The exterior differentiation operator d maps r-form fields linearly to (r+ l)-form fields. Acting on a zero-form field (i.e. a function)/, it gives the one-form field d/defined by (cf. §2.2)

and acting on the r-form field

A = Aab...d&xa adz6 a ... a dxd it gives the (r+ l)-form field dA defined by dA = d^^ .¿Ada^Ada^A ... Ada;d. (2.9)

To show that this (r + l)-form field is independent of the coordinates {x"} used in its definition, consider another set of coordinates {a;«"}.

where the components Ag.v d. are given by

Bxa dxh 8xd

Thus the (r+ l)-form dA defined by these coordinates is dA = dAg?,, ¿.dx"' A cLr6' A ... A da^'

= dAgj, d f\dx° a cLr6 a ... a da;d as 82xaj8xa' dxf- is symmetric in a' and e', but da?' a dz°' is skew. Note that this definition only works for forms; it would not be independent of the coordinates used if the a product were replaced by a tensor product. Using the relation d(fg) = gdf+fdg, which holds for arbitrary functions /, g, it follows that for any r-form A and form B, d(A a B) = dA a B + (— )r A a dB. Since (2.8) implies that the local coordinate expression for df is d/= (df/Bx1) dxl, it follows that d(d/) = (d2f/Bxi dx}) da^ a = 0, as the first term is symmetric and the second skew-symmetric. Similarly it follows from (2.9) that d(dA) = 0

holds for any r-form field A.

The operator d commutes with manifold maps, in the sense: if is a CT (r > 2) map and A is a Ck (k > 2) form field on then (by (2.7))

(which is equivalent to the chain rule for partial derivatives).

The operator d occurs naturally in the general form of Stokes' theorem on a manifold. We first define integration of n-forms: let be a compact, orientable «-dimensional manifold with boundary d^tf and let {/0} be a partition of unity for a finite oriented atlas {<25f0,0O}. Then if A is an n-form field on the integral of A over is defined as

[ A=(n!)"'sf /«^ii...»daIda«...das», (2.10)

where A12 n are the components of A with respect to the local coordinates in the coordinate neighbourhood and the integrals on the right-hand side are ordinary multiple integrals over open sets <}>a{fya) of Rn. Thus integration of forms on is defined by mapping the form, by local coordinates, into R71 and performing standard multiple integrals there, the existence of the partition of unity ensuring the global validity of this operation.

The integral (2.10) is well-defined, since if one chose another atlas (y^, and partition of unity {¡fy} for this atlas, one would obtain the integral

(w!)"1 £ f g,Alv...n-d*1'da;2'...da;»', /} J tpi-Tp)

where xv are the corresponding local coordinates. Comparing these two quantities in the overlap (<%a n of coordinate neighbourhoods belonging to two atlases, the first expression can be written a 0 J4>A»f(] and the second can be written

Comparing the transformation laws for the form A and the multiple integrals in Rn, these expressions are equal at each point, so A is

Jur independent of the atlas and partition of unity chosen.

Similarly, one can show that this integral is invariant under diffeomorphisms:

if <p is a Cr diffeomorphism (r > 1) from ^ to .

Using the operator d, the generalized Stokes' theorem can now be written in the form: if B is an (n— l)-form field on then which can be verified (see e.g. Spivak (1965)) from the definitions above; it is essentially a general form of the fundamental theorem of calculus. To perform the integral on the left, one has to define an orientation on the boundary d^tf oi^K. This is done as follows: if <ifa is a coordinate neighbourhood from the oriented atlas of such that intersects cL^, then from the definition of ¿L^, 4>Jf%ia H lies in the plane x1 = 0 in 2?" and n lies in the lower half x1 < 0.

The coordinates (x2,x3 xn) are then oriented coordinates in the neighbourhood <25f0 n d^ of ¿L^. It may be verified that this gives an oriented atlas on d^tf.

The other type of differentiation defined naturally by the manifold structure is Lie differentiation. Consider any CT (r > 1) vector field X on By the fundamental theorem for systems of ordinary differential equations (Burkill (1956)) there is a unique maximal curve A(t) through each pointy of such that A(0) = p and whose tangent vector at the point A(i) is the vector X|A((). If {a;1} are local coordinates, so that the curve A(<) has coordinates xi(t) and the vector X has components Xi, then this curve is locally a solution of the set of differential equations

This curve is called the integral curve of X with initial pointy. For each point q oi^K, there is an open neighbourhood % of q and an e > 0 such that X defines a family of diffeomorphisms <f)t: whenever

|t| < e, obtained by taking each pointy in tfl a parameter distance t along the integral curves of X (in fact, the <p, form a one-parameter local group of diffeomorphisms, as <p,+s = <p, o <f>s = <f>s o <f>, for \t\, |«|, |i + «| < e, so <f>-t—(<Pt)~x and <f>0 is the identity). This diffeomorphism maps each tensor field T at p of type (r,s) into

The Lie derivative LXT of a tensor field T with respect to X is

defined to be minus the derivative with respect to t of this family of tensor fields, evaluated at t — 0, i.e.

From the properties of it follows that

(1) Lx preserves tensor type, i.e. if T is a tensor field of type (r,s), then LxT is also a tensor field of type (r, s);

(2) Lx maps tensors linearly and preserves contractions. As in ordinary calculus, one can prove Leibniz' rule:

(3) For arbitrary tensors S, T, LX(S ® T) = LXS ® T+ S ® LXT. Direct from the definitions:

Under the map <pt, the point q — <f>-t{p) is mapped into p. Therefore <f>m is a map from Tq to Tp. Thus, by (2.6),

If {a;'} are local coordinates in a neighbourhood ofp, the coordinate components of Y at p are

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