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Theorem in vector calculus
Stokes' theorem, also known as the Kelvin–Stokes theorem, is a theorem in vector calculus that relates the behavior of a vector field along the edge of
Stokes'_theorem
Statement about integration on manifolds
the generalized Stokes theorem (sometimes with apostrophe as Stokes' theorem or Stokes's theorem), also called the Stokes–Cartan theorem, is a statement
Generalized_Stokes_theorem
British mathematician and physicist (1819–1903)
mathematician, he popularised Stokes' theorem in vector calculus and contributed to the theory of asymptotic expansions. Stokes, along with Felix Hoppe-Seyler
Sir George Stokes, 1st Baronet
Sir_George_Stokes,_1st_Baronet
Theorem in calculus relating line and double integrals
special case of Stokes' theorem (surface in R 3 {\displaystyle \mathbb {R} ^{3}} ). In one dimension, it is equivalent to the fundamental theorem of calculus
Green's_theorem
Geometric model of the physical space
\,\mathbf {q} ]}\nabla \varphi (\mathbf {r} )\cdot d\mathbf {r} .} Stokes' theorem relates the surface integral of the curl of a vector field F over a
Three-dimensional_space
Equations of motion for viscous fluids
Navier and George Gabriel Stokes, who developed them over a few decades of progressive work, from 1822 (Navier) to 1842–1850 (Stokes). Siméon Denis Poisson
Navier–Stokes_equations
Theorem in calculus
In vector calculus, the divergence theorem, also known as Gauss's theorem or Ostrogradsky's theorem, is a theorem relating the flux of a vector field through
Divergence_theorem
Concept of complex analysis
integral theorem and Cauchy's integral formula. The residue theorem should not be confused with special cases of the generalized Stokes' theorem; however
Residue_theorem
Operation on differential forms
a natural, metric-independent generalization of Stokes' theorem, Gauss's theorem, and Green's theorem from vector calculus. If a differential k {\displaystyle
Exterior_derivative
Book by Michael Spivak
letter from Lord Kelvin to Sir George Stokes containing the first disclosure of the classical Stokes' theorem. Calculus on Manifolds aims to present
Calculus_on_Manifolds_(book)
Theorem in topology
Brouwer's fixed-point theorem is a fixed-point theorem in topology, named after L. E. J. Brouwer. It states that for any continuous function f {\displaystyle
Brouwer_fixed-point_theorem
Relationship between derivatives and integrals
The fundamental theorem of calculus is a theorem that links the concept of differentiating a function (calculating its slopes, or rate of change at every
Fundamental theorem of calculus
Fundamental_theorem_of_calculus
Circulation density in a vector field
vector fields. The corresponding form of the fundamental theorem of calculus is Stokes' theorem, which relates the surface integral of the curl of a vector
Curl_(mathematics)
Operation in calculus
and Stokes' theorem simultaneously generalizes the three theorems of vector calculus: the divergence theorem, Green's theorem, and the Kelvin-Stokes theorem
Integral
Calculus of vector-valued functions
2-forms, respectively, and the key theorems of vector calculus are all special cases of the general form of Stokes' theorem. From the point of view of both
Vector_calculus
Expression that may be integrated over a region
theorem of calculus, the divergence theorem, Green's theorem, and Stokes' theorem as special cases of a single general result, the generalized Stokes
Differential_form
Provides integral formulas for all derivatives of a holomorphic function
The proof of Cauchy's integral theorem for higher dimensional spaces relies on the using the generalized Stokes theorem on the quantity G ( r , r ′ ) f
Cauchy's_integral_formula
Map from multiple vectors to an underlying field of scalars, linear in each argument
Stokes' theorem can be further generalized to arbitrary smooth manifolds-with-boundary and even certain "rough" domains (see the article on Stokes' theorem
Multilinear_form
Integration over a non-flat region in 3D space
and vector calculus, such as the divergence theorem, magnetic flux, and its generalization, Stokes' theorem. Let us notice that we defined the surface
Surface_integral
single-variable fundamental theorem of calculus to higher dimensions, in a different vein than the generalization that is Stokes' theorem. Let G {\displaystyle
Darboux_derivative
