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...

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

\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...

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...

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...

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...

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...

specified divergence and a specified curl, and if it also vanishes at infinity, it is uniquely specified by its divergence and curl. This theorem is of great...

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

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

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

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

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...

} Integral form: by the integral remainder form of Taylor's Theorem, a Bregman divergence can be written as the integral of the Hessian of F {\displaystyle...

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

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...

Stokes' theorem simultaneously generalizes the three theorems of vector calculus: the divergence theorem, Green's theorem, and the Kelvin-Stokes theorem. The...

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

amplitudes in quantum scattering theory. Divergence theorem Gradient theorem Methods of contour integration Nachbin's theorem Line element Surface integral Volume...

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...

\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...

mathematical statistics, the Kullback–Leibler (KL) divergence (also called relative entropy and I-divergence), denoted D KL ( P ∥ Q ) {\displaystyle D_{\text{KL}}(P\parallel...

} 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...

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

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

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

mechanics, the equipartition theorem relates the temperature of a system to its average energies. The equipartition theorem is also known as the law of...

by considering any polyhedron with vector sides and invoking the Divergence Theorem. Half-side formula Law of sines Law of tangents Law of cotangents...

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...

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