See Semimanifold and PseudoAffine for the methods.

This is the class underlying manifolds. (Manifold only precludes boundaries and adds an extra constraint that would be circular if it was in a single class. You can always just use Manifold as a constraint in your signatures, but you must define only PseudoAffine for manifold types – the Manifold instance follows universally from this, if 'Interior x ~ x.)

The interface is (boundaries aside) almost identical to the better-known AffineSpace class, but we don't require associativity of .+~^ with ^+^ – except in an asymptotic sense for small vectors.

That innocent-looking change makes the class applicable to vastly more general types: while an affine space is basically nothing but a vector space without particularly designated origin, a pseudo-affine space can have nontrivial topology on the global scale, and yet be used in practically the same way as an affine space. At least the usual spheres and tori make good instances, perhaps the class is in fact equivalent to manifolds in their usual maths definition (with an atlas of charts: a family of overlapping regions of the topological space, each homeomorphic to the Needle vector space or some simply-connected subset thereof).

A pathwise connected subset of a manifold m, whose tangent space has scalar s.

Represent a Region by a smooth function which is positive within the region, and crosses zero at the boundary.

The category of differentiable functions between manifolds over scalar s.

As you might guess, these offer automatic differentiation of sorts (basically, simple forward AD), but that's in itself is not really the killer feature here. More interestingly, we actually have the (à la Curry-Howard) proof built in: the function f has at x₀ derivative f'ₓ₀, if, for¹ ε>0, there exists δ such that |f x − (f x₀ + x⋅f'ₓ₀)| < ε for all |x − x₀| < δ.

Observe that, though this looks quite similar to the standard definition of differentiability, it is not equivalent thereto – in fact it does not prove any analytic properties at all. To make it equivalent, we need a lower bound on δ: simply δ gives us continuity, and for continuous differentiability, δ must grow at least like √ε for small ε. Neither of these conditions are enforced by the type system, but we do require them for any allowed values because these proofs are obviously tremendously useful – for instance, you can have a root-finding algorithm and actually be sure you get all solutions correctly, not just some that are (hopefully) the closest to some reference point you'd need to laborously define!

Unfortunately however, this also prevents doing any serious algebra etc. with the category, because even something as simple as division necessary introduces singularities where the derivatives must diverge. Not to speak of many trigonometric e.g. trigonometric functions that are undefined on whole regions. The PWDiffable and RWDiffable categories have explicit handling for those issues built in; you may simply use these categories even when you know the result will be smooth in your relevant domain (or must be, for e.g. physics reasons).

¹(The implementation does not deal with ε and δ as difference-bounding reals, but rather as metric tensors that define a boundary by prohibiting the overlap from exceeding one; this makes the concept actually work on general manifolds.)

Category of functions that almost everywhere have an open region in which they are continuously differentiable, i.e. PieceWiseDiff'able.

Category of functions that, where defined, have an open region in which they are continuously differentiable. Hence RegionWiseDiff'able. Basically these are the partial version of PWDiffable.

Though the possibility of undefined regions is of course not too nice (we don't need Java to demonstrate this with its everywhere-looming null values...), this category will propably be the “workhorse” for most serious calculus applications, because it contains all the usual trig etc. functions and of course everything algebraic you can do in the reals.

The easiest way to define ordinary functions in this category is hence with its AgentValues, which have instances of the standard classes Num through Floating. For instance, the following defines the binary entropy as a differentiable function on the interval ]0,1[: (it will actually know where it's defined and where not! – and I don't mean you need to exhaustively isNaN-check all results...)

hb :: RWDiffable ℝ ℝ ℝ
hb = alg (\p -> - p * logBase 2 p - (1-p) * logBase 2 (1-p) )

The word “metric” is used in the sense as in general relativity. Cf. HerMetric.

A Riemannian metric assigns each point on a manifold a scalar product on the tangent space. Note that this association is not continuous, because the charts/tangent spaces in the bundle are a priori disjoint. However, for a proper Riemannian metric, all arising expressions of scalar products from needles between points on the manifold ought to be differentiable.

The RealFloat class plus manifold constraints.

The AffineSpace class plus manifold constraints.

Basically just an “updated” version of the VectorSpace class. Every vector space is a manifold, this constraint makes it explicit.

(Actually, LinearManifold is stronger than VectorSpace at the moment, since HasMetric requires FiniteDimensional. This might be lifted in the future.)

Require some constraint on a manifold, and also fix the type of the manifold's underlying field. For example, WithField ℝ HilbertSpace v constrains v to be a real (i.e., Double-) Hilbert space. Note that for this to compile, you will in general need the -XLiberalTypeSynonyms extension (except if the constraint is an actual type class (like Manifold): only those can always be partially applied, for type constraints this is by default not allowed).

A Hilbert space is a complete inner product space. Being a vector space, it is also a manifold.

(Stricly speaking, that doesn't have much to do with the completeness criterion; but since Manifolds are at the moment confined to finite dimension, they are in fact (trivially) complete.)

An euclidean space is a real affine space whose tangent space is a Hilbert space.

Interpolate between points, approximately linearly. For points that aren't close neighbours (i.e. lie in an almost flat region), the pathway is basically undefined – save for its end points.

A proper, really well-defined (on global scales) interpolation only makes sense on a Riemannian manifold, as geodesics. This is a task to be tackled in the future.