Counts 0s of a vector field on a differentiable manifold using its Euler characteristic
Poincaré–Hopf theorem (also known as the Poincaré–Hopf index formula, Poincaré–Hopf index theorem, or Hopf index theorem) is an important theorem in differential
Poincaré–Hopf_theorem
Four-vector analogue of the gradient operation
calculus, and more generally differential geometry, Stokes' theorem (also called the generalized Stokes' theorem) is a statement about the integration of differential
Four-gradient
Topics referred to by the same term
Stokes shift Stokes stream function Stokes' theorem Stokes wave Campbell–Stokes recorder Navier–Stokes equations Stokes Bay (disambiguation) Stokes Township
Stokes
Theorem regarding circulation in a barotropic ideal fluid
first term, we substitute from the governing equation, and then apply Stokes' theorem, thus: ∮ C D u D t ⋅ d s = ∫ A ∇ × ( − 1 ρ ∇ p + ∇ Φ ) ⋅ n d S = ∫
Kelvin's_circulation_theorem
Statement relating differentiable symmetries to conserved quantities
theorem can be seen as a consequence of the fundamental theorem of calculus (known by various names in physics such as the Generalized Stokes theorem
Noether's_theorem
Textbook
discussion of the implicit and inverse function theorems, differential forms, the generalized Stokes theorem, and the Lebesgue integral. Locascio, Andrew
Principles of Mathematical Analysis
Principles_of_Mathematical_Analysis
less computational power than if a uniformly fine mesh were used. Stokes' theorem relates the integral of a differential (n − 1)-form ω over the boundary
Discrete_exterior_calculus
Navier–Stokes equations, see section on fluid dynamics Navier–Stokes existence and smoothness Stokes' theorem Kelvin–Stokes theorem Generalized Stokes theorem
List of things named after George Gabriel Stokes
List_of_things_named_after_George_Gabriel_Stokes
Line integral of the fluid velocity around a closed curve
{\displaystyle {\boldsymbol {\omega }}=\nabla \times \mathbf {V} .} By Stokes' theorem, the flux of curl or vorticity vectors through a surface S is equal
Circulation_(physics)
Calculus of functions of several variables
is embodied by the integral theorems of vector calculus: Gradient theorem Stokes' theorem Divergence theorem Green's theorem. In a more advanced study of
Multivariable_calculus
Type of fluid flow
Stokes flow (named after George Gabriel Stokes), also named creeping flow or creeping motion, is a type of fluid flow where advective inertial forces are
Stokes_flow
Topics referred to by the same term
Stokes' formula can refer to: Stokes' law for friction force in a viscous fluid. Stokes' law (sound attenuation) law describing attenuation of sound in
Stokes_formula
this lemma. Stokes' theorem. It is named after Sir George Gabriel Stokes (1819–1903), although the first known statement of the theorem is by William
List_of_misnamed_theorems
Basic law of electromagnetism
and time t. It can also be written in an integral form by the Kelvin–Stokes theorem: ∮ ∂ Σ E ⋅ d l = − ∬ Σ ∂ B ∂ t ⋅ d A {\displaystyle \oint _{\partial
Faraday's_law_of_induction
Concept in classical electromagnetism
form". The forms are exactly equivalent, and related by the Kelvin–Stokes theorem (see the "proof" section below). Forms using SI units, and those using
Ampère's_circuital_law
Concept in 3-dimensional geometry
is entirely determined by the boundary. These are consequences of Stokes' theorem. The vector area of a parallelogram is given by the cross product of
Vector_area
Evaluates a line integral through a gradient field using the original scalar field
The gradient theorem, also known as the fundamental theorem of calculus for line integrals, says that a line integral through a gradient field can be evaluated
Gradient_theorem
Electromagnetic quantum-mechanical effect in regions of zero magnetic and electric field
the electromagnetic four-potential, (Φ, A), must be used instead. By Stokes' theorem, the magnitude of the Aharonov–Bohm effect can be calculated using
Aharonov–Bohm_effect
Equations describing classical electromagnetism
of the Gauss divergence theorem and the Kelvin–Stokes theorem. According to the (purely mathematical) Gauss divergence theorem, the electric flux through
Maxwell's_equations
British physicist, engineer and mathematician (1824–1907)
Kelvin wave Kelvin's heat death paradox Kelvin's circulation theorem Kelvin–Stokes theorem Kelvin–Varley divider The SI unit of temperature, kelvin Mount
Lord_Kelvin
Surface specified with parameters
Surfaces that occur in two of the main theorems of vector calculus, Stokes' theorem, and the divergence theorem, are frequently given in a parametric form
Parametric_surface
Indefinite integral
instance double integrals, polar coordinates, the Jacobian and the Stokes' theorem) Numerical integration (a technique for approximating a definite integral
Antiderivative
Mathematical identities
\iint _{S}\left(\nabla \times \mathbf {A} \right)\cdot d\mathbf {S} } (Stokes' theorem) ∮ ∂ S ψ d ℓ = − ∬ S ∇ ψ × d S {\displaystyle \oint _{\partial
Vector_calculus_identities
Theorem in measure theory
disintegration theorem can also be seen as justifying the use of a "restricted" measure in vector calculus. For instance, in Stokes' theorem as applied to
Disintegration_theorem
Discrete (i.e., incremental) version of infinitesimal calculus
lemma, the first proof of the general Stokes Theorem, and a lot more L. E. J. Brouwer: simplicial approximation theorem Élie Cartan, Georges de Rham: the
Discrete_calculus
horn Jacobian matrix Hessian matrix Curvature Green's theorem Divergence theorem Stokes' theorem Vector Calculus Infinite series Maclaurin series, Taylor
List_of_calculus_topics
field Solenoidal vector field Stokes' theorem Submersion Surface integral Symmetry of second derivatives Taylor's theorem Total derivative Vector field
List of multivariable calculus topics
List_of_multivariable_calculus_topics
Media. p. 782. ISBN 978-3-642-22421-8. Victor J. Katz, The History of Stokes' Theorem, Mathematics Magazine Vol. 52, No. 3 (May, 1979), pp. 146–156, at p
Timeline_of_bordism
Differentiation under the integral sign formula
\cdot \mathbf {v} +\mathbf {v} \cdot \nabla )\mathbf {F} ,} and that Stokes theorem equates the surface integral of the curl over Σ with a line integral
Leibniz_integral_rule
Aspects of fluid mechanics involving fluid flow
control volume. Differential formulations of the conservation laws apply Stokes' theorem to yield an expression that may be interpreted as the integral form
Fluid_dynamics
Mathematical method in calculus
v)-(-1)^{k}\int \limits _{M}u\wedge dv.} An application of generalized Stokes' theorem gives the result: ∫ M d u ∧ v = ∮ ∂ M u ∧ v − ( − 1 ) k ∫ M u ∧ d v
Integration_by_parts
Seven mathematical problems with a US$1 million prize for each solution
September 2026. "On the Navier–Stokes Millennium Prize Problem". OpenAI. 2026-09-08. Retrieved 2026-09-09. "On the Navier–Stokes Millennium Prize Problem"
Millennium_Prize_Problems
Velocity field as the gradient of a scalar function
simply-connected contour C {\displaystyle C} is zero. This can be shown using the Stokes theorem, Γ ≡ ∮ C v ⋅ d l = ∫ ω ⋅ d f = 0 {\displaystyle \Gamma \equiv \oint
Potential_flow
Series of two mathematics textbooks
multivariable calculus, including topics in vector calculus like Green's theorem and Stokes' theorem, as well as linear differential equations and the theory of probability
Calculus_(Apostol_books)
of essential singular points, 1850 - George Gabriel Stokes rediscovers and proves Stokes' theorem, 1861 - Karl Weierstrass starts to use the language
Timeline of calculus and mathematical analysis
Timeline_of_calculus_and_mathematical_analysis
Method of mathematical integration
of differential forms on manifolds, and to Stokes' theorem as the generalization of the fundamental theorem of calculus. By contrast, Lebesgue integration
Lebesgue_integral
Millennium Prize Problem
named four instances of the Navier–Stokes existence and smoothness problem, two of which dealing with the Navier–Stokes equation with no external force and
Navier–Stokes existence and smoothness
Navier–Stokes_existence_and_smoothness
Conditions for switching order of integration in calculus
Fubini's theorem gives the conditions under which a double integral can be computed as an iterated integral, i.e. by integrating in one variable at a
Fubini's_theorem
Theorem in magnetohydrodynamics
In ideal magnetohydrodynamics, Alfvén's theorem, or the frozen-in flux theorem, states that electrically conducting fluids and embedded magnetic fields
Alfvén's_theorem
Prize from University of Cambridge in mathematics and theoretical physics
Stokes included an examination question on a particular theorem that William Thomson had written to him about, which is now known as Stokes' theorem.
