The
Fundamental theorem of vector calculus, also known as
Helmholtz decomposition http://en.wikipedia.org/wiki/Helmholtz_decomposition , states that any sufficiently smooth, rapidly decaying vector field in three dimensions
can be constructed with the sum of an irrotational (curl-free) vector field and a solenoidal (divergence-free) vector field (scalar potential
and a vector potential
(1)
However, the gradient of scalar function does not form the vector field. As well known from textbook [1, p. 15] «
… under co-ordinate change the gradient of function transforms differently from a vector »: hence the theory requiring (1) must be false. The next unpleasant things we can see for such well-known classical rules. In mathematics and physics the
(or curl) is an operation which takes the vector field
and produces another vector field
. However it is well-known that
is an Antisymmetric Tensor . Therefore under co-ordinate change the tensor
transforms differently from a true vector. For elimination of these contradictions the
Fundamental theorem of vector calculus can be written as follows:
(2)
This formula completely corresponds to transformed
Navier–Stokes equations (NSE) for incompressible fluids (
)
(3)
Here,
- vectors sum of a given, externally applied forces (e.g. gravity
, magnetic
and other),
- pressure (scalar function),
- velocity vector,
- acceleration vector,
- density,
- viscosity,
- Laplace operator.
Equations (3) and (2) are consistent. Hence there is no reason to say that the theory requiring (2) must be false. As we can see from NSE the sum
forms the vector field.
Note that we will receive the formula (2) also after similar transformation of the
Navier–Stokes for a compressible fluid and after transformation of the
Lame equations for an elastic media.
From this brief note follows that Helmholtz decomposition is wrong and demands major revision.
Therefore I ask to formulate own position on this problem.
1. Dubrovin, B. A.; Fomenko, A. T.; Novikov, Sergeĭ Petrovich (1992). Modern Geometry--methods and Applications: The geometry of surfaces, transformation groups, and fields (2nd ed.). Springer. (p. 15). ISBN 0387976639
http://books.google.com/books?id=FC0QFlx12pwC&pg=PA15P.S. Я надеюсь, что участники Международного Научного Форума и читатели Рунета простят меня за изложение темы только на одном официальном языке Форума. Изложенная информация вряд ли требует перевода. В случае необходимости такой перевод можно выполнить.