on Ruminations exterior algebra 1.1 Bases, once and for all. In preparation for moving to differential forms, let's settle once and for all on a notation for the basis of a vector space V and its dual V . These are fairly standard notations. For V , rather than having to say ". . . and suppose {v1 , . . . , vn } is a basis for V . . . ", unless otherwise specified, we'll use either , ,... n 1 x2 x x or {e1 , e2 , . . . en } as our basis the reason for the first of these will, I hope, become apparent next week. Of course, the first of these is a bit unwieldy, so we'll occasionally revert to the second. So the vector that might have been denoted 3i + 4j - 5k in R3 will now be 3 or 3e1 + 4e2 - 5e3 . For the dual basis, we'll always use {dx1 , dx2 , . . . , dxn }. Sometimes, if we have two different coordinate systems or some kind of change of basis going on, we might also have dy 1 , etc (and /y 1 , . . . as well), but you get the picture. This isn't unwieldy, so we don't need an alternative notation. In this notation we have, for example, dx1 3 and (2dx1 - 4dx2 + dx4 )(e1 - 2e2 + e3 ) = 2dx1 (e1 - 2e2 + e3 ) - 4dx2 (e1 - 2e2 + e3 ) + dx4 (e1 - 2e2 + e3 ) = 2 + 8 + 0 = 10. 1.2 One-forms, two-forms, red-forms, blue-forms. . . To construct the exterior algebra V means to construct a sequence of vector spaces 0 V = R, 1 V = V , 2 V, . . . n V . There are many ways to do this, but perhaps the simplest is to proceed +4 2 -5 3 1 x x x =3 +4 2 -5 3 x1 x x
2
parallel transport
as we did in class and begin by defining p V as the set (vector space) of alternating p-linear functions on V . While that might sound a little unnerving, it just means that an element of p V is a map that takes p vectors v1 , . . . , vp V and returns a real number (v1 , . . . , vp ). The "p-linear" part means that is linear in each variable separately, in other words if all the vectors but vi are kept fixed, and vi is replaced by a1 w1 + a2 w2 (where a1 and a2 are numbers and w1 and w2 are vectors in V ), then (v1 , . . . ,vi-1 , a1 w1 + a2 w2 , vi+1 , . . . , vp ) = a1 (v1 , . . . , vi-1 , w1 , vi+1 , . . . , vp ) + a2 (v1 , . . . , vi-1 , w2 , vi+1 , . . . , vp ). "Alternating" means that the order in which the vectors v1 , . . . , vp appear inside matters, and matters in a particular way. In an alternating function, if two of the vectors v1 , . . . , vp are interchanged, then the value of becomes the negative of what it had been before the interchange (in contrast to a symmetric function for which the interchange would have no effect, or a more general tensor where there is no rule governing what happens). Elements of p V are called p-forms (on V ). We'll need some notation so that we can write down a few examples. Let's start with 2-forms. If and are 1-forms, then we make a 2-form (called the "wedge product" of and ) by defining (v, w) = (v)(w) - (w)(v). This is clearly bilinear and alternating. For instance, dx1 dx2 is the 2-form for which dx1 dx2 (e1 , e2 ) = 1 (so dx1 dx2 (e2 , e1 ) = -1) and dx1 dx2 (ei , ej ) = 0 unless {i, j} = {1, 2}. In the same manner we can determine dxk dx for any k and . Note that if k = we must have dxk dx = 0 (the zero bilinear function) since (using k for both k and , since they're equal) dxk dxk (v, w) = dxk (v)dxk (w) - dxk (w)dxk (v) = 0. More generally, = 0 for any 1-form . A bilinear form on V is determined by its values on all pairs of basis elements of V , i.e., by the n2 numbers (ei , ej ) as i, j = 1, . . . , n. Bescuse of the skew-symmetry, a 2-form is determined by its values on the n numbers (e1 , ej ) for i < j, since 2 we already know that (ei , ei ) = 0 and since (ei , ej ) = -(ej , ei ). An easy linear algebra exercise, then, is to show that this implies that the set {dxi dxj | i = 1, . . . , n - 1, j = i + 1, . . . , n} is a basis for 2 V , so we can write any 2-form as
n-1 n
=
i=1 j=i+1
aij dxi dxj .
on S 3 and H 3
3
Equivalently, we could set aji = -aij if i > j (and aii = 0) and write 1 n n = aij dxi dxj . 2 i=1 j=1 We can continue in this way with p-forms for p > 2. A 3-form is an alternating tri-linear function, so it's determined by the values (ei , ej , ek ) for i < j < k. This is because we can get any permutation of ei , ej , ek by a sequence of swaps of two of them at a time, for instance: (ek , ei , ej ) = -(ei , ek , ej ) (ei = , ej , ek ). As is to be expected from this, for any permutation of ijk, we'll get (e(i) , e(j) , e(k) ) = (ei , ej , ek ) according to the sign of the permutation (plus if is an even permutation and minus of is odd). If we start with three one-forms , and , we can make the 3-form by setting (v1 , v2 , v3 ) = (v1 )(v2 )(v3 ) + (v2 )(v3 )(v1 ) + (v3 )(v1 )(v2 ) - (v1 )(v3 )(v2 ) - (v3 )(v2 )(v1 ) - (v2 )(v1 )(v3 ). As with 2-forms, we get that a 3-form is determined by the (ei , ej , ek ) for i < j < k, so we can write
n-2 n-1 n n 3
numbers
=
i=1 j=i+1 k=j+1
aijk dxi dxj dxk ,
or, after suitably defining aijk so that it is completely skew-symmetric, = 1 n n n aijk dxi dxj dxk . 6 i=1 j=1 k=1
1.3 Formalism. We could continue with the point of view of the preceding section, and regard elements of p V as alternating p-linear functions, and there are certainly situations where it is profitable to take this point of view. Another perspective that is also useful in many situations is simply to regard the construction of p V as a kind of algebraic game:
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To play this game, we simply declare 0 V = R, then 1 V is the vector space with basis {dx1 , . . . , dxn }, in other words, the set of expressions of the form a1 dx1 + + an dxn for any choice of constants a1 , . . . , an . In this version, we don't worry about what dxi stands for, it's just a symbol. We then declare that we can multiply the dxi 's together in such a way that (1) the multiplication is associative, (2) the multiplication is alternating, (3) real coefficients always factor out of multiplications in the usual way (that is to say, we are constructing an R-algebra), and (4) there aren't any other rules. The upshot of all this is that we can multiply forms together the same way we multiply polynomials together, except for one small details: we're not allowed to permute the dxi 's in a product unless we remember to apply the appropriate plus or minus sign (i.e., whenever we swap two of them we get a minus). To remind ourselves that the multiplication is a little odd, instead of just writing dxi dxj dxk , we'll use the wedge symbol to denote the product: dxi dxj dxk . The entire algebra that we construct in this way is called V . The elements of p are simply the (homogeneous) degree-p elements of the algebra. Working in the formalism becomes natural after a while, and you start to see patterns that you can take advantage of. For example, = 0 for any 1-form , and more generally, 1 2 p = 0 if the i are linearly dependent (to prove this recall that if the 's are linearly dependent, then...