Package rational brings rational complex, split-complex, and dual numbers to Go. It borrows heavily from the math, math/cmplx, and math/big packages. Indeed, it is built on top of the big.Rat type.
This package contains three two-dimensional types (e.g. complex numbers), five four-dimensional types (e.g. quaternions), and seven eight-dimensional types (e.g. octonions). Each type is either an elliptic, a parabolic, or a hyperbolic Cayley-Dickson construct.
Given elements from what I will call a seed algebra, the Cayley-Dickson construction allows you to build elements of a higher-dimensional algebra with certain interesting properties. Let a, b, c, and d be elements in the seed algebra. Let p = (a, b) and q = (c, d) be elements in the construct algebra. Thus,
Add(p, q) = (Add(a, c), Add(b, d))
That is, addition is element-wise. The multiplication operation can be any of three kinds. Before we look into these, we should mention that the seed algebra must also include two involutions, Conj and Neg, such that the construct algebra also has two involutions Conj and Neg given by:
Neg(p) = (Neg(a), Neg(b))
Conj(p) = (Conj(a), Neg(b))
With Neg you can define subtraction:
Sub(p, q) = Add(p, Neg(q))
The three multiplication operations are named after conic sections. Each has the form
Mul(p, q) = (F(a, b, c, d), G(a, b, c, d))
With F and G being a linear combination of bilinear terms.
The elliptic multiplication operation is:
F(a, b, c, d) = Sub(Mul(a, c), Mul(Conj(d), b))
G(a, b, c, d) = Add(Mul(d, a), Mul(b, Conj(c)))
The order of the arguments in all Mul calls is very important.
The parabolic multiplication operation is:
F(a, b, c, d) = Mul(a, c)
G(a, b, c, d) = Add(Mul(d, a), Mul(b, Conj(c)))
The order of the arguments in all Mul calls is very important. Note that if a and c are both zero, then Mul(p, q) = (0, 0) with p and q not necessarily being zero. That is, there are zero divisors with this multiplication operation.
The hyperbolic multiplication operation is:
F(a, b, c, d) = Add(Mul(a, c), Mul(Conj(d), b))
G(a, b, c, d) = Add(Mul(d, a), Mul(b, Conj(c)))
The order of the arguments in all Mul calls is very important. Although it is not obvious, this multiplication operation also leads to zero divisors.
The three "conic" multiplication operations above all use the Conj involution of the seed algebra. One can also consider non-sesquilinear multiplication rules. If the seed algebra is commutative and associative, then the resulting construct algebra will also be commutative and associative.
The non-sesquilinear elliptic multiplication operation is:
F(a, b, c, d) = Sub(Mul(a, c), Mul(b, d))
G(a, b, c, d) = Add(Mul(a, d), Mul(b, c))
The non-sesquilinear parabolic multiplication operation is:
F(a, b, c, d) = Mul(a, c)
G(a, b, c, d) = Add(Mul(a, d), Mul(b, c))
The non-sesquilinear hyperbolic multiplication operation is:
F(a, b, c, d) = Add(Mul(a, c), Mul(b, d))
G(a, b, c, d) = Add(Mul(a, d), Mul(b, c))
Note the different ordering in some of the Mul calls. The resulting construct algebras are very different from the familiar Cayley-Dickson constructs.
There are three two-dimensional types. The (binary) multiplication operation for all two-dimensional types is commutative and associative.
The rational.Complex type represents a rational complex number. It corresponds to an elliptic Cayley-Dickson construct with big.Rat values. The imaginary unit element is denoted i. The multiplication rule is:
Mul(i, i) = -1
This type can be used to study Gaussian integers.
The rational.Perplex type represents a rational perplex number. It corresponds to a hyperbolic Cayley-Dickson construct with big.Rat values. The split unit element is denoted s. The multiplication rule is:
Mul(s, s) = +1
Perplex numbers are more commonly known as split-complex numbers, but "perplex" is used here for brevity and symmetry with "complex".
The rational.Infra type represents a rational infra number. It corresponds to a parabolic Cayley-Dickson construct with big.Rat values. The dual unit element is denoted α. The multiplication rule is:
Mul(α, α) = 0
Infra numbers are useful for computing numerical first-order derivatives. If z = a + 1α, and f is a function, then
f(z) = f(a) + f'(a)α
Infra numbers are more commonly known as dual numbers, but "infra" is used here along with "supra" and "ultra" for the higher-dimensional analogs.
There are five four-dimensional types. The (binary) multiplication operation for all four-dimensional types is noncommutative but associative.
The rational.Hamilton type represents a rational Hamilton quaternion. It corresponds to an elliptic Cayley-Dickson construct with rational.Complex values. The imaginary unit elements are denoted i, j, and k. The multiplication rules are:
Mul(i, i) = Mul(j, j) = Mul(k, k) = -1
Mul(i, j) = -Mul(j, i) = k
Mul(j, k) = -Mul(k, j) = i
Mul(k, i) = -Mul(i, k) = j
Hamilton quