior derivative. This formulation captures the essence of how local differentiation, via d, manifests globally through boundary integrals, providing a versatile tool for computations on abstract geometric objects beyond Euclidean space.^[5] As a unifying principle, the generalized Stokes' theorem encompasses and generalizes classical results from vector calculus. It recovers the fundamental theorem of calculus in one dimension (relating integrals over intervals to endpoint evaluations), Green's theorem in two dimensions (linking line integrals around regions to double integrals of curls), Stokes' theorem in three dimensions (connecting surface integrals to boundary curves), and the divergence theorem (equating volume integrals of divergences to flux through enclosing surfaces). By recasting these in the coordinate-free language of differential forms, the theorem reveals their common topological and analytic structure, applicable to manifolds of arbitrary dimension.^[5] The theorem's significance extends across mathematics and physics, enabling key advancements in multiple domains. In topology, it forms the basis for de Rham cohomology, which equates smooth invariants derived from closed forms to singular homology groups, facilitating the study of manifold invariants like Betti numbers. In physics, it underpins conservation laws, such as those encoded in Maxwell's equations for electromagnetism, where integrals of field forms over regions relate to boundary fluxes representing physical quantities like charge or current. In analysis, it supports techniques for integration over curved domains and change-of-variables formulas, essential for solving partial differential equations on non-flat spaces. This broad applicability arises from the need to extend consistent integral calculus to higher dimensions and general geometries, bridging local and global properties in a rigorous manner.^[5]^[5]^[5]

Historical Context

The generalized Stokes' theorem traces its roots to the fundamental theorem of calculus, independently developed by Isaac Newton and Gottfried Wilhelm Leibniz in the late 17th century, which relates differentiation and integration for functions on the real line.^[6] This foundational result laid the groundwork for higher-dimensional integral theorems by establishing the principle of relating boundary integrals to interior derivatives. In the 19th century, these ideas extended to two dimensions with George Green's theorem of 1828, which equates a line integral around a closed curve to a double integral over the enclosed region, as presented in his seminal essay on electricity and magnetism.^[7] Further advances in the mid-19th century included the divergence theorem, first formulated by Joseph-Louis Lagrange in 1762 in his paper on the propagation of sound, though without a complete proof, and later independently rediscovered and proved by Carl Friedrich Gauss in 1813 and Mikhail Ostrogradsky in 1831.^[8] George Gabriel Stokes contributed his theorem in 1854, linking surface integrals of curl fields to line integrals over boundaries, initially posed as a Cambridge examination question and later proved by Hermann Hankel in 1861.^[4] These vector calculus theorems—Green's, Stokes', and the divergence theorem—unified partial results from earlier works, with Vito Volterra providing a more general framework in 1889 that anticipated broader integrations.^[4] The modern generalization emerged in the late 19th and early 20th centuries through the theory of differential forms, pioneered by Élie Cartan, who introduced exterior differential forms in 1899 and formulated the unified Stokes' theorem in 1945, encompassing all classical cases on manifolds.^[4] This development built on influences from Bernhard Riemann's 1851 work on complex analysis and Riemann surfaces, which explored multi-dimensional integrals, and Henri Poincaré's introduction of homology theory around 1895, linking topological invariants to integration. In the 1930s, Georges de Rham established the connection to cohomology in his 1931 thesis, showing that closed forms modulo exact forms yield topological information isomorphic to singular cohomology.^[9] Post-World War II refinements by Henri Cartan and collaborators integrated these ideas into sheaf theory and general manifold frameworks, solidifying the theorem's role in differential geometry and topology during the mid-20th century.^[4]

Mathematical Prerequisites

Differential Forms

A differential form on a smooth manifold M is a mathematical object that generalizes the notion of scalars, vectors, and higher-dimensional analogs in a coordinate-independent way, serving as the fundamental integrands in the generalized Stokes' theorem. Specifically, a k-form \omega on M is a smooth section of the k-th exterior power of the cotangent bundle, \Lambda^k T^*M, which can be understood as a smooth, alternating multilinear map from the k-fold product of the tangent space T_pM at each point p \in M to the real numbers \mathbb{R}.^[10] In local coordinates (x^1, \dots, x^n) around a point, \omega takes the expression

\omega = \sum_{i_1 < \cdots < i_k} f_{i_1 \dots i_k} \, dx^{i_1} \wedge \cdots \wedge dx^{i_k},

where the f_{i_1 \dots i_k} are smooth functions and the dx^i are basis 1-forms.^[11] The key properties of differential forms stem from the exterior algebra structure. Antisymmetry ensures that \omega(v_1, \dots, v_k) = -\omega(v_{\sigma(1)}, \dots, v_{\sigma(k)}) for any permutation \sigma with odd sign, making forms skew-symmetric tensors. The wedge product \wedge defines the tensor product in this algebra: for a k-form \alpha and an \ell-form \beta, \alpha \wedge \beta is a (k+\ell)-form satisfying \deg(\alpha \wedge \beta) = \deg(\alpha) + \deg(\beta), with the operation being graded commutative, i.e., \beta \wedge \alpha = (-1)^{k\ell} \alpha \wedge \beta. These properties allow forms to model oriented volumes without reference to a specific basis.^[12] Central to the theory is the exterior derivative operator d: \Omega^k(M) \to \Omega^{k+1}(M), which maps k-forms to (k+1)-forms and captures the notion of differentiation in a coordinate-free manner. It satisfies d^2 = 0, the Poincaré lemma condition for closed forms, and obeys the graded Leibniz rule: for forms \alpha of degree k and \beta,

d(\alpha \wedge \beta) = d\alpha \wedge \beta + (-1)^k \alpha \wedge d\beta.

In local coordinates, for a k-form \omega = \sum_I f_I \, dx^I with multi-index I = (i_1 < \cdots < i_k),

d\omega = \sum_{I,j} \frac{\partial f_I}{\partial x^j} \, dx^j \wedge dx^I,

where the sum is over all indices, and terms with repeated dx's vanish due to antisymmetry. This operator generalizes the gradient, curl, and divergence in vector calculus.^[5] The pullback operation facilitates the transfer of forms between manifolds. For a smooth map f: M \to N between manifolds, the pullback f^*: \Omega^k(N) \to \Omega^k(M) is defined by f^*\omega (v_1, \dots, v_k) = \omega (df_p(v_1), \dots, df_p(v_k)) at p \in M, extending multilinearly. It preserves the wedge product and commutes with the exterior derivative, f^*(d\omega) = d(f^*\omega), enabling change of variables in integration and relating local computations on M to global structures on N.^[11] Examples illustrate these concepts concretely. Zero-forms (k=0) are simply smooth functions f: M \to \mathbb{R}, with df being the 1-form representing the differential. One-forms correspond to covector fields and, via a Riemannian metric, can identify with vector fields; for instance, on \mathbb{R}^3, the 1-form \omega = P\, dx + Q\, dy + R\, dz relates to the vector field (P, Q, R). On an n-dimensional oriented manifold, an n-form is a volume form, locally \mu = g(x) \, dx^1 \wedge \cdots \wedge dx^n, which measures oriented n-dimensional content up to the positive function g.^[12]

Oriented Manifolds with Boundary

A smooth n-dimensional manifold M is a Hausdorff, second-countable topological space that is locally homeomorphic to the Euclidean space \mathbb{R}^n, equipped with a smooth atlas consisting of charts whose transition maps are C^\infty-diffeomorphisms.^[13] This structure ensures that M admits a well-defined notion of differentiability, allowing for the study of tangent spaces and differential forms. Manifolds may be compact, meaning they are closed and bounded in every chart, or non-compact, which permits unbounded or open structures like \mathbb{R}^n itself.^[14] An orientation on a smooth manifold M is a consistent choice of ordered basis for the tangent space T_pM at each point p \in M, such that the choice varies continuously across M.^[15] Equivalently, it corresponds to the existence of a nowhere-vanishing n-form \omega on M defined up to multiplication by a positive smooth function, which selects one equivalence class of bases in the top exterior power \Lambda^n T_p^*M./2.19/orientation.pdf) This can be specified via an oriented atlas, where the Jacobian determinants of transition maps between overlapping charts are positive, ensuring global consistency.^[16] A manifold with boundary, denoted (M, \partial M), is a smooth manifold where each point in the boundary \partial M has a chart mapping to the closed half-space \mathbb{H}^n = \{(x_1, \dots, x_n) \in \mathbb{R}^n \mid x_n \geq 0\}, with \partial M corresponding to the hyperplane x_n = 0.^[14] Here, \partial M itself forms a smooth (n-1)-dimensional manifold without boundary. At boundary points, inward and outward normals are defined relative to the half-space model, pointing towards the interior or exterior of M, respectively. The collar neighborhood theorem guarantees that \partial M admits an open neighborhood in M diffeomorphic to \partial M \times [0,1), providing a tubular structure around the boundary essential for defining integrations and extensions.^[17] The induced orientation on \partial M follows the right-hand rule convention: for an oriented basis of T_pM with outward normal as the last vector, the first n-1 vectors induce the orientation on T_p(\partial M). For example, on the n-ball B^n with the standard orientation, the outward normal pairs with the counterclockwise orientation on the boundary sphere S^{n-1}.^[18] For integration over non-smooth domains, singular chains provide a topological framework, consisting of formal sums of continuous maps from standard simplices into the space, which generate homology groups without requiring smoothness.^[19]

Integration of Forms

The integration of a differential form \omega over a compact oriented n-manifold M without boundary is defined for a top-degree n-form \omega, where the integral \int_M \omega is constructed using an oriented atlas \{(U_i, \psi_i)\} and a partition of unity \{f_i\} subordinate to the atlas.^[20] In each chart (U_i, \psi_i), the pullback \psi_i^* \omega = f(x_1, \dots, x_n) \, dx_1 \wedge \cdots \wedge dx_n, and the integral over the chart domain \psi_i(U_i) \subset \mathbb{R}^n is the standard Lebesgue integral \int_{\psi_i(U_i)} f(x_1, \dots, x_n) \, dx_1 \cdots dx_n, assuming absolute convergence; the global integral is then \int_M \omega = \sum_i \int_{\psi_i(U_i)} f_i \cdot (\psi_i^* \omega).^[20] This construction ensures independence from the choice of atlas and partition of unity, provided the charts respect the orientation of M.^[21] Under a change of coordinates given by an orientation-preserving diffeomorphism \phi: U \to V between open sets in \mathbb{R}^n, the integral transforms consistently via the pullback: \int_U \omega = \int_V \phi^* \omega, where \phi^* \omega = f(\phi(y)) \det(D\phi(y)) \, dy_1 \wedge \cdots \wedge dy_n and \det(D\phi) > 0 to preserve orientation.^[20] This accounts for the Jacobian determinant, ensuring that the integral measures the signed volume scaled by the form's coefficients in the new coordinates.^[22] For a compact oriented n-manifold M with boundary \partial M, which is an oriented (n-1)-manifold, the integral \int_{\partial M} \omega for an (n-1)-form \omega is defined analogously using charts on \partial M, where near the boundary, M is modeled on the half-space \mathbb{H}^n = \{x \in \mathbb{R}^n : x_n \geq 0\}.^[21] The induced orientation on \partial M is such that a basis v_1, \dots, v_{n-1} on \partial M is positive if the outward-pointing normal \nu together with v_1, \dots, v_{n-1} forms a positive basis for T_p M, ensuring consistency between interior and boundary orientations.^[21]^[23] The integration operation satisfies linearity: \int_M (a \omega + b \eta) = a \int_M \omega + b \int_M \eta for scalars a, b and forms \omega, \eta of the same degree, and additivity over disjoint unions: if M = M_1 \sqcup M_2 with compatible orientations, then \int_M \omega = \int_{M_1} \omega + \int_{M_2} \omega.^[21] When \omega is a volume form on an oriented n-manifold M, the integral \int_M \omega computes the signed volume of M with respect to that form.^[22] As an example, consider integrating a 1-form \alpha = P \, dx + Q \, dy over an oriented curve \gamma: [a, b] \to \mathbb{R}^2, parametrized by \gamma(t) = (x(t), y(t)); the integral is \int_\gamma \alpha = \int_a^b \left( P(\gamma(t)) x'(t) + Q(\gamma(t)) y'(t) \right) dt, which represents the line integral of the vector field (P, Q) along \gamma.^[21] Similarly, for a 2-form \omega = R \, dx \wedge dy over an oriented surface S in \mathbb{R}^3 parametrized by \sigma(u, v) = (x(u,v), y(u,v), z(u,v)), the integral is \int_S \omega = \iint_D R(\sigma(u,v)) \det \begin{pmatrix} x_u & x_v \\ y_u & y_v \end{pmatrix} \, du \, dv, where D \subset \mathbb{R}^2 is the parameter domain, corresponding to the flux through S if \omega is induced by a vector field.^[21]

Core Formulation

Statement of the Theorem

The generalized Stokes' theorem asserts that if M is a compact, oriented n-dimensional smooth manifold with boundary \partial M, where \partial M is equipped with the induced orientation from M, and \omega is a smooth (n-1)-form on an open set containing M, then

\int_M d\omega = \int_{\partial M} \omega,

where d denotes the exterior derivative operator on differential forms, distinct from the total differential in other contexts.^[2]^[24] This formulation requires M to be compact to guarantee the existence and finiteness of the integrals, and smooth to allow the use of local coordinate charts for computation.^[2] The theorem holds in this domain for smooth manifolds, though extensions to piecewise smooth or Lipschitz boundaries exist in more advanced settings.^[24] A more general version applies to singular chains in the context of algebraic topology: if c is an oriented (k+1)-dimensional simplicial chain with boundary \partial c, and \omega is a smooth k-form defined on an open set containing the image of c, then

\int_c d\omega = \int_{\partial c} \omega.

^[25] This chain formulation unifies the manifold case as a special instance where c = M.

Proof Outline

The proof of the generalized Stokes' theorem proceeds by reducing the global statement to local integrals over coordinate charts and then patching these using a partition of unity subordinate to an atlas of the manifold.^[2]^[26] This approach leverages the smoothness of the manifold and form to ensure well-defined integrals, while compactness guarantees finiteness.^[27] Locally, consider a boundary chart where a neighborhood in M maps diffeomorphically to an open set in the half-space H^n = \{ (x^1, \dots, x^n) \in \mathbb{R}^n \mid x^n \geq 0 \}, with boundary \partial M corresponding to the hyperplane \{x^n = 0\}.^[26] For a smooth (n-1)-form \omega with compact support in this chart, the exterior derivative in coordinates is d\omega = \sum_I df_I \wedge dx^I, where \omega = \sum_I f_I \, dx^I and the sum runs over increasing multi-indices I of length n-1.^[2] Integrating d\omega over the half-space yields \int_M d\omega = \sum_i (-1)^{i-1} \int \frac{\partial f_I}{\partial x^i} \, dx^1 \wedge \cdots \wedge dx^n (up to sign and permutation factors), which by Fubini's theorem and integration in the x^n-direction reduces to the boundary integral via the fundamental theorem of calculus; there are no contributions from the "face at infinity" due to compact support, matching \int_{\partial M} \omega under the induced orientation.^[26]^[27] Boundary evaluation involves the limit as x^n \to 0^+, preserving the orientation via the chart's Jacobian.^[2] A fundamental lemma establishes the result for the half-space model: for the upper half-space H^n = \{ (x^1, \dots, x^n) \in \mathbb{R}^n \mid x^n \geq 0 \} with boundary the hyperplane \{x^n = 0\}, and compactly supported \omega, Fubini's theorem allows iterated integration, reducing \int_{H^n} d\omega to a 1-dimensional integral over the x^n-direction, where the fundamental theorem of calculus equates it to \int_{\{x^n=0\}} \omega (up to orientation sign).^[26] This holds by smoothness of \omega, ensuring partial derivatives exist and integrals converge.^[27] To globalize, select a finite atlas \{ (U_\alpha, \phi_\alpha) \} covering the compact oriented manifold M with boundary, and a partition of unity \{ \rho_\alpha \} subordinate to \{U_\alpha\}, so \sum_\alpha \rho_\alpha = 1 smoothly.^[2] Then, \int_M d\omega = \sum_\alpha \int_M d(\rho_\alpha \omega) = \sum_\alpha \int_{\partial M} \rho_\alpha \omega, since each \rho_\alpha \omega has support in U_\alpha where the local theorem applies, and orientation consistency across charts ensures boundary integrals align without overlap errors.^[26]^[27] Compactness of M justifies the finite sum and finite integrals, while smoothness of the atlas and form guarantees the existence of all local expressions and derivatives.^[2]

Interpretations and Principles

Underlying Topological Principle

The generalized Stokes theorem is deeply rooted in algebraic topology, particularly through the lens of chain complexes and their associated homology theories. The space of smooth differential forms \Omega^*(M) on an oriented manifold M, equipped with the exterior derivative d: \Omega^k(M) \to \Omega^{k+1}(M), forms a cochain complex satisfying d^2 = 0. This structure mirrors the chain complex of singular chains C_*(M), where the boundary operator \partial: C_k(M) \to C_{k-1}(M) also satisfies \partial^2 = 0. The duality between these complexes is captured by the integration pairing \langle c, \omega \rangle = \int_c \omega for a k-chain c \in C_k(M) and a k-form \omega \in \Omega^k(M), which is compatible with the differentials via the generalized Stokes theorem: \int_{\partial c} \omega = \int_c d\omega.^[28]^[29] This compatibility implies a profound connection to homology. For a closed form \omega (i.e., d\omega = 0), the integral \int_{\partial c} \omega = 0 for any chain c, meaning such integrals vanish over boundaries and thus descend to homology classes in H_k(M). Conversely, the theorem ensures that exact forms (i.e., \omega = d\alpha) integrate to zero over cycles (boundaries of zero). This pairing induces the de Rham homomorphism from de Rham cohomology H^*_{dR}(M) = \ker d / \operatorname{im} d to singular cohomology H^*(M; \mathbb{R}), which the de Rham theorem proves is an isomorphism, linking analytic properties of forms to topological invariants.^[30]^[29] Locally, on contractible open sets U \subset \mathbb{R}^n, the Poincaré lemma asserts that every closed form is exact (d\alpha = \omega if d\omega = 0), rendering the generalized Stokes theorem trivial in such regions since boundaries integrate to zero by direct computation. Globally, however, the theorem reveals non-trivial topology: integrals of closed forms over cycles detect cohomology classes that may not vanish. For instance, on the 2-torus T^2 = S^1 \times S^1, the de Rham cohomology H^1_{dR}(T^2) \cong \mathbb{R} \oplus \mathbb{R} is spanned by classes of the closed 1-forms d\theta_1 and d\theta_2 (angular coordinates), whose periods \int_{\gamma_1} d\theta_1 = 2\pi and \int_{\gamma_2} d\theta_2 = 2\pi over the generating loops \gamma_1, \gamma_2 are non-zero, signaling the two independent 1-dimensional holes.^[28]^[31] Philosophically, the generalized Stokes theorem unifies differential analysis and topology by providing a natural integration pairing between chains and cochains, where the theorem enforces the required compatibility \langle \partial c, \omega \rangle = \langle c, d\omega \rangle. This duality underpins de Rham–Hodge theory, transforming topological questions about cycles and boundaries into analytic problems solvable via forms and their derivatives.^[32]

