DpDqf(r) = lim t→0 Dqf(r+tp)−Dqf(r) t = lim t→0 f, q(r+tp)n−1/(n−1)! − f, qrn−1/(n−1)! t = lim t→0f, q(n−1)r n−2 tp+O(t2)/(n−1)!t =f, pq rn−2/(n−2)!.
Each differentiation thus sets one factor of the n-site s with which the n-form f eventually gets paired. Differentiating k times leaves us with an (n − k)-form, which we can then evaluate at an anchor r by setting the remaining n−k factors of s to rn−k/(n−k)! . Ifwe differentiate n times,
we get the constant Dp1· · ·Dpnf(r) = f, p1· · ·pn, independent of r. If we
further specialize to the case p1 =· · ·=pn =p in which alln directions are
the same, we find that (Dp)nf = f, pn. Since the rule for evaluation at p
is f(p) = f, pn/n!, we see that the relationship between evaluating an n-ic and differentiating it n times is indeed as we claimed in Formula 7.6-5.
For the record, here is the formula for differentiating an n-form k times and then evaluating the blossom ofthe resulting (n−k)-form at the anchors r1 through rn−k:
(Dp1· · ·Dpkf)∼(r1, . . . , rn−k) =f, p1· · ·pkr1· · ·rn−k/(n−k)!.
7.8
The contraction operators
Ifwe set k ofthe factors ofthen-site with which an n-form will eventually get paired, we have essentially converted that n-form into an (n−k)-form. The operator that does that conversion is called contraction.
Let A be an affine space, let f be an n-form on A, and let s be a k-site over A, where k ≤ n. In the special case k = n, we know how to combine f with s to produce a real number: the pairing value f, s. When k < n, we can’t get a real number. But we can produce, from f and s, a mapping that takes (n−k)-sites to real numbers: the mapping t → f, st, for any (n −k)-site t. This mapping is an element ofthe dual space Symn−k( ˆA)∗, which we are representing as the space Symn−k( ˆA∗) of (n−k)-forms. Thus, the n-form f and the k-site s together determine an (n−k)-form, which is written f s and called the contraction of f on s or the s-contraction of f. The terms “internal product” and “inner product” are also used. Note that, in the expression f s, the vertical stroke ofthe operator symbol is next to the operand ofhigher degree.
96 CHAPTER 7. THE PAIRED-ALGEBRAS FRAMEWORK
Formally speaking, the contraction f s of f on s is completely defined by the equation
f s, t=f, st. (7.8-1)
Intuitively, contraction is a flavor ofpartial evaluation. We can think of our n-form f as a function that accepts an n-site as its input and returns a real number. When we contract f on a k-site s, we are declaring that we are interested in the values ofthat function only on those n-sites that are multiples ofs.
In the special case k=n, contracting ann-formf on ak-site s results in a 0-f orm f s, that is, in a scalar. By setting t in Equation 7.8-1 to be the 0-sitet:= 1, we find thatf s,1=f, s·1=f, s. Since pairing a 0-form with a 0-site simply multiplies the two scalars, as discussed in Exercise 7.3-3, we conclude that f s = f s,1 =f, s. Thus, when k =n, contraction reduces to pairing.
It is convenient to extend the contraction operatorf sto the case k > n by settingf s= 0. To support this, we make the convention that 0, which we have already agreed is anm-form for every nonnegative m, is also anm-form — in fact, is the uniquem-form — whenm=n−kis negative. Extending the contraction operator in this way makes the valuef swell-defined whenever f and s are homogeneous, whatever their degrees. We further extend to those cases where the arguments f and s are inhomogeneous in the unique way that preserves linearity. Having done so, the sitef sis now well-defined for any formf and any sites — even inhomogeneous ones.
Successive contractions commute with each other. Indeed, we have the identity (f s) t = (f t) s=f (st). Whenf,s, andt, are all homogeneous, this follows because all three expressions denote the unique form of degree deg(f)−deg(s)−deg(t) that, when paired with any site u ofthat degree, returns the real number f, stu. When f, s, or t are inhomogeneous, the result follows by linearity.
Just as we can contract a form on a site, we can contract a site on a form. Ifsis an m-site andg is ak-form, the expressiong s denotes an (m−k)-site called the g-contraction of s or the contraction of s on g. It is the unique (m−k)-site that makesf, g s=f g , s, for all (m−k)-formsf onA. We extend this dual contraction operator also to return zero when k > m, and we further extend it by linearity to the inhomogeneous case.
If f is an n-form and s is an m-site, don’t get the two contractions f s and f s confused. The operator with its vertical bar on the left, the form side, produces an (n −m)-form, while the one with its vertical bar on the right, the site side, produces an (m−n)-site. If nandm are distinct, at least one ofthe two results will have negative degree and hence will perforce be zero. Ifn =m, we have f s=f s =f, s.
7.9. DIFFERENTIATION AS CONTRACTION 97