Stokes' theorem
E155868
Stokes' theorem is a fundamental result in vector calculus that relates the surface integral of the curl of a vector field over a surface to the line integral of the field around the surface’s boundary.
All labels observed (6)
| Label | Occurrences |
|---|---|
| Stokes' theorem canonical | 4 |
| Green's theorem | 1 |
| Kelvin–Stokes theorem | 1 |
| Stokes theorem | 1 |
| Stokes' theorem in vector calculus | 1 |
| generalized Stokes' theorem on manifolds | 1 |
How this entity was disambiguated
This entity first appeared as the object of triple T1367415 — resolving that mention is where its identity was fixed. The disambiguator weighed these candidate entities and picked the highlighted one (or “None”, minting a new entity). This is how homonymy is resolved: the same surface form can point to different entities.
NED1
Entity disambiguation (via context triple)
gpt-5-mini-2025-08-07
Target entity: Stokes' theorem Context triple: [George Gabriel Stokes, knownFor, Stokes' theorem]
-
A.
Gauss–Bonnet theorem (early form)
The Gauss–Bonnet theorem (early form) is an early version of the fundamental result in differential geometry that links the total curvature of a surface to its topological characteristics, originally developed by Carl Friedrich Gauss.
-
B.
Gauss’s law
Gauss’s law is a fundamental principle of electromagnetism that relates the electric flux through a closed surface to the electric charge enclosed within that surface.
-
C.
Maxwell stress tensor
The Maxwell stress tensor is a mathematical construct in classical electromagnetism that represents how electric and magnetic fields transmit mechanical stresses, such as pressure and tension, through space and matter.
-
D.
Gauss's law for magnetism
Gauss's law for magnetism is the Maxwell equation stating that magnetic monopoles do not exist and that magnetic field lines always form closed loops with zero net flux through any closed surface.
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E.
Theorema Egregium
Theorema Egregium is Gauss’s celebrated theorem in differential geometry showing that the Gaussian curvature of a surface is an intrinsic property independent of how the surface is embedded in space.
- F. None of above. chosen
- G. Unsure - the case is ambiguous/there is not enough information to decide.
NED2
Entity disambiguation (via description)
gpt-5-mini-2025-08-07
Target entity: Stokes' theorem Target entity description: Stokes' theorem is a fundamental result in vector calculus that relates the surface integral of the curl of a vector field over a surface to the line integral of the field around the surface’s boundary.
-
A.
Gauss–Bonnet theorem (early form)
The Gauss–Bonnet theorem (early form) is an early version of the fundamental result in differential geometry that links the total curvature of a surface to its topological characteristics, originally developed by Carl Friedrich Gauss.
-
B.
Gauss’s law
Gauss’s law is a fundamental principle of electromagnetism that relates the electric flux through a closed surface to the electric charge enclosed within that surface.
-
C.
Maxwell stress tensor
The Maxwell stress tensor is a mathematical construct in classical electromagnetism that represents how electric and magnetic fields transmit mechanical stresses, such as pressure and tension, through space and matter.
-
D.
Gauss's law for magnetism
Gauss's law for magnetism is the Maxwell equation stating that magnetic monopoles do not exist and that magnetic field lines always form closed loops with zero net flux through any closed surface.
-
E.
Theorema Egregium
Theorema Egregium is Gauss’s celebrated theorem in differential geometry showing that the Gaussian curvature of a surface is an intrinsic property independent of how the surface is embedded in space.
- F. None of above. chosen
Statements (46)
| Predicate | Object |
|---|---|
| instanceOf |
mathematical theorem
ⓘ
theorem in vector calculus ⓘ |
| alsoKnownAs |
Stokes' theorem
ⓘ
surface form:
Kelvin–Stokes theorem
|
| appliesTo |
oriented smooth surfaces
ⓘ
piecewise smooth boundaries ⓘ vector fields with continuous partial derivatives ⓘ |
| assumes |
sufficient smoothness of the surface
ⓘ
sufficient smoothness of the vector field ⓘ |
| category |
theorems in analysis
ⓘ
theorems in calculus ⓘ |
| connects | local properties of a vector field to global circulation ⓘ |
| dimension | three-dimensional Euclidean space (classical form) ⓘ |
| expresses | equivalence between a surface integral and a boundary line integral ⓘ |
| field |
differential geometry
ⓘ
multivariable calculus ⓘ vector calculus ⓘ |
| formalStatement | ∮_{∂S} F · dr = ∬_S (∇ × F) · n dS ⓘ |
| generalizes |
Green's theorem
ⓘ
Fundamental Theorem of Calculus ⓘ
surface form:
fundamental theorem of calculus
|
| hasGeneralization | Stokes' theorem for differential forms on manifolds ⓘ |
| historicalAttribution | communicated by George Gabriel Stokes in the 19th century ⓘ |
| implies | circulation around a closed curve equals flux of curl through spanning surface ⓘ |
| involves |
curl of a vector field
ⓘ
line integral ⓘ oriented boundary curve ⓘ surface integral ⓘ |
| isSpecialCaseOf |
Stokes' theorem
self-linksurface differs
ⓘ
surface form:
generalized Stokes' theorem on manifolds
|
| mathematicalObject | relation between integrals over a manifold and its boundary ⓘ |
| namedAfter |
George Stokes
ⓘ
surface form:
George Gabriel Stokes
|
| relates |
line integral of a vector field around a closed curve
ⓘ
surface integral of the curl of a vector field ⓘ |
| relatesConcept |
circulation
ⓘ
flux ⓘ vorticity ⓘ |
| requires |
compatibility of orientations between surface and boundary
ⓘ
orientation of the boundary curve ⓘ orientation of the surface ⓘ |
| role | bridge between local differential operators and global integrals ⓘ |
| type | integral theorem ⓘ |
| usedFor |
converting difficult line integrals into surface integrals
ⓘ
converting difficult surface integrals into line integrals ⓘ deriving integral forms of Maxwell's equations ⓘ |
| usedIn |
differential forms
ⓘ
electromagnetism ⓘ fluid dynamics ⓘ theory of manifolds ⓘ |
How these facts were elicited
The pipeline generated the facts above by prompting gpt-5.1 with this entity's name + description and the instruction below.
Instruction
You are a knowledge base construction expert. Given a subject entity and a description of it, return factual statements that you know for the subject as a JSON list of dictionaries(triples), where keys must be "subject", "predicate" and "object". The number of facts may be very high, between 25 to 50 or more, for very popular subjects. For less popular subjects, the number of facts can be very low, like 5 or 10. # Requirements - If you don't know the subject at all, return an empty list. - If the subject is not a named entity, return an empty list. - Include at least one triple where predicate is "instanceOf". - Do not get too wordy. - Separate several objects into multiple triples with one object.
Input
Subject: Stokes' theorem Description of subject: Stokes' theorem is a fundamental result in vector calculus that relates the surface integral of the curl of a vector field over a surface to the line integral of the field around the surface’s boundary.
Referenced by (9)
Full triples — surface form annotated when it differs from this entity's canonical label.
subject surface form:
George Gabriel Stokes
subject surface form:
George Gabriel Stokes
this entity surface form:
Stokes' theorem in vector calculus
this entity surface form:
Green's theorem
this entity surface form:
generalized Stokes' theorem on manifolds
this entity surface form:
Kelvin–Stokes theorem
this entity surface form:
Stokes theorem
subject surface form:
George Gabriel Stokes