Smith's_Prize
On converting relations to functions of several real variables
In multivariable calculus, the implicit function theorem is a theorem that provides sufficient conditions under which a planar curve specified by F ( x
Implicit_function_theorem
Type of infinitesimal in calculus
Q ) = 0 {\displaystyle \nabla \times (\nabla Q)=\mathbf {0} } and Stokes' theorem. ∮ ∂ Σ ∇ Q ⋅ d r = ∬ Σ ( ∇ × ∇ Q ) ⋅ d a = 0 {\displaystyle \oint _{\partial
Exact_differential
Physical condition
taken to go from A {\displaystyle A} to B {\displaystyle B} . From Stokes' theorem, the integral of a second order tensor along a closed path is given
Compatibility_(mechanics)
American mathematician (1907–1989)
book Geometric Integration Theory he gives a theoretical basis for Stokes' theorem applied with singularities on the boundary. Later, his work on such
Hassler_Whitney
Generalization of the product rule in calculus
Leibniz rule bears a strong resemblance to the binomial theorem, and in fact the binomial theorem can be proven directly from the Leibniz rule by taking
General_Leibniz_rule
Generalization of definite integrals to functions of multiple variables
distribution. Main analysis theorems that relate multiple integrals: Divergence theorem Stokes' theorem Green's theorem Stewart, James (2008). Calculus:
Multiple_integral
Calculus of functions generalization
which is the usual form of the Stokes' theorem on surfaces. Green’s theorem is also a special case of Stokes’ formula. Stokes' formula also yields a general
Calculus_on_Euclidean_space
went on to prove Stokes' theorem, which earned that name after Stokes asked students to prove it in the Smith's Prize exam in 1854. Stokes learned it from
Mathematics, science, technology and engineering of the Victorian era
Mathematics,_science,_technology_and_engineering_of_the_Victorian_era
Physics theorem for symmetries of action
physics, Noether's second theorem relates symmetries of an action functional with a system of differential equations. The theorem is named after its discoverer
Noether's_second_theorem
Vector field that is the gradient of some function
\mathbf {v} } as conservative). This can be proved directly by using Stokes' theorem, ∮ P c v ⋅ d r = ∬ A ( ∇ × v ) ⋅ d a = 0 {\displaystyle \oint _{P_{c}}\mathbf
Conservative_vector_field
Physics theorem about a swimmer's displacement
consequences of the linearity of Stokes equations. To summarize, the linearity of Stokes equations allows us to use the reciprocal theorem to relate the swimming
Scallop_theorem
Theorem in mathematics
In calculus and real analysis, the mean value theorem (or Lagrange's mean value theorem) is a theorem about differentiable functions, roughly stating
Mean_value_theorem
Vector operator in vector calculus
from the original on 2020-02-20. Adam Powell (12 April 2010). "The Navier-Stokes Equations" (PDF). Wilson, Edwin B. (1913). "On the unification of vectorial
Divergence
Instantaneous rate of change (mathematics)
constant, because the derivative of a constant is zero. The fundamental theorem of calculus shows that finding an antiderivative of a function gives a
Derivative
Formula for the derivative of a ratio of functions
Laplacian Directional derivative Identities Theorems Gradient Green's Stokes' Divergence Generalized Stokes Helmholtz decomposition Multivariable Formalisms
Quotient_rule
Approximation of a function by a polynomial
In calculus, Taylor's theorem gives an approximation of a k {\textstyle k} -times differentiable function around a given point by a polynomial of degree
Taylor's_theorem
theorem (logic) Diaconescu's theorem (mathematical logic) Easton's theorem (set theory) Erdős–Dushnik–Miller theorem (set theory) Erdős–Rado theorem (set
List_of_theorems
Integral of sin(x)/x from 0 to infinity
particularly when it is not useful to directly apply the fundamental theorem of calculus due to the lack of an elementary antiderivative for the integrand
Dirichlet_integral
Mathematics of real numbers and real functions
Arzelà-Ascoli theorem, the Stone-Weierstrass theorem, the Banach fixed-point theorem, the inverse and implicit function theorems, and Stokes' theorem. More advanced
Real_analysis
Special mathematical functions defined on the surface of a sphere
{\partial }{\partial r}}+r^{-2}\Delta _{S^{n-1}}} It follows from the Stokes theorem and the preceding property that the spaces Hℓ are orthogonal with respect
Spherical_harmonics
Manifold upon which it is possible to perform calculus
fundamental theorems of integral calculus in several variables—namely Green's theorem, the divergence theorem, and Stokes' theorem—generalize to a theorem (also
