differential geometry book that uses infinitesimals

#Differential geometry book that uses infinitesimals tv

If f is differentiable, then as t becomes smaller, the path becomes closer and closer to a short line segment which we can draw as a vector. In f(x,t) we can think of t as time and each point x on the manifold flows along a path t→f(x,t). A finite flow on a manifold, M, is simply a function f: R×M→M such that f(x,0)=0. In differential geometry a vector field is intuitively thought of as an 'infinitesimal flow' on a manifold. These numbers are in some sense infinitesimal. Let D be the subset of R consisting of numbers whose square is zero. In fact, there is a way to construct tangent spaces and vector fields directly from this - but instead I want to now consider Kock's construction which is slightly different.

The points of the form (a*d 1,0) do, indeed, form a line tangent to the circle. In fact, Protagoras argued that there was a short line segment in common between the circle and the line and that because Pythogoras's theorem gave only one solution the whole of geometry was at fault! Using R instead of the reals makes sense of this intuition. Does this mean anything? Actually, it fits some people's intuition of what the intersection should look like better than the usual idea that there is only one intersection point. In fact, we find that onS (a*d 1,0) & onL (a*d 1,0) gives True for any Float value a. Notice that in R this has many solutions. (I hope the double use of 'R' won't be confusing, but I want to be consistent-ish with Kock.)īefore we do any more calculus, consider the following geometric figure:Ī point is in the intersection of the circle and the line if it satisfies the equations: I'll use R as mathematical notation for the type given by R Float. We can use these compute partial derivatives. Apart from this, they act just like before. Note that we have (d i)*(d j)=0 for any i and j. We now have an infinite set of independent dual numbers, d i, one for each natural i. The d i provide a basis for these infinitesimals. > exp (R a a') = let e = exp a in R e (e. > instance Floating a => Floating (R a) where > instance (Num a,Fractional a) => Fractional (R a) where > negate (R a a') = R (negate a) (map negate a')Īnd some more functions for good measure: > instance (Eq a,Num a) => Eq (R a) whereĪ slightly more extended version of the Num interface: > zipWith' f l r (a:as) (b:bs) = (f a b) : zipWith' f l r as bs We can directly read off f(1)=1 and f'(1)=4.īut, as Kock points out, the dual numbers don't provide 'enough' infinitesimals, we just get one, e, from which all of the others are built. For example if we defineĪnd evaluate f (1+e) we get D 1 4. The important point to note is that e^2=0. They're called dual numbers because this particular algebra was apparently first described by Clifford and that's what he called them. Clearly we aren't talking about real numbers here, but instead an extension to the real numbers. Kock's work is based on the idea of infinitesimal numbers whose square is zero but which aren't themselves zero.

#Differential geometry book that uses infinitesimals tv

(Unfortunately this is going to be very sketchy as there's a lot I could say and I only have a small amount of time while I sit at home waiting for a new TV to be delivered.)

Normally it takes a significant amount of mathematical machinery to define these things but in the framework of SDG it becomes very simple. My goal here is to illustrate how the definition of vector field in Anders Kock's book gives a nice functional definition of a vector field and how that definition leads naturally to the Lie bracket. This is a pity because SDG allows you to do an amazing thing - write computer programs that easily manipulate objects such as vector fields and differential forms on manifolds without doing symbolic algebra. I've looked at a variety of SDG papers but none of them seem to make the connection with AD. My own formulation of the technique is more algebraic than the description that is usually given, and recently it's begun to dawn on me that all I've done is rediscover Synthetic Differential Geometry (SDG). Previously I've talked about Automatic Differentiation (AD).