Divergence theorem

Divergence theorem   (Redirected from Gauss theorem) Jump to navigation Jump to search "Gauss's theorem" redirects here. For Gauss's theorem concerning the electric field, see Gauss's law. "Ostrogradsky theorem" redirects here. For Ostrogradsky's theorem concerning the linear instability of the Hamiltonian associated with a Lagrangian dependent on higher time derivatives than the first, see Ostrogradsky instability. Part of a series of articles about Calculus Fundamental theorem Leibniz integral rule Limits of functionsContinuity Mean value theoremRolle's theorem show Differential show Integral show Series hide Vector GradientDivergenceCurlLaplacianDirectional derivativeIdentities Theorems GradientGreen'sStokes'Divergencegeneralized Stokes show Multivariable show Advanced show Specialized show Miscellaneous vte In vector calculus, the divergence theorem, also known as Gauss's theorem or Ostrogradsky's theorem,[1] is a theorem which relates the flux of a vector field through a closed surface to the divergence of the field in the volume enclosed.

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. Intuitively, it states that "the sum of all sources of the field in a region (with sinks regarded as negative sources) gives the net flux out of the region".

The divergence theorem is an important result for the mathematics of physics and engineering, particularly in electrostatics and fluid dynamics. In these fields, it is usually applied in three dimensions. However, it generalizes to any number of dimensions. In one dimension, it is equivalent to integration by parts. In two dimensions, it is equivalent to Green's theorem.

Contents 1 Explanation using liquid flow 2 Mathematical statement 3 Proofs 3.1 For bounded open subsets of Euclidean space 3.2 For compact Riemannian manifolds with boundary 4 Informal derivation 5 Corollaries 6 Example 7 Applications 7.1 Differential and integral forms of physical laws 7.1.1 Continuity equations 7.2 Inverse-square laws 8 History 9 Worked examples 9.1 Example 1 9.2 Example 2 10 Generalizations 10.1 Multiple dimensions 10.2 Tensor fields 11 See also 12 References 13 External links Explanation using liquid flow Vector fields are often illustrated using the example of the velocity field of a fluid, such as a gas or liquid. A moving liquid has a velocity—a speed and a direction—at each point, which can be represented by a vector, so that the velocity of the liquid at any moment forms a vector field. Consider an imaginary closed surface S inside a body of liquid, enclosing a volume of liquid. The flux of liquid out of the volume is equal to the volume rate of fluid crossing this surface, i.e., the surface integral of the velocity over the surface.

Since liquids are incompressible, the amount of liquid inside a closed volume is constant; if there are no sources or sinks inside the volume then the flux of liquid out of S is zero. If the liquid is moving, it may flow into the volume at some points on the surface S and out of the volume at other points, but the amounts flowing in and out at any moment are equal, so the net flux of liquid out of the volume is zero.

However if a source of liquid is inside the closed surface, such as a pipe through which liquid is introduced, the additional liquid will exert pressure on the surrounding liquid, causing an outward flow in all directions. This will cause a net outward flow through the surface S. The flux outward through S equals the volume rate of flow of fluid into S from the pipe. Similarly if there is a sink or drain inside S, such as a pipe which drains the liquid off, the external pressure of the liquid will cause a velocity throughout the liquid directed inward toward the location of the drain. The volume rate of flow of liquid inward through the surface S equals the rate of liquid removed by the sink.

If there are multiple sources and sinks of liquid inside S, the flux through the surface can be calculated by adding up the volume rate of liquid added by the sources and subtracting the rate of liquid drained off by the sinks. The volume rate of flow of liquid through a source or sink (with the flow through a sink given a negative sign) is equal to the divergence of the velocity field at the pipe mouth, so adding up (integrating) the divergence of the liquid throughout the volume enclosed by S equals the volume rate of flux through S. This is the divergence theorem.[2] The divergence theorem is employed in any conservation law which states that the total volume of all sinks and sources, that is the volume integral of the divergence, is equal to the net flow across the volume's boundary.[3] Mathematical statement A region V bounded by the surface {displaystyle S=partial V} with the surface normal n Suppose V is a subset of {displaystyle mathbb {R} ^{n}} (in the case of n = 3, V represents a volume in three-dimensional space) which is compact and has a piecewise smooth boundary S (also indicated with {displaystyle partial V=S} ). If F is a continuously differentiable vector field defined on a neighborhood of V, then:[4][5] {displaystyle iiint _{V}left(mathbf {nabla } cdot mathbf {F} right),mathrm {d} V=} {displaystyle scriptstyle S} {displaystyle (mathbf {F} cdot mathbf {hat {n}} ),mathrm {d} S.} The left side is a volume integral over the volume V, the right side is the surface integral over the boundary of the volume V. The closed manifold {displaystyle partial V} is oriented by outward-pointing normals, and {displaystyle mathbf {hat {n}} } is the outward pointing unit normal at each point on the boundary {displaystyle partial V} . ( {displaystyle mathrm {d} mathbf {S} } may be used as a shorthand for {displaystyle mathbf {n} mathrm {d} S} .) In terms of the intuitive description above, the left-hand side of the equation represents the total of the sources in the volume V, and the right-hand side represents the total flow across the boundary S.

Proofs For bounded open subsets of Euclidean space We are going to prove the following: Theorem — Let {displaystyle Omega subset mathbb {R} ^{n}} be open and bounded with {displaystyle C^{1}} boundary. If {displaystyle u} is {displaystyle C^{1}} on an open neighborhood {displaystyle O} of {displaystyle {overline {Omega }}} , that is, {displaystyle uin C^{1}(O)} , then for each {displaystyle iin {1,dots ,n}} , {displaystyle int _{Omega }u_{x_{i}},dV=int _{partial Omega }unu _{i},dS,} where {displaystyle nu :partial Omega to mathbb {R} ^{n}} is the outward pointing unit normal vector to {displaystyle partial Omega } . Equivalently, {displaystyle int _{Omega }nabla u,dV=int _{partial Omega }unu ,dS.} Proof of Theorem. [6] (1) The first step is to reduce to the case where {displaystyle uin C_{c}^{1}(mathbb {R} ^{n})} . Pick {displaystyle phi in C_{c}^{infty }(O)} such that {displaystyle phi =1} on {displaystyle {overline {Omega }}} . Note that {displaystyle phi uin C_{c}^{1}(O)subset C_{c}^{1}(mathbb {R} ^{n})} and {displaystyle phi u=u} on {displaystyle {overline {Omega }}} . Hence it suffices to prove the theorem for {displaystyle phi u} . Hence we may assume that {displaystyle uin C_{c}^{1}(mathbb {R} ^{n})} .

(2) Let {displaystyle x_{0}in partial Omega } be arbitrary. The assumption that {displaystyle {overline {Omega }}} has {displaystyle C^{1}} boundary means that there is an open neighborhood {displaystyle U} of {displaystyle x_{0}} in {displaystyle mathbb {R} ^{n}} such that {displaystyle partial Omega cap U} is the graph of a {displaystyle C^{1}} function with {displaystyle Omega cap U} lying on one side of this graph. More precisely, this means that after a translation and rotation of {displaystyle Omega } , there are {displaystyle r>0} and {displaystyle h>0} and a {displaystyle C^{1}} function {displaystyle g:mathbb {R} ^{n-1}to mathbb {R} } , such that with the notation {displaystyle x'=(x_{1},dots ,x_{n-1}),} it holds that {displaystyle U={xin mathbb {R} ^{n}:|x'|

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