Divergence theorem

Gauss's theoremGauss theoremdivergent-freeGaussGauss divergence theoremGauss' divergence theoremGauss' theoremGauss's divergence theoremGauss–Ostrogradsky theoremGauss’ theorem
In vector calculus, the divergence theorem, also known as Gauss's theorem or Ostrogradsky's theorem, is a result that relates the flux of a vector field through a closed surface to the divergence of the field in the volume enclosed.wikipedia
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Fluid dynamics

hydrodynamicshydrodynamicfluid flow
The divergence theorem is an important result for the mathematics of physics and engineering, in particular in electrostatics and fluid dynamics.
The differential form of the continuity equation is, by the divergence theorem:

Vector calculus

vector analysisvectorvector algebra
In vector calculus, the divergence theorem, also known as Gauss's theorem or Ostrogradsky's theorem, is a result that relates the flux of a vector field through a closed surface to the divergence of the field in the volume enclosed.

Vector field

vector fieldsvectorgradient flow
In vector calculus, the divergence theorem, also known as Gauss's theorem or Ostrogradsky's theorem, is a result that relates the flux of a vector field through a closed surface to the divergence of the field in the volume enclosed. Vector fields are often illustrated using the example of the velocity field of a fluid, such as a liquid.
The divergence at a point represents the degree to which a small volume around the point is a source or a sink for the vector flow, a result which is made precise by the divergence theorem.

Flux

flux densityion fluxflow
In vector calculus, the divergence theorem, also known as Gauss's theorem or Ostrogradsky's theorem, is a result that relates the flux of a vector field through a closed surface to the divergence of the field in the volume enclosed.
The divergence theorem states that the net outflux through a closed surface, in other words the net outflux from a 3D region, is found by adding the local net outflow from each point in the region (which is expressed by the divergence).

Electrostatics

electrostaticelectrostatic repulsionelectrostatic interactions
The divergence theorem is an important result for the mathematics of physics and engineering, in particular in electrostatics and fluid dynamics. Three examples are Gauss's law (in electrostatics), Gauss's law for magnetism, and Gauss's law for gravity.
The divergence theorem allows Gauss's Law to be written in differential form:

Green's identities

Green's second identityGreen's first identityGreen's formula
, in which case the theorem is the basis for Green's identities.
This identity is derived from the divergence theorem applied to the vector field

Surface integral

surface elementsurface integralsarea element
More precisely, the divergence theorem states that the surface integral of a vector field over a closed surface, which is called the flux through the surface, is equal to the volume integral of the divergence over the region inside the surface.
Various useful results for surface integrals can be derived using differential geometry and vector calculus, such as the divergence theorem, and its generalization, Stokes' theorem.

Volume integral

integral over spacevolumecomputing the volume
More precisely, the divergence theorem states that the surface integral of a vector field over a closed surface, which is called the flux through the surface, is equal to the volume integral of the divergence over the region inside the surface.

Gauss's law for magnetism

Gauss' law for magnetismGauss's lawfor magnetism
Three examples are Gauss's law (in electrostatics), Gauss's law for magnetism, and Gauss's law for gravity.
These forms are equivalent due to the divergence theorem.

Fundamental theorem of calculus

First Fundamental Theorem Of Calculusfundamental theorem of real calculusfundamental theorem of the calculus
In one dimension, it is equivalent to the fundamental theorem of calculus.
The most familiar extensions of the fundamental theorem of calculus in higher dimensions are the divergence theorem and the gradient theorem.

Continuity equation

Mass continuitycontinuityConservation of mass
Continuity equations offer more examples of laws with both differential and integral forms, related to each other by the divergence theorem.
By the divergence theorem, a general continuity equation can also be written in a "differential form":

Divergence

divergence operatorconverge or divergeConvergence
More precisely, the divergence theorem states that the surface integral of a vector field over a closed surface, which is called the flux through the surface, is equal to the volume integral of the divergence over the region inside the surface.

Green's theorem

Cauchy–GreenGreen–Stokes formula
In two dimensions, it is equivalent to Green's theorem.
Considering only two-dimensional vector fields, Green's theorem is equivalent to the two-dimensional version of the divergence theorem:

Three-dimensional space

three-dimensional3Dthree dimensions
represents a volume in three-dimensional space) which is compact and has a piecewise smooth boundary S (also indicated with
is a continuously differentiable vector field defined on a neighborhood of V, then the divergence theorem says:

Gauss's law

Gauss' lawGauss lawGauss
Three examples are Gauss's law (in electrostatics), Gauss's law for magnetism, and Gauss's law for gravity.
The law can be expressed mathematically using vector calculus in integral form and differential form; both are equivalent since they are related by the divergence theorem, also called Gauss's theorem.

Vector calculus identities

vector calculus identityvector identitiesidentity
vector identities).

Gauss's law for gravity

Gauss's lawfor gravityGauss' law for gravity
Three examples are Gauss's law (in electrostatics), Gauss's law for magnetism, and Gauss's law for gravity.
The divergence theorem states:

Stokes' theorem

Stokes theoremStokes's theoremKelvin–Stokes theorem
One can use the general Stokes' Theorem to equate the n-dimensional volume integral of the divergence of a vector field
This classical statement, along with the classical divergence theorem, the fundamental theorem of calculus, and Green's theorem are simply special cases of the general formulation stated above.

Mikhail Ostrogradsky

Mikhail Vasilievich OstrogradskyOstrogradskyMikhail Ostrogradski
In 1826, Ostrogradsky gave the first general proof of the divergence theorem, which was discovered by Lagrange in 1762.

Surface (mathematics)

surfacesurfaces2-dimensional shape
In vector calculus, the divergence theorem, also known as Gauss's theorem or Ostrogradsky's theorem, is a result that relates the flux of a vector field through a closed surface to the divergence of the field in the volume enclosed.

Physics

physicistphysicalphysicists
The divergence theorem is an important result for the mathematics of physics and engineering, in particular in electrostatics and fluid dynamics.

Engineering

engineerengineersengineered
The divergence theorem is an important result for the mathematics of physics and engineering, in particular in electrostatics and fluid dynamics.

Velocity

velocitiesvelocity vectorlinear velocity
Vector fields are often illustrated using the example of the velocity field of a fluid, such as a liquid.

Fluid

fluidsanalysis of fluidsenergy fluids
Vector fields are often illustrated using the example of the velocity field of a fluid, such as a liquid.

Vector (mathematics and physics)

vectorvectorsvectorial
A moving liquid has a velocity, a speed and direction, at each point which can be represented by a vector, so the velocity of the liquid forms a vector field.