A system of skew coordinates is a
curvilinear coordinate system where the
coordinate surfaces
In geometry, a coordinate system is a system that uses one or more numbers, or coordinates, to uniquely determine the position of the points or other geometric elements on a manifold such as Euclidean space. The order of the coordinates is si ...
are not
orthogonal, in contrast to
orthogonal coordinates In mathematics, orthogonal coordinates are defined as a set of ''d'' coordinates q = (''q''1, ''q''2, ..., ''q'd'') in which the coordinate hypersurfaces all meet at right angles (note: superscripts are indices, not exponents). A coordinate su ...
.
Skew coordinates tend to be more complicated to work with compared to orthogonal coordinates since the
metric tensor will have nonzero off-diagonal components, preventing many simplifications in formulas for
tensor algebra
In mathematics, the tensor algebra of a vector space ''V'', denoted ''T''(''V'') or ''T''(''V''), is the algebra of tensors on ''V'' (of any rank) with multiplication being the tensor product. It is the free algebra on ''V'', in the sense of being ...
and
tensor calculus
In mathematics, tensor calculus, tensor analysis, or Ricci calculus is an extension of vector calculus to tensor fields (tensors that may vary over a manifold, e.g. in spacetime).
Developed by Gregorio Ricci-Curbastro and his student Tullio Levi ...
. The nonzero off-diagonal components of the metric tensor are a direct result of the non-orthogonality of the basis vectors of the coordinates, since by definition:
[
]
:
where
is the metric tensor and
the (covariant)
basis vector
In mathematics, a set of vectors in a vector space is called a basis if every element of may be written in a unique way as a finite linear combination of elements of . The coefficients of this linear combination are referred to as components ...
s.
These coordinate systems can be useful if the geometry of a problem fits well into a skewed system. For example, solving
Laplace's equation in a
parallelogram will be easiest when done in appropriately skewed coordinates.
Cartesian coordinates with one skewed axis
The simplest 3D case of a skew coordinate system is a
Cartesian one where one of the axes (say the ''x'' axis) has been bent by some angle
, staying orthogonal to one of the remaining two axes. For this example, the ''x'' axis of a Cartesian coordinate has been bent toward the ''z'' axis by
, remaining orthogonal to the ''y'' axis.
Algebra and useful quantities
Let
,
, and
respectively be unit vectors along the
,
, and
axes. These represent the
covariant basis; computing their dot products gives the
metric tensor:
:
where
:
and
:
which are quantities that will be useful later on.
The contravariant basis is given by
[
:
:
:
The contravariant basis isn't a very convenient one to use, however it shows up in definitions so must be considered. We'll favor writing quantities with respect to the covariant basis.
Since the basis vectors are all constant, vector addition and subtraction will simply be familiar component-wise adding and subtraction. Now, let
:
where the sums indicate summation over all values of the index (in this case, ''i'' = 1, 2, 3). The contravariant and covariant components of these vectors may be related by
:
so that, explicitly,
:
:
:
The ]dot product
In mathematics, the dot product or scalar productThe term ''scalar product'' means literally "product with a scalar as a result". It is also used sometimes for other symmetric bilinear forms, for example in a pseudo-Euclidean space. is an alge ...
in terms of contravariant components is then
:
and in terms of covariant components
:
Calculus
By definition,[
] the gradient
In vector calculus, the gradient of a scalar-valued differentiable function of several variables is the vector field (or vector-valued function) \nabla f whose value at a point p is the "direction and rate of fastest increase". If the gr ...
of a scalar function ''f'' is
:
where are the coordinates ''x'', ''y'', ''z'' indexed. Recognizing this as a vector written in terms of the contravariant basis, it may be rewritten:
:
The divergence
In vector calculus, divergence is a vector operator that operates on a vector field, producing a scalar field giving the quantity of the vector field's source at each point. More technically, the divergence represents the volume density of t ...
of a vector is
:
and of a tensor
:
The Laplacian of ''f'' is
:
and, since the covariant basis is normal and constant, the vector Laplacian
In mathematics, the Laplace operator or Laplacian is a differential operator given by the divergence of the gradient of a scalar function on Euclidean space. It is usually denoted by the symbols \nabla\cdot\nabla, \nabla^2 (where \nabla is the ...
is the same as the componentwise Laplacian of a vector written in terms of the covariant basis.
While both the dot product and gradient are somewhat messy in that they have extra terms (compared to a Cartesian system) the advection operator which combines a dot product with a gradient turns out very simple:
:
which may be applied to both scalar functions and vector functions, componentwise when expressed in the covariant basis.
Finally, the curl of a vector is
:
::
References
{{Reflist
Coordinate systems