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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]
As the name implies, the divergence is a (local) measure of the degree to which vectors in the field diverge. The divergence of a tensor field of non-zero order k is written as =, a contraction of a tensor field of order k − 1. Specifically, the divergence of a vector is a scalar.
In mathematics, the comparison test, sometimes called the direct comparison test to distinguish it from similar related tests (especially the limit comparison test), provides a way of deducing whether an infinite series or an improper integral converges or diverges by comparing the series or integral to one whose convergence properties are known.
Divergence; Divergence theorem; Double integral; Equipotential surface; Euler's theorem on homogeneous functions; Exterior derivative; Flux; Frenet–Serret formulas; Gauss's law; Gradient; Green's theorem; Green's identities; Harmonic function; Helmholtz decomposition; Hessian matrix; Hodge star operator; Inverse function theorem; Irrotational ...
More precisely, the divergence theorem states that the outward flux of a vector field through a closed 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 minus the sum of all sinks gives the net flow out of a region .
Vector calculus or vector analysis is a branch of mathematics concerned with the differentiation and integration of vector fields, primarily in three-dimensional Euclidean space, . [1] The term vector calculus is sometimes used as a synonym for the broader subject of multivariable calculus, which spans vector calculus as well as partial differentiation and multiple integration.
We assume that is well behaved and that we can reverse the order of integration. Also, recall that flow is normal to the unit area of the cell. Now, since in one dimension , we can apply the divergence theorem, i.e. =, and substitute for the volume integral of the divergence with the values of () evaluated at the cell surface (edges / and + /) of the finite volume as follows:
Plugging this into the divergence theorem produces Green's theorem, = ^. Suppose that the linear differential operator L is the Laplacian , ∇ 2 , and that there is a Green's function G for the Laplacian.