# 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.