Divergence theorem

The mathematics is completed by one additional theorem relating the divergence of the gradient of the electrical potential at a given point to the charge density at that point through Poisson s equation... 

This means that the discrete solution nearly conserves the Hamiltonian H and, thus, conserves H up to 0 t ). If H is analytic, then the truncation index N in (2) is arbitrary. In general, however, the above formal series diverges as jV —> 00. The term exponentially close may be specified by the following theorem. 

Analogous to vector operations the tensorial form of the divergence theorem is written as... 

The convergence or divergence of an infinite series is unaffected by the removal of a finite number of finite terms. This is a trivial theorem but useful to remember, especially when using the comparison test to be described in the subsection Tests for Convergence and Divergence. ... 

From (A.48), (A.49), and (A. 104) s grad v = s k. Integrating this equation over the region and using the divergence theorem, it is found that... 

Invoking the divergence theorem from calculus we can rewrite this as... 

It is now necessary to derive analogous relations for the divergence of a vector, viz. V - A. The calculation can be carried out in at least two ways. The direct analytic approach is long, but does not involve any methods other than those of vector algebra. Otherwise, it is necessary to develop the diver-geoce (Gauss s) theorem, after which the desired result is easily obtained (see Appendix VI). In either case it is given by... 

The divergence theorem has many applications. A very important specified by Eq. (5-66), one of the four equations of Maxwell. It is i... 

If a series is conditionally convergent, its sums can be made to have any arbitrary value by a suitable rearrangement of the series it can in fact be made divergent or oscillatory (Riemanns theorem). 

THEOREM 6.1 It is not partially decidable whether a Turing machine diverges on the blank initial tape. 

THEOREM 6.3 The following problems are not partially decidable for two-tape oneway deterministic finite state acceptors M even if M is guaranteed not to diverge on (w,w) for w ultimately periodic ... 

The other way to calculate the volume inside the triangulated surface is to use the Ostrogradski-Gauss theorem. It relates the surface integral from a vector field j to a volume integral from its divergence ... 

The divergence of the flux vector is therefore the net rate of accumulation of the quantity which is transported in and out of the volume element dK This can be integrated over an arbitrary volume Cl limited by the surface I to give the divergence theorem of Gauss... 

Exchanging the order of derivation and integration on the left-hand side integral and using the divergence theorem on the right-hand side, we get... 

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