Helmholtz decomposition contradictions
Alexandr Kozachok
Kiev, Ukraine, a-kozachok1@yandex.ua
http://continuum-paradoxes.narod.ru/
Abstract. In this paper we can read the simple proofs that Helmholtz decomposition
(Fundamental theorem of vector calculus) is wrong and requires major revision. New formula for
Fundamental theorem completely corresponds to Navier–Stokes equations and Lame (Navier)
equations (elastic media).
Key Words and Phrases: vector field, gradient, Navier – Stokes equations, Lame equations,
Helmholtz decomposition, antisymmetric tensor
2010 Mathematics Subject Classifications: 35Qxx, 35Q30, 76D05
1. Introduction.
The Fundamental theorem of vector calculus
(so called Helmholtz
decomposition) states that any sufficiently smooth, rapidly decaying vector field in
three dimensions F can be constructed with the sum of an irrotational (curl-free)
vector field and a solenoidal (divergence-free) vector field (so called scalar
potential  and a vector potential A )
F   grad   rot A  F  grad   rot A .
(1)
As well known from this university textbook [1, p. 15] « … under co-ordinate
change the gradient of function transforms differently from a vector ». Therefore
the gradient of scalar function does not construct a true vector field.
The next unpleasant things we can see for such well-known classical rule. In
mathematics and physics the rot (curl) is an operation which takes any vector field
A and produces another vector field rot A . However it is known that rot A is
equivalent to Antisymmetric Tensor [2, p. 183]. Therefore under co-ordinate
change the Antisymmetric Tensor transforms differently from true vector. The
author of textbook [3, p. 50] paid attention that «… pseudo-vector … from the point
of view of its vector product on other true vector is equivalent to antisymmetric
tensor, but as the vector cannot be equal to tensor ».
Hence the theory requiring (1) must be false.
Counterexample. As we well know the divergence of any vector field on
Euclidean space is a scalar field. Therefore as an example let's calculate the
divergence of an acceleration vector divu . In expanded form the components of
acceleration vector u can be written so [2, p. 39]
du x u x
u
u
u

 ux x  u y x  uz x ,
dt
t
x
y
z
du
u
u
u
u
u y  y  y  ux y  u y y  uz y ,
dt
t
x
y
z
du
u
u
u
u
uz  z  z  ux z  u y z  uz z .
dt
t
x
y
z
ux 
After taking of an operator div we have
div u 
u x u y u z 





 div u  u x div u  u y div u  u z div u 
x
y
z t
x
y
z
 u  2  u y  2  u  2
 u x u y u y u z u x u z  
x
z
 


.
 
  2
 
z y
z x  
 y x
 x   y   z 
(2)
This formula can be written as
div u 
d
div u  (div u ) 2
dt
(3)
if and only if such equality is true
 u x u y u y u z u x u z 
 u x   u y   u z 



 (div u ) 2 .

 
  2


z y
z x 
 x   y   z 
 y x
2
2
2
(4)
The realization of equality (4) requires another one:
ux u y u y uz ux u z u x u y u y u z u x u z





.
y x
z y
z x
x y
y z
x z
(5)
Note that equality (4) can make sense only if rot u  0 . In the case rot u  0 all
terms (in brackets) of left-hand side (4) are positive and div u  0 is impossible.
Thus the requirements rot u  0, div u  0 for true vector field
are
inconsistent. As we well know u  grad ,  2  0 , if rot u  0, div u  0 .
Therefore the vector fields cannot be constructed out of scalar fields using the
gradient operator u  grad  . Therefore so-called Laplacian field is not true
vector field.
2. Helmholtz decomposition major revision
For elimination of above contradictions the Fundamental theorem of vector
calculus can be written as follows:
F  grad   rot rot A .
(6)
This formula completely corresponds to Navier–Stokes equations (NSE) for
incompressible fluids. As we well know
 2u  grad div u  rotrot u .
Therefore we obtain (if div u  0 )
u   grad p   2u  F  ( F  u )  grad p  rotrot u .
(7)
Here, F  F1  F2  ... - vectors sum of a given, externally applied forces
r
(e.g. gravity F1 , magnetic F2 and other), p - pressure (scalar function), u& r& r&
velocity vector, u&
 du / dt - acceleration vector,  - density (const),  - viscosity
2
(const),  - Laplace operator.
Equations (6) and (7) coincide. Hence there is no reason to say that the theory
requiring (6) must be false. From NSE we can see that this sum
 grad p   2u  (grad p  rotrot u )
must construct true vector field.
Note that we will obtain formula (6) also after similar transformation of the
Navier–Stokes for a compressible fluid and after transformation of the Lame
equations for an elastic media. According [4, p. 300] the Lame equations can be
written so
u  (  2)grad div u   rot rot u  F .
(8)
After minor transformation we can see that forms of equations (6), (7) and below
equations completely coincide
( F  u )  graddiv  (  2)u   rot rot u .
r
Here u - displacement vector,  ,  - Lame constants.
3. Revision consequences
The vector fields cannot be constructed out of scalar fields using the gradient
operator. Therefore so-called Laplacian field is not a true vector field. Thus, the
requirements rot u  0,div u  0 are incompatible for true vector fields. This result
confirms the proof about impossibility of irrotational velocity field in this old
university textbook [5, p. 100-101].
Other unpleasant things we can see for many well-known classical equations.
For example the Euler equations (fluid dynamics) can be written as follows
 grad p  (div u  F ) .
Note
that
such
equations
have
no
sense
as
exact
vector
equations because left-hand side gradp is not the true vector. As we well know
this theorem is of great importance in electrostatics (Maxwell's equations).
Therefore we can continue a list of similar incorrect mathematical
physics equations.
4. Conclusion
From this brief note follows that Helmholtz decomposition is total wrong and
demands major revision. For elimination of contradictions the Fundamental
theorem of vector calculus can be written as follows from (6). This formula
completely corresponds to transformed Navier–Stokes equations for compressible
and incompressible fluids and the Lame equations for an elastic media. Discussion
of this problem we can see here [6].
References
[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, Eng). ISBN 0387976639.
[2] Sedov L.I. (1970) Continuum Mechanics, v. 1. Textbook, Science, Moscow (p. 183, Rus).
ISBN-13:978-9971507282(Eng.)
[3] Lojtsjansky L.G. (1970). Fluid and Gas Mechanics. Manual, Science, Moscow (p. 50, Rus)
[4] Truesdell C. (1972). A First Course in Rational Continuum Mechanics. The Jons Hopkins
University. Baltimore, Maryland (Eng), (p 300, Rus)
[5] Slezkin N.A. (1955) Dynamics of the Viscous Incompressible Fluids. Textbook, State
publishing house, Moscow (p. 100-101, Rus)
[6] Helmholtz decomposition is wrong
http://en.wikipedia.org/wiki/Wikipedia_talk:WikiProject_Mathematics/Archive/2012/Mar#Helm
holtz_decomposition_is_wrong