above for the
notation [!j].)
Given a scalar function f(y), the gradient df = f[!i], a 1-form.
Define a related function g(x) = f(y(x)).
What is the gradient of g? By the chain rule:
dg[j] = sum (i = 1..n) f[!i] J[i][j].
A ``covariant tensor'' is an object whose coordinate representation is
transformed similarly to a gradient (going from Y to X), by matrix
multiplying it (going from Y to X) by the Jacobian of the map function. A
``contravariant tensor'' is transformed by multiplying it by the inverse of
the Jacobian; vectors are typical.
Summation Convention
When covariant and contravariant objects appear in the same term
they almost always are summed as in the above expression for dg[j].
When the same subscript letter is repeated in a covariant and a
contravariant position, summation is assumed but the sum symbol is
not actually written.
Christoffel Symbols
How do you transform the derivative of a tensor? It isn't a tensor;
the derivatives of the Jacobian get mixed in. However, the ordinary
derivative of a tensor can be separated into two parts; one is a tensor
called the ``covariant derivative'' and the other is a matrix-type product
of the original object with a set of functions called the ``Christoffel
symbols''. Let g[i][j] represent the metric tensor
and g'[i][j] represent its matrix inverse. The definitions of the two kinds
of Christoffel symbols are:
[ij,k] = 1/2 (g[i][k][!j] + g[j][k][!i] - g[i][j][!k])
{ k }
{i j} = g'[k][m] [ij,m]
In this document the notation C[i][j][k] is used in place of [ij,k].
The ordinary derivative of a 1-form q, of which a
gradient is a typical example, is split up like this:
q[i][!j] = q[i][/j] + q[m] g'[m][k] [ij,k]
More normally one sees the covariant derivative defined in terms of the
ordinary derivative (which is calculated by ordinary means):
q[i][/j] = q[i][!j] - q[m] g'[m][k] [ij,k]
The analogous formula if q is contravariant is:
q[i][/j] = q[i][!j] + q[m] g'[i][k] [mj,k]
Vector
Consider a curve c(t) on the manifold. For any function f(p) on the
manifold, you can get d/dt f(c(t)), the directional derivative.
Focus on one point q at a time. Frequently different curves through q
will produce the same directional derivative, no matter which function is
differentiated -- if the curves are tangent.
Thus make equivalence classes among curves (through q).
These equivalence classes are called vectors (at q).
They form a vector space.
p-Vector
An ordered list of p (in 0 to n)
vectors is referred to as a p-vector; it defines an element of area (p =
n-1) or volume (p = n). The vectors in a p-vector are combined
antisymmetrically with the Grassman product such that v^w = -w^v.
The ``cross'' product of vectors is the Grassman product in 3 dimensions.
Form
A linear functional on p-vectors is called a p-form. While the value of
a form (when a vector is fed to it) is typically a scalar, a composite
object (form or vector or matrix) could also be the value. Forms can also
be combined with the Grassman product. Any function f (defined at point q)
defines a 1-form at q, the gradient of f, wherein the argument vector takes
the directional derivative of the function, and the coordinate functions
define a basis of all the 1-forms, notated dx[i] (i = 1..n). For
calculation, suppose the components of a form A are A[i] and a vector v is
v[i]. Then to compute, the functional value A(v) = sum (i = 1..n)
A[i]v[i]. This is often abbreviated A.v
Metric Dual (represented by *)
For a p-form A, suppose you take its metric product with another p-form
B. Point by point, this is a linear functional of p-forms, that is, it is
a p-vector. Suppose you wished to integrate the function value relative to
the volume element in the space. An alternative way to express the kernel
of the integral would be to obtain a (n-p)-form *A and take its Grassman
product with B. This is the metric dual of A. It has a simple form: it
has the same components (some changing signs) multiplied by G, the positive
square root of the determinant of the metric tensor. (It is unclear what
the metric dual would be if the determinant were negative, for an illegal
``distance function''.) On an even
dimension space, ** (metric dual twice) reverses the sign of odd degree
forms but is an identity transformation for even degree forms. In odd
dimensions it also either is the identity or reverses the sign, but it's
more complicated to determine for which degrees this happens.
Current
A linear functional on fields of forms. One example of a current is a
shape (volume, area, etc.), the form being integrated over it. The various
differentiation operators are also currents. It's a
theorem that every current T can be realized as the limit of a sequence of
C-infinity forms A(j), where (B being the argument form field) T(B) =
integral of *A(j) ^ B.
ð (Co-derivative)
ð = -*d* on a space of even dimension. When the dimension is odd, the
sign is negative only for forms of odd degree.
Ð (Laplacian)
>Ð = (dð + ðd). This is the ``generalized Laplacian''. Green's function
is the kernel of its inverse.
Harmonic form
A form H for which ÐH = 0.
Compact
For a set S to be compact means that every subset of S with infinitely
many points has (at least one) limit point in S. (Corollary: every limit
point is in S.) A subset of a metric space is compact if and only if it is
closed and bounded (from the Heine-Borel theorem). A function has compact
support if the set of points where it is nonzero is compact.