Differentiable_manifold
Concept in physics
{\mathcal {S}}} , the closed-path Berry phase can be rewritten using Stokes' theorem as γ n = ∫ S d S ⋅ Ω n ( R ) . {\displaystyle \gamma _{n}=\int _{\mathcal
Berry connection and curvature
Berry_connection_and_curvature
{S} =\oint _{C}\nabla \psi \cdot \mathbf {n} dl} where we used the Stokes theorem for circulation and ω = − ∇ 2 ψ {\displaystyle \omega =-\nabla ^{2}\psi
Prandtl–Batchelor_theorem
Operator generalizing the Laplacian in differential geometry
X\operatorname {vol} _{n}} where the last equality is an application of Stokes' theorem. Dualizing gives for all compactly supported functions f {\displaystyle
Laplace–Beltrami_operator
Undergraduate math course at Harvard University
algebra, tensors, differential forms, manifolds, and the generalized Stokes theorem. Although both were demanding courses that presented calculus from a
Math_55
\mathbf {dr} } For a simply-connected surface S {\displaystyle S} , Stokes theorem holds, and the circulation vanishes, as the velocity can be expressed
Quantum_turbulence
Cohomology with real coefficients computed using differential forms
{\displaystyle H_{\mathrm {dR} }^{k}(M)\simeq H_{\mathrm {dR} }^{k}(S^{1}).} Stokes' theorem is an expression of duality between de Rham cohomology and the homology
De_Rham_cohomology
of essential singular points. 1850 – George Gabriel Stokes rediscovers and proves Stokes' theorem. 1854 – Bernhard Riemann introduces Riemannian geometry
Timeline_of_mathematics
Change of variable for integrals involving trigonometric functions
Laplacian Directional derivative Identities Theorems Gradient Green's Stokes' Divergence Generalized Stokes Helmholtz decomposition Multivariable Formalisms
Tangent half-angle substitution
Tangent_half-angle_substitution
Quantized unit of magnetic flux
{q}{\hbar }}\mathbf {A} .} Integrating around the hole/loop using Stokes' theorem and ∇ × A = B gives: Φ B = ∮ A ⋅ d l = ℏ q ∮ ∇ θ ⋅ d l . {\displaystyle
Magnetic_flux_quantum
Second-order partial differential equation
may be defined by a line integral. The integrability condition and Stokes' theorem implies that the value of the line integral connecting two points is
Laplace's_equation
Derivative defined on normed spaces
Let D {\displaystyle D} be any linear functional. Riesz Representation Theorem tells us that D {\displaystyle D} could be defined by D v = ⟨ a , v ⟩ {\displaystyle
Fréchet_derivative
Connected open subset of a topological space
functions defined on the domain to hold, such as integral theorems (Green's theorem, Stokes theorem), properties of Sobolev spaces, and to define measures
Domain (mathematical analysis)
Domain_(mathematical_analysis)
Theorem in mathematics
In mathematical analysis, the inverse function theorem gives sufficient conditions for a function to have an inverse function. The essential idea is that
Inverse_function_theorem
Complex-differentiable (mathematical) function
_{\gamma }f\,\mathrm {d} z.} In light of the Jordan curve theorem and the generalized Stokes' theorem, F γ ( z ) {\displaystyle F_{\gamma }(z)} is independent
Holomorphic_function
2026 scientific priority controversy
special case of Navier–Stokes equations where the viscosity of the fluid is zero, a property called superfluidity. The Navier–Stokes existence and smoothness
Navier–Stokes priority controversy
Navier–Stokes_priority_controversy
Element in Riemannian geometry
vector field has compact support. In that case it is immediate from Stokes' theorem that d d t vol ( f t ) = − ∫ S ⟨ W t , H ( f t ) ⟩ g ω t + ∫ ∂ S
First variation of area formula
First_variation_of_area_formula
Force due to magnetic field
magnetization, the problem can be simplified in two different ways, using Stokes' theorem. Upon integration along the direction of magnetization, all dipoles
Force_between_magnets
embedding theorem Critical value Sard's theorem Saddle point Morse theory Lie derivative Hairy ball theorem Poincaré–Hopf theorem Stokes' theorem De Rham
List of differential geometry topics
List_of_differential_geometry_topics
Method for estimating new data within known data points
that vector calculus identities are satisfied, including Stokes' theorem and the divergence theorem. As a result, mimetic interpolation conserves line, area
Interpolation
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STOKES THEOREM
STOKES THEOREM
STOKES THEOREM
STOKES THEOREM
STOKES THEOREM
STOKES THEOREM
STOKES THEOREM
STOKES THEOREM
STOKES THEOREM
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