Relation to de Rham Cohomology

The de Rham complex of a smooth manifold M is the cochain complex

$0 \to \Omega^0(M) \xrightarrow{d} \Omega^1(M) \xrightarrow{d} \cdots \xrightarrow{d} \Omega^n(M) \to 0,

where n = \dim M, \Omega^k(M) is the vector space of smooth k-forms on M, and d is the exterior derivative satisfying d \circ d = 0. The de Rham cohomology groups are the cohomology of this complex: H^k_{\mathrm{dR}}(M) = \ker(d: \Omega^k(M) \to \Omega^{k+1}(M)) / \operatorname{im}(d: \Omega^{k-1}(M) \to \Omega^k(M)).^[30] The de Rham theorem states that there exists a natural isomorphism H^k_{\mathrm{dR}}(M) \cong H^k(M; \mathbb{R}) for each k, where the right-hand side denotes the k-th singular cohomology group of M with real coefficients. This isomorphism arises from the integration pairing that maps a cohomology class [\omega] \in H^k_{\mathrm{dR}}(M), represented by a closed k-form \omega (i.e., d\omega = 0), to the homology class of a k-cycle c via \int_c \omega. The generalized Stokes' theorem guarantees that the pairing descends to cohomology, as it vanishes on boundaries: for any (k+1)-chain \sigma, \int_{\partial \sigma} \omega = \int_\sigma d\omega = 0 when d\omega = 0.^[30]^[33] A standard proof establishes the isomorphism by first verifying that the induced map is a chain map between cochain complexes and well-defined on cohomology via Stokes' theorem. Injectivity follows from the non-degeneracy of the pairing, which pairs de Rham cohomology non-trivially with homology generators. Surjectivity is shown using partitions of unity to locally solve the Poincaré lemma (every closed form is exact on contractible opens) and extend primitives globally, ensuring every cohomology class pairs non-trivially with some form.^[30] Key consequences include the equality of Betti numbers b_k(M) = \dim H_k(M; \mathbb{R}) = \dim H^k_{\mathrm{dR}}(M), allowing computation of these topological invariants purely from smooth data on M. The theorem also underpins applications to characteristic classes, where topological invariants like Euler or Chern classes on vector bundles over M are represented by closed differential forms, with the isomorphism preserving their cohomology classes.^[30] The theorem was originally proved by Georges de Rham in 1931 using homology theory on oriented manifolds.^[33]^[30]

Special Cases in Vector Calculus

Fundamental Theorem of Calculus

The fundamental theorem of calculus emerges as the zeroth-dimensional special case of the generalized Stokes theorem when applied to a one-dimensional oriented manifold with boundary. Consider the closed interval M = [a, b] \subset \mathbb{R}, which serves as a compact oriented 1-manifold. Its boundary is given by \partial M = \{b\} - \{a\}, where the points a and b are oriented 0-manifolds with the induced orientation: the endpoint b carries positive orientation, while the starting point a carries negative orientation.^[22]^[34] Let f be a smooth 0-form on M, which is simply a smooth real-valued function f: [a, b] \to \mathbb{R}. The exterior derivative of this 0-form is the 1-form df = f'(x) \, dx, where f' denotes the ordinary derivative of f and dx is the standard basis 1-form on \mathbb{R}. The generalized Stokes theorem then asserts that the integral of df over M equals the integral of f over the boundary \partial M:

\int_{[a,b]} df = \int_{[a,b]} f'(x) \, dx = f(b) - f(a) = \int_{\partial M} f.

This equality holds under the assumption that f is smooth (continuously differentiable) on [a, b], ensuring the existence of the derivative and the validity of integration by parts in this context.^[22]^[34] The left-hand side represents the net accumulation of the rate of change f' along the interval, while the right-hand side evaluates the antiderivative f at the endpoints, capturing the fundamental relation between differentiation and integration.^[35] This formulation interprets the fundamental theorem as the evaluation of a function at boundary points, unifying it with higher-dimensional integral theorems under the Stokes framework. For paths in higher-dimensional Euclidean spaces, the result extends via parametrization: if \gamma: [a, b] \to \mathbb{R}^n is a smooth curve, then \int_\gamma df = f(\gamma(b)) - f(\gamma(a)), where the pullback of df along \gamma yields the line integral of the gradient of f. This assumes f is smooth on the image of \gamma.^[22]^[34]

Green's Theorem

Green's theorem arises as a special case of the generalized Stokes theorem when applied to an oriented 2-manifold M \subset \mathbb{R}^2 equipped with a piecewise smooth boundary \partial M, where the boundary curve is positively oriented, typically counterclockwise. In this setting, consider a smooth 1-form \omega = P \, dx + Q \, dy defined on M, with P and Q being smooth functions on an open set containing M. The exterior derivative of \omega is the 2-form d\omega = \left( \frac{\partial Q}{\partial x} - \frac{\partial P}{\partial y} \right) dx \wedge dy. By the generalized Stokes theorem, the integral of d\omega over M equals the integral of \omega over \partial M:

\int_M d\omega = \int_{\partial M} \omega.

Substituting the expressions for d\omega and \omega, this yields

\int_M \left( \frac{\partial Q}{\partial x} - \frac{\partial P}{\partial y} \right) \, dA = \oint_{\partial M} P \, dx + Q \, dy,

where the left side is a double integral over the area of M and the right side is a line integral along the boundary. This formulation, which first appeared in George Green's 1828 essay on electricity and magnetism, provides a foundational relation between area integrals and boundary line integrals in the plane.^[36]^[37] In the language of vector calculus, Green's theorem corresponds to the circulation form for a vector field \mathbf{F} = \langle P, Q \rangle, equating the line integral of \mathbf{F} around the closed boundary curve to the double integral of the scalar curl component over the region:

\oint_{\partial M} \mathbf{F} \cdot d\mathbf{r} = \iint_M \left( \frac{\partial Q}{\partial x} - \frac{\partial P}{\partial y} \right) \, dA = \iint_M (\nabla \times \mathbf{F}) \cdot \mathbf{k} \, dA,

where \nabla \times \mathbf{F} = \left( \frac{\partial Q}{\partial x} - \frac{\partial P}{\partial y} \right) \mathbf{k} and \mathbf{k} is the unit vector in the z-direction. This vector interpretation highlights how the theorem measures the net circulation of \mathbf{F} around \partial M in terms of the rotation (curl) within M. The exterior derivative here captures the curl operation intrinsic to differential forms.^[38]^[37] The theorem holds under the assumptions that M is a bounded, simply connected region in \mathbb{R}^2 with piecewise smooth boundary \partial M, and that P and Q (or \mathbf{F}) have continuous first partial derivatives on an open set containing M. For example, consider the unit disk M = \{ (x,y) \mid x^2 + y^2 \leq 1 \} with boundary the unit circle \partial M oriented counterclockwise, and the 1-form \omega = -y \, dx + x \, dy. Then d\omega = 2 \, dx \wedge dy, so \int_M d\omega = \iint_M 2 \, dA = 2\pi. The boundary integral \int_{\partial M} \omega parametrizes the unit circle as x = \cos \theta, y = \sin \theta for \theta \in [0, 2\pi], yielding \oint_{\partial M} -y \, dx + x \, dy = \int_0^{2\pi} 1 \, d\theta = 2\pi, confirming the theorem. This example illustrates the theorem's utility for computing circulations in planar domains.^[38]^[39]

Stokes' Theorem

Stokes' theorem represents a special case of the generalized Stokes theorem applied to oriented surfaces in three-dimensional Euclidean space, connecting the integral of the curl of a vector field over the surface to the line integral of the field along the surface's boundary. Consider an oriented surface S \subset \mathbb{R}^3, which is a 2-dimensional manifold with boundary \partial S, a closed curve. The orientation of S induces an orientation on \partial S via the right-hand rule: if the fingers of the right hand curl in the direction of the boundary traversal, the thumb points in the direction of the positive normal to the surface. To formulate the theorem using differential forms, associate the vector field \mathbf{F} = (F_1, F_2, F_3) with the 1-form \omega = F_1 \, dx + F_2 \, dy + F_3 \, dz. The exterior derivative is the 2-form

d\omega = \left( \frac{\partial F_3}{\partial y} - \frac{\partial F_2}{\partial z} \right) dy \wedge dz + \left( \frac{\partial F_1}{\partial z} - \frac{\partial F_3}{\partial x} \right) dz \wedge dx + \left( \frac{\partial F_2}{\partial x} - \frac{\partial F_1}{\partial y} \right) dx \wedge dy.

The components of d\omega correspond to those of \nabla \times \mathbf{F}, the curl of \mathbf{F}, such that integrating d\omega over S yields the flux of the curl through S: \int_S d\omega = \int_S (\nabla \times \mathbf{F}) \cdot d\mathbf{S}. By the generalized Stokes theorem, this equals the line integral \int_{\partial S} \omega = \int_{\partial S} \mathbf{F} \cdot d\mathbf{r}. Thus, Stokes' theorem states

\int_S (\nabla \times \mathbf{F}) \cdot d\mathbf{S} = \int_{\partial S} \mathbf{F} \cdot d\mathbf{r},

where the orientations are compatible.^[40] This result generalizes Green's theorem, which applies to flat regions in the plane, to arbitrary oriented surfaces in \mathbb{R}^3.^[12] A illustrative example is the upper hemisphere S defined by x^2 + y^2 + z^2 = 4, z \geq 0, with boundary \partial S the circle x^2 + y^2 = 4, z = 0, oriented counterclockwise when viewed from above. For the vector field \mathbf{F} = (z - y, x, -x), the curl is \nabla \times \mathbf{F} = (0, 2, 2). Parameterizing the boundary as \mathbf{r}(t) = (2\cos t, 2\sin t, 0), $0 \leq t \leq 2\pi, the line integral is \int_{\partial S} \mathbf{F} \cdot d\mathbf{r} = 8\pi. The surface integral \int_S (\nabla \times \mathbf{F}) \cdot d\mathbf{S}, computed using the projection onto the xy-plane and polar coordinates, also equals $8\pi, verifying the theorem.^[41]

Divergence Theorem

The divergence theorem emerges as a special case of the generalized Stokes theorem when applied to three-dimensional oriented manifolds in \mathbb{R}^3. Consider an oriented 3-manifold V \subset \mathbb{R}^3 that is compact with boundary, where the boundary \partial V is a closed oriented 2-manifold (surface) equipped with the induced outward orientation.^[2]^[42] To recover the classical vector calculus statement, associate a smooth vector field \mathbf{F} = (F_1, F_2, F_3) on \mathbb{R}^3 with a corresponding 2-form \omega = F_1 \, dy \wedge dz + F_2 \, dz \wedge dx + F_3 \, dx \wedge dy.^[42] The generalized Stokes theorem then yields \int_V d\omega = \int_{\partial V} \omega. The exterior derivative computes as

d\omega = \left( \frac{\partial F_1}{\partial x} + \frac{\partial F_2}{\partial y} + \frac{\partial F_3}{\partial z} \right) dx \wedge dy \wedge dz = (\operatorname{div} \mathbf{F}) \, dx \wedge dy \wedge dz,

so the left-hand integral is the volume integral \int_V (\operatorname{div} \mathbf{F}) \, dV. The right-hand side corresponds to the flux integral \int_{\partial V} \mathbf{F} \cdot d\mathbf{S}, where d\mathbf{S} denotes the outward-oriented surface element.^[42]^[2] Thus, the divergence theorem states \int_V (\operatorname{div} \mathbf{F}) \, dV = \int_{\partial V} \mathbf{F} \cdot d\mathbf{S}, linking the total divergence within the volume to the net outward flux through its boundary.^[42] A representative example illustrates this for the unit ball V = \{ (x,y,z) \in \mathbb{R}^3 : x^2 + y^2 + z^2 \leq 1 \} with spherical boundary \partial V = S^2. Take \mathbf{F} = (x, y, z), so \operatorname{div} \mathbf{F} = 3. The volume integral is $3 \times \frac{4}{3} \pi = 4\pi. On the boundary, parametrize using spherical coordinates where \mathbf{F} = \mathbf{r} (position vector) and the outward normal aligns with \mathbf{r}/|\mathbf{r}|, yielding flux \int_{S^2} 1 \, dS = 4\pi, confirming equality.^[43]

Advanced Generalizations

Extension to Currents

In the theory of currents, developed within geometric measure theory, a current is defined as a continuous linear functional on the space of compactly supported smooth differential forms of degree k, denoted \mathcal{D}'(\Omega^k(M)), where M is a smooth manifold and the continuity is with respect to the appropriate topology on forms.^[44] This formulation generalizes the notion of integration over oriented submanifolds to more singular geometric objects, allowing for the treatment of boundaries and singularities in a distributional sense. The boundary operator \partial T on a k-current T is naturally defined by \partial T(\omega) = T(d\omega) for any (k-1)-form \omega, mirroring the classical boundary in simplicial chains but extended to this functional setting.^[45] The generalized Stokes theorem for currents states that for a k-current T and a compactly supported (k-1)-form \omega,

\langle T, d\omega \rangle = \langle \partial T, \omega \rangle,

where \langle \cdot, \cdot \rangle denotes the pairing between currents and forms. This identity holds by the very definition of the boundary operator and extends to all currents via density arguments: smooth approximations of singular currents converge in the appropriate topology, preserving the theorem under limits of mass and boundary mass.^[44] Unlike the classical version, which requires smooth manifolds and forms, this formulation applies to weak objects without assuming regularity, enabling analysis of non-smooth domains. Prominent examples include Dirac currents, which represent integration over points or submanifolds; for a 0-dimensional Dirac current at a point p, T(\omega) = \omega(p) for a 0-form \omega, while higher-dimensional versions integrate forms tangent to the submanifold.^[46] Rectifiable currents, a key class of integer-multiplicity currents, approximate varifold-like structures with finite Hausdorff measure on countably \mathcal{H}^k-rectifiable sets, equipped with an integer-valued multiplicity function and orientation; these capture the geometry of "almost smooth" singular sets.^[46] This extension finds significant applications in the Plateau problem, where minimizers of mass among currents with prescribed boundary yield area-minimizing surfaces, including minimal surfaces that may develop singularities.^[46] The framework accommodates non-compact supports and currents with infinite total mass, provided the boundary has finite mass, by restricting to normal currents (finite mass and comass-bounded boundary) or using compactness theorems in the flat topology for integral currents.^[44] Smooth oriented manifolds embed naturally as regular currents via integration, T(\omega) = \int_M \omega, recovering the classical Stokes theorem as a special case when the manifold and forms are smooth.^[45]

Applications in Physics

In electromagnetism, Faraday's law of induction states that the electromotive force around a closed loop is equal to the negative rate of change of magnetic flux through the surface bounded by that loop, expressed as \oint_{\partial S} \mathbf{E} \cdot d\mathbf{l} = -\frac{d}{dt} \int_S \mathbf{B} \cdot d\mathbf{S}. This integral form directly follows from the differential version \nabla \times \mathbf{E} = -\frac{\partial \mathbf{B}}{\partial t} via Stokes' theorem, which equates the surface integral of the curl to the line integral over the boundary, thereby linking local field behavior to global circulation effects.^[47]^[48] Gauss's law for electricity, \oint_{\partial V} \mathbf{E} \cdot d\mathbf{S} = \frac{Q_{\text{encl}}}{\epsilon_0}, asserts that the electric flux through a closed surface equals the enclosed charge divided by the permittivity of free space. This arises from the differential form \nabla \cdot \mathbf{E} = \frac{\rho}{\epsilon_0} using the divergence theorem, which converts the volume integral of the divergence into a surface flux, quantifying how charge sources produce field divergence within a region./16:_Vector_Calculus/16.08:_The_Divergence_Theorem)^[49] The generalized Stokes' theorem underpins conservation laws in physics by relating the integral of an exterior derivative over a manifold to its boundary integral, \int_M d\omega = \int_{\partial M} \omega, implying that the flux of a field through a boundary equals the integrated source inside the domain. This structure connects to Noether's theorem, where continuous symmetries of the Lagrangian yield conserved currents, with the theorem ensuring their global conservation through topological boundaries.^[50]^[51] In fluid dynamics, the divergence theorem applies to mass conservation, yielding the continuity equation \int_V \nabla \cdot (\rho \mathbf{v}) \, dV = \oint_{\partial V} \rho \mathbf{v} \cdot d\mathbf{S}, where \rho is density and \mathbf{v} is velocity; this equates the net mass flux out of a volume to the rate of change of mass inside plus any sources, forming the basis for the local form \frac{\partial \rho}{\partial t} + \nabla \cdot (\rho \mathbf{v}) = 0.^[52]^[53] In relativistic contexts, Maxwell's equations in curved spacetime employ differential forms for covariance, with the homogeneous equation d\mathbf{F} = 0 (where \mathbf{F} is the Faraday 2-form) implying \mathbf{F} = d\mathbf{A} locally via Poincaré's lemma, while the inhomogeneous equation d \star \mathbf{F} = \mathbf{J} (with \star the Hodge star and \mathbf{J} the current 3-form) uses Stokes' theorem to integrate over manifolds, adapting flat-space laws to gravitational curvature without explicit Christoffel symbols.^[54]^[55] The Aharonov-Bohm effect exemplifies the theorem's role in non-trivial topology, where charged particles acquire phase shifts from vector potential \mathbf{A} in regions of zero field but non-zero magnetic flux through inaccessible paths, tied to the de Rham cohomology of the configuration space; the phase \exp(i \frac{e}{\hbar} \oint \mathbf{A} \cdot d\mathbf{l}) depends on the non-exactness of \mathbf{A}, as d\mathbf{A} = \mathbf{B} integrates to flux via Stokes, revealing global geometric phases.^[56]^[57]

References

[1]
7.3: C- Differential Forms and Stokes' Theorem
Sep 5, 2021 · The main point of differential forms is to find the proper context for the Fundamental Theorem of Calculus, 1) ∫ a b f ′ ⁡ ( x ) d x = f ⁡ ⁡ We ...
[2]
[PDF] the generalized stokes' theorem
Aug 24, 2012 · Now that we understand the idea of a differential form, our next objective is to generate differential forms from ones that we already have.
[3]
[PDF] the generalized stokes' theorem
4. Differential forms. 4.1. Tensors: the Wedge Product. Tensors. Let us begin by defining the tensor product: given a k-linear ...
[4]
[PDF] The History of Stokes' Theorem - Harvard Mathematics Department
The first published proof of the theorem seems to have been in a monograph of Hermann Hankel in 1861 [10]. Hankel gives no credit for the theorem, only a ...
[5]
[PDF] Differential Forms - MIT Mathematics
Feb 1, 2019 · One of the goals of this text on differential forms is to legitimize this interpretation of equa- tion (1) in 𝑛 dimensions and in fact, ...
[6]
Calculus history - MacTutor - University of St Andrews
Newton's work on Analysis with infinite series was written in 1669 and circulated in manuscript. It was not published until 1711. Similarly his Method of ...
[7]
George Green (1793 - Biography - MacTutor History of Mathematics
George Green was an English mathematician best-known for Green's function and Green's theorems in potential theory. Biography. George Green's father, also ...
[8]
[PDF] A History of the Divergence, Green's, and Stokes' Theorems
Lagrange was the first to be on the hunt for a proof of the Divergence Theorem, and although he was unable to prove it, he began a journey that would not be ...
[9]
Georges de Rham - Biography - MacTutor - University of St Andrews
But in 1931 de Rham set out to give a rigorous proof. The technical problems were considerable at the time, as both the general theory of manifolds and the ...Missing: cohomology | Show results with:cohomology
[10]
[PDF] Manifolds and Differential Forms Reyer Sjamaar - Cornell Mathematics
First definition. There are several different ways to define differential forms on manifolds. In this section we present a practical, workaday definition. A ...<|control11|><|separator|>
[11]
[PDF] spivak-calculus-on-manifolds.pdf - Cimat
The area of differential geometry is one in which recent developments have effected great changes. That part of differential geometry centered about Stokes' ...
[12]
[PDF] Introduction to differential forms - Purdue Math
May 6, 2016 · The calculus of differential forms give an alternative to vector calculus which is ultimately simpler and more flexible.
[13]
[PDF] 1 Manifolds: definitions and examples - MIT Mathematics
Definition 1.3. A smooth n-dimensional manifold is a Hausdorff, second count- able, topological space X together with an atlas, A. 1.1 examples. R n or any ...
[14]
[PDF] NOTES ON MANIFOLDS (239) - UC Davis Mathematics
A set M with n-dimensional differential structure is called a (smooth) n-dimensional manifold. Note that instead of a class of equivalent atlases we can speak ...
[15]
[PDF] Math 396. Orientations In the theory of manifolds there will be a ...
Orientations arise from certain notions in linear algebra, applied to tangent and cotangent spaces of manifolds. The aim of this handout is to develop these ...
[16]
[PDF] 1 Introduction 2 Orientations of Smooth Manifolds
A choice of orientations for all tangent spaces at points p of M such that the orientations depend continuously on p is called an orientation of M. We say M is ...
[17]
[PDF] 1. Collar neighbourhood theorem Definition 1.0.1. Let M n be a ...
Let Mn be a manifold with boundary. Then there exists a smooth vector field V on M such that V (p) is inward for any p ∈ ∂M . Proof ...
[18]
[PDF] Orientation The line and surface integrals we meet in multi-variable ...
In this note we give a precise definition of orientation and induced orientation on a boundary. A line is oriented by choice of direction, a plane is oriented ...
[19]
(PDF) Stokes' theorem for nonsmooth chains - ResearchGate
Aug 9, 2025 · We give a definition of a singular integral over non-rectifiable curves and fractals which generalizes the known one. We define ...
[20]
[PDF] Integration on Manifolds - MIT OpenCourseWare
To fix an orientation, we just need to say which local coordinate sys- tems (or bases of tangent spaces) are right-handed, and do so in a consistent way. But ...
[21]
[PDF] Differential Forms Lecture Notes Liam Mazurowski
It is a meta principle that every operation on differential forms has a dual operation on cells. For example, the exterior derivative operator on forms is.
[22]
[PDF] Differential Forms and Integration - UCLA Mathematics
The integration on forms concept is of fundamental importance in differ- ential topology, geometry, and physics, and also yields one of the most important.
[23]
[PDF] LECTURE 18: INTEGRATION ON MANIFOLDS 1. Orientations and ...
Orientations and Integration. Before we integrate an n-form on manifold, we need an orientation. Definition 1.1. Let M be a smooth manifold of dimension n. (1) ...
[24]
[PDF] The Generalized Stokes Theorem and its applications
May 15, 2023 · ω = P(x,y,z)dx < dy + Q(x,y,z)dy < dz + R(x,y,z)dz < dx, whereas 3-form is the expression ω = f (x,y,z)dx < dy < dz. Dmytro Kulish, Yaroslav ...
[25]
[PDF] Stokes' Theorem - Integration of Differential Forms Over Chains
theorem called the Generalized Stokes' theorem, or simply Stokes' theorem. We begin with the foundation of differential forms – the tensors on a vector ...
[26]
https://math.libretexts.org/Bookshelves/Analysis/Tasty_Bits_of_Several_Complex_Variables_(Lebl)/07:_Appendix/7.03:_C-_Differential_Forms_and_Stokes'_Theorem
[27]
[PDF] Stokes's Theorem and Whitney Manifolds
Title: Stokes's Theorem and Whitney Manifolds. A Sequel to Basic Real Analysis. Cover: An example of a Whitney domain in two-dimensional space.
[28]
[PDF] V.2 : Generalized Stokes' Formula (Conlon, §§ 2.6, 8.1–8.2
homology we obtain an isomorphism from the de Rham cohomology of U to the ordinary singular cohomology of U with real coefficients. The aim of this section ...
[29]
[PDF] homology, cohomology, and the de rham theorem - UChicago Math
The de Rham Theorem proves an isomorphism between singular cohomology groups,. Date: AUGUST 28, 2021. 1. Page 2. 2. CHAD BERKICH.
[30]
[PDF] De Rham's Theorem
In this paper we review basic notions of differential forms, singular simplexes and chain complexes. We then introduce both the de rham and singu- lar ...
[31]
[PDF] A Short Course on deRham Cohomology - University of Oregon
Oct 7, 2021 · Theorem 9 The only non-zero cohomology groups of the n-dimensional sphere are. H0(Sn) = R. Hn(Sn) = R. Proof: Suppose n is at least two. Let U ...
[32]
[PDF] Undergraduate Lecture Notes in De Rham–Hodge Theory - arXiv
May 15, 2011 · The fundamental topological duality is based on the Stokes theorem,. Z∂Cω = ZC dω or h∂C, ωi = hC, dωi , where ∂C is the boundary of the p ...
[33]
[PDF] Sur l'analysis situs des variétés à n dimensions - Numdam
I .es propriétés à*Analysis situs étudiées ici sont celles qui se rattachent aux systèmes de variétés orientées ou champs d'intégration et aux.
[34]
[PDF] Differential Forms Crash Course
Nov 6, 2024 · ... Stokes'. Notice how the expression. ∫. 𝑆. 𝑑𝜔 = ∫. 𝜕𝑆. 𝜔 is now the same for both the Stokes' Theorem and the Fundamental Theorem of Calculus.
[35]
[PDF] an introduction to differential forms, stokes' theorem and gauss ...
This paper serves as a brief introduction to differential geome- try. It first discusses the language necessary for the proof and applications of a powerful ...<|control11|><|separator|>
[36]
[PDF] Green 1828 - The Application of Mathematical Analysis
MATHEMATICAL ANALYSIS TO THE THEORIES OF ELECTRICITY. AND MAGNETISM. INTRODUCTORY OBSERVATIONS. A HE object of this Essay is to submit to Mathematical Analysis ...
[37]
[PDF] Differential Forms: Unifying the Theorems of Vector Calculus
We are now ready to state Stokes's Theorem in its most general form: Theorem [Stokes's general theorem]. If R is a p-dimensional region and η a (p − 1)-form ...
[38]
6.4 Green's Theorem - Calculus Volume 3 | OpenStax
Mar 30, 2016 · In this section, we examine Green's theorem, which is an extension of the Fundamental Theorem of Calculus to two dimensions. Green's theorem ...
[39]
https://math.libretexts.org/Bookshelves/Calculus/Calculus_(OpenStax)/16%3A_Vector_Calculus/16.04%3A_Greens_Theorem
[40]
[PDF] Vector fields and differential forms - Arizona Math
Sep 25, 2008 · The classical Stokes's theorem in three dimensions is obtained by applying. Stokes's theorem for 1-forms to this particular 1-form. This ...
[41]
[PDF] Section 17.7 Stokes' Theorem - CSUN
EXAMPLE 1. Verifying Stokes' Theorem. Confirm that Stokes' Theorem holds for the vector field F = 〈z - y, x, -x〉, where S is the hemisphere x2 + y2 + z2 = 4, ...
[42]
[PDF] Calculus on Manifolds - Strange beautiful grass of green
Differential, 91. Differential form, 8 absolute, 126 closed, 92 eontinuous, 88 differentiable, 88 exact, 92. Index. Differential form, on a manifold,. 117.
[43]
6.8 The Divergence Theorem - Calculus Volume 3 | OpenStax
Mar 30, 2016 · Let S r S r denote the boundary sphere of B r . B r . We can approximate the flux across S r S r using the divergence theorem as follows: ∬ S r ...
[44]
Geometric Measure Theory - SpringerLink
Book Title: Geometric Measure Theory · Authors: Herbert Federer · Editors: B. · Series Title: Classics in Mathematics · Publisher: Springer Berlin, Heidelberg.
[45]
[PDF] Herbert Federer - National Academy of Sciences
Because a current is defined as a continuous linear functional on a space of differential k forms, there is a natural definition of boundary, namely ∂T:=T°d, ...
[46]
Normal and Integral Currents - jstor
ordinary functions to differential forms of arbitrary degree. The elegant linear function approach to measure theory, which almost ignores that. 2 Abstracts ...
[47]
12.5 Electromagnetic Induction: Faraday's Law
Faraday observed that changing magnetic fields produced modifications to the laws of electrostatics: in particular, changing magnetic fields lead to ...
[48]
Faraday's Law — Electromagnetic Geophysics - EM GeoSci
Faraday's law is named after English scientist Michael Faraday (1791-1867), and describes the manner in which time-varying magnetic fields induce the rotational ...
[49]
[PDF] Gauss's Divergence Theorem
Gauss's Divergence Theorem tells us that the flux of F across ∂S can be found by integrating the divergence of F over the region enclosed by ∂S. ⇀. ⇀. ⇀. ⇀. ∂S ...
[50]
Noether Symmetries and Covariant Conservation Laws in Classical ...
Generalized Stokes theorem in this context establishes a bridge between conserved quantities and (co)homology of forms (and hence with topology and global ...
[51]
[PDF] 3 Classical Symmetries and Conservation Laws
Noether theorem reduces to proving the existence of a locally conserved current. In the following sections we will prove Noether's theorem for internal and ...
[52]
[PDF] 11–Applications of the Divergence Theorem - UC Davis Math
Conservation of Mass: The time rate of change of the mass in volume r equals minus the mass flux through the boundary ∂r ρu ρu. Conclude: The continuity ...
[53]
2.4 Conservation of mass - Notes on CFD: General Principles
Apr 11, 2022 · The law of conservation of mass states that the rate of mass increase inside a volume equals the rate of mass inflow across its surface.<|separator|>
[54]
[PDF] GRAVITATION F10 Lecture 12 1. Maxwell's Equations in Curved ...
1. Maxwell's Equations in Curved Space-Time 1.1. Recall that Maxwell equations in Lorentz covariant form are. ∂µFµν = jν, Fµν = ∂µAν − ∂νAµ.Missing: differential | Show results with:differential
[55]
[PDF] Differential Forms in Physics II - Maxwell's Equations
In this monograph we rewrite Maxwell's Equations in the language of differential forms, showcasing yet again (as in the Stokes's Theorem paper) the elegance and ...<|separator|>
[56]
[PDF] The Aharonov-Bohm effect: reality and folklore - arXiv
Sep 25, 2025 · This space is called the first de Rham cohomology of Σ, and it is non-trivial only if there are loops in Σ that are not contractible to a point: ...
[57]
[PDF] Electromagnetism, local covariance, the Aharonov-Bohm effect and ...
The term current was introduced by de Rham (cf. [14]) because, in the setting of electromagnetism, such objects can be interpreted as electromagnetic currents.