Contents, with illustrations of the denotations introduced:
Since the central focus of my studies (for all that I get distracted by mathematics a lot) is theoretical physics, especially General Relativity, I need my notation to provide a practical way of discussing tensor algebra. There are several challenges here: the result is a compromise between familiarity to the orthodox, minimalism in the forms the reader must learn to understand and minimalism in the amount of typing I have to do to keep what I write correct and readable.
In particular, my denotations for non-trivial tensor contraction, multiplication and permutation are related to, but different from, Einstein's. My hope is that the similarities shall be enough to make the result accessible to those familiar with the orthodoxy that has sprung from Einstein, without compromising my other goals. Like Einstein, I use subscripts (denoting gradient, or covector, ranks) and superscripts (for tangent, or vector, ranks), collectively indices, with a convention that repetition of a symbol, within a formula, is only allowed if the symbol is used once each as a subscript and as a superscript, in which case there is an implicit contraction performed (Einstein's summation convention); within either the subscripts or the superscripts (independently) square brackets denote antisymmetrisation while parentheses denote symmetrisation. However: the non-repeated indices in any expression shall always be (or stand for) the members of some natural number; the metric is always explicitly given, there is no raising/lowering convention; and an expression using indices is actually a denotation for a tensor quantity, not one of its components with respect to some basis or co-ordinates. When I want co-ordinates, I'll introduce a basis and its dual, contract these with tensors and thus obtain the co-ordinates as scalar fields.
All of the standard notational forms
of arithmetic denotations carry over into the
tensor algebra, where applicable. [For the meaning of rank
and other
denotations used in the explanations, see the next section.]
In particular,
(both binary and unary) apply to tensors just the same as for scalars and
vectors, with the constraint that tensor algebra always demands that
the two operands of a binary additive operator be of the same type
, which
generally means tensors of the same rank.
are used in a specialized way (see below) consistent with the general forms.
Likewise, the use of pointwise extension and all denotations for relations, mappings and collections retain their standard forms.
The letters d, D and ∂ are given special meaning: in particular, although they stand for operators, I also prefix other (usually single-letter) names with them to denote the affect of applying the operator to, or composing it after, the value denoted by the unprefixed name. More precisely, where the following denotational forms are instantiated, if name is: a simple name; or is an instance of the function invocation template, relation ( [ early , …] last ), in which a simple name is used for relation; the d, D or ∂ may simply be prefixed to that name; otherwise, name must begin with ( and end with ), subject to the usual enclosure-nesting rules.
This is meaningful when (N: name :M) is a mapping between smooth manifolds: it denotes a linear map from M's tangents, at each point p of (: name |), to N's tangents at name(p); its transpose is thus a linear map from N's gradients at name(p) to M's gradients at p; and it induces a chain of mappings from the alternating algebra of N to that of M.
This is known as partial differentiation: it has meaning in two cases, each involving a chart-forming family of scalar fields; coord either is this family or is a member of it; in this last case, context must make clear which family this is. Each member of such a family can be construed as a co-ordinate in a description of a region of the smooth manifold and the partial differential operator denotes differentiation with respect to these co-ordinates. The comments above relating to the text used to match name, in the denotations above, apply equally to expr and coord in this denotational form. (Thus dcoord is meaningful.) In each case, expr must be a tensor field or a family of tensor fields (so Dexpr is meaningful, in so far as D is).
In the simplest case, where expr is a scalar field and coord is one scalar field of a chart-forming family, ({scalar fields}: x :), the gradients dx form a basis of the gradient bundle at each point in some neighbourhood and we can induce a dual basis p of the tangent bundle at each such point. Let q be the member of the family p that corresponds to dcoord, so that q·dcoord = 1 while each p(i) other than q has p(i)·dcoord = 0 and q·dx(i) = 0 (with x(i) not being coord, since p(i) isn't q, which corresponds to dcoord). Then ∂expr / ∂coord simply stands for q·dexpr, a scalar field representing the rate of variation of expr as coord varies, with the other co-ordinates in its family held constant.
