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...

as Gauss's flux theorem or sometimes Gauss's theorem, is one of Maxwell's equations. It is an application of the divergence theorem, and it relates the...

In vector calculus, divergence is a vector operator that operates on a vector field, producing a scalar field giving the quantity of the vector field's...

\mathbb {R} ^{3},} and the divergence theorem is the case of a volume in R 3 . {\displaystyle \mathbb {R} ^{3}.} Hence, the theorem is sometimes referred to...

theorem of calculus. In three dimensions, it is equivalent to the divergence theorem. Let C be a positively oriented, piecewise smooth, simple closed curve...

independent combinations of the Lagrangian expressions are divergences. The main idea behind Noether's theorem is most easily illustrated by a system with one coordinate...

Ostrogradsky gave the first general proof of the divergence theorem, which was discovered by Lagrange in 1762. This theorem may be expressed using Ostrogradsky's...

In physics and mathematics, the Helmholtz decomposition theorem or the fundamental theorem of vector calculus states that certain differentiable vector...

mathematician George Green, who discovered Green's theorem. This identity is derived from the divergence theorem applied to the vector field F = ψ ∇φ while using...

extensions of the fundamental theorem of calculus in higher dimensions are the divergence theorem and the gradient theorem. One of the most powerful generalizations...

differentiable vector field defined on a neighborhood of V, then the divergence theorem says: ∭ V ( ∇ ⋅ F ) d V = {\displaystyle \iiint _{V}\left(\mathbf...

\mathbf {X} )}} In vector calculus, the divergence theorem, also known as Gauss's theorem or Ostrogradsky's theorem, is a result that relates the flow (that...

This is Theorem 7.25 in. Applying this theorem to KL-divergence yields the Donsker–Varadhan representation. Attempting to apply this theorem to the general...

invoked when discussing the continuity equation, the divergence of the field and the divergence theorem. The analogy sometimes includes swirls and saddles...

\cdot d\mathbf {S} \ =\ \iiint _{V}\nabla \cdot \mathbf {A} \,dV} (divergence theorem) ∂ V {\displaystyle \scriptstyle \partial V} A × d S   =   − ∭ V ∇...

differential equation that contain a divergence term are converted to surface integrals, using the divergence theorem. These terms are then evaluated as...

horn Jacobian matrix Hessian matrix Curvature Green's theorem Divergence theorem Stokes' theorem Vector Calculus Infinite series Maclaurin series, Taylor...

corresponding theorems which generalize the fundamental theorem of calculus to higher dimensions: In two dimensions, the divergence and curl theorems reduce...

} where n is the outward unit normal to the boundary of V. By the divergence theorem, ∫ V div ⁡ ∇ u d V = ∫ S ∇ u ⋅ n d S = 0. {\displaystyle \int _{V}\operatorname...

this property is to say that the field has no sources or sinks. The divergence theorem gives an equivalent integral definition of a solenoidal field; namely...

forms of Gauss's law for gravity are mathematically equivalent. The divergence theorem states: ∮ ∂ V g ⋅ d A = ∫ V ∇ ⋅ g d V {\displaystyle \oint _{\partial...

geometry and vector calculus, such as the divergence theorem, magnetic flux, and its generalization, Stokes' theorem. Let us notice that we defined the surface...

form and an integral form. These forms are equivalent due to the divergence theorem. The name "Gauss's law for magnetism" is not universally used. The...

consequence of the Gauss divergence theorem and the Kelvin–Stokes theorem. According to the (purely mathematical) Gauss divergence theorem, the electric flux...

phase space. A proof of Liouville's theorem uses the n-dimensional divergence theorem. The proof is based on the fact that the evolution of ρ {\displaystyle...

{P}{2}}\oint {\hat {\mathbf {n} }}\cdot \mathbf {r} \,dA.} By the divergence theorem, ∮ n ^ ⋅ r d A = ∫ ∇ ⋅ r d V = 3 ∫ d V = 3 V {\textstyle \oint {\hat...

interval, its boundary being its extrema, then the divergence theorem reduces to the fundamental theorem of calculus: ∫ x m x m + 1 F ( x ′ ) d x ′ = 0 ...

calculus, the Reynolds transport theorem (also known as the Leibniz–Reynolds transport theorem), or simply the Reynolds theorem, named after Osborne Reynolds...

the electric field, and ⋅ is the dot product). Using the divergence theorem, Poynting's theorem can also be written in integral form: − d d t ∫ V u   d...

complex volume integrals, and higher order integrals), we must use the divergence theorem. For now, let ∇ ⋅ {\displaystyle \nabla \cdot } be interchangeable...