When expr is a tensor field and coord is one scalar field of a chart-forming family, as before, with gradients dx forming a basis with dual p, these bases induce a basis of each tensor rank on the smooth manifold, using which we can express expr in terms of its components with respect to the induced basis of its rank. Each such component is a scalar field, to which we can apply the simplest case above and reconstitute a tensor by using the results as components of a new tensor. Thus, if c is the induced basis of expr's rank and e is the dual basis, we have a family h of scalar fields defined by h(i) = e(i)·expr, so that expr = sum(: h(i).c(i) ←i :). With q being the tangent field dual to coord's member of dx, as before, ∂expr / ∂coord stands for sum(: q·h(i).c(i) ←i :). Applying this to the special case of the scalar fields, one-dimensional at each point of the manifold, the induced basis doesn't depend on dx or p, the scalar field expr is its own only component with respect to the basis whose sole member is the constant scalar field ({1}::) and we recover exactly the same meaning as given above.
As an alternative use of this
denotation, coord may be the chart-forming family
of scalar fields; in this
case, ∂expr
/ ∂coord stands for sum(:
dcoord(i)×∂expr
/ ∂coord(i) ←i :). In this case,
(: ∂expr
/ ∂coord
←expr :) is the differential operator induced
by using the chart implied by coord to
represent expr in a vector space, differentiating
the resulting tensor using the usual differential operator of that vector space
and then mapping the resulting derivative back to the manifold via the chart.
This is known as the co-ordinate differential operator
associated
with coord.
In the special case of a one-dimensional manifold, a single (nowhere stationary) scalar field can be construed as a chart-forming family; and all members of a one-dimensional space (e.g. the space of gradients or tangents at any point of such a manifold) are multiples of any non-sero member of that space, making division by such a member meaningful in such a space. In this case, wherever a scalar field x has non-zero gradient, dφ/dx is actually meaningful; and coincides (give or take a tensor factor of an identity linear map) with ∂φ/∂x, for any tensor (or scalar) field, or list of such, φ.
To give a concrete example of the comments on name, above: if φ is a scalar field, its gradient is officially d(φ) and equal to D(φ) but we can write these as dφ and Dφ respectively (when context has introduced a general differential operator, D). In the case of a list, x, of scalar fields, dx is a matching list of gradient fields: this may also be written d&on;x = (: d(x(i)) ←i :) and, again, is necessarily equal to Dx = D&on;x. This action of D can likewise be applied to fields, or lists of fields, of any rank, not just to scalar fields.
Contraction, tensor multiplication and the antisymmetric form of the latter.
a / b means that c for which a = c * b; likewise, b \ a means that c for which a = b * c (the other way round). In each case there may be ambiguity when b is singular; in particular, neither is defined in terms of an inverse for b – precisely to allow them to have meaning when b is singular but there is some such c – but when b does have an inverse, q, a / b is just a·q and b \ a is q·a; and q can be written as (1/b) or (b\1). When the result of such a division is contracted with something on the denominator's side, e·(b\a) or (a/b)·e, I write the result using the other division operator, e/b\a or a/b\e respectively.
Introduce τ as both a function and a notational extension for the trace-permute operator. As a function, its first parameter is a list S of linear spaces, its second is a list I of naturals and symbols; τ(S, I) is then a linear map from (any of the linear spaces canonically isomorphic to) the tensor product of the entries in S to a space derived therefrom by permuting the entries in S corresponding to naturals in I, omitting those corresponding to symbols, and taking the tensor product of the resulting list of linear spaces. Each symbol, other than a natural number (or expression denoting a natural number), that appears in I must appear exactly twice in positions that correspond to mutually dual entries in S. The naturals in I must all be distinct and should be exactly all of the members of some natural (i.e. they should form an initial sub-set of {naturals}). As a notational extension τ[…] is a short-hand for the above with the list […] as I and S inferred from context's characterisation of the parameter passed to this linear map.
Contraction and tensor multiplication can be combined with permutation of the ranks of the resulting tensor and with tracing out some of its ranks – indeed, contraction is just tensor multiplication combined with such a tracing and the antisymmetric version of the tensor multiplication is just the tensor multiplication combined with an averaging over permutations. Throwing in some differentiation can complicate the matter a little more, too – particularly in the antisymmetrised case, where it begets an extension of d, the d^ operator, on tensors whose rank has only gradient factors.
In subscript/superscript notations, [ square brackets ] or ( parentheses ) may be used around some indices to indicate symmetrisation or antisymmetrisation (TODO: remind myself of orthodox usage and match it). There is also an antisymmetric multiplication, on a smooth manifold (Jacobi bracket ?), that uses [ square brackets ] to combine two vector fields to make a third. Take care, when defining, to avoid confusion with the use of square brackets in lists, in τ and in other extensions; the crucial difference is their use in subscripts and superscripts.
Written by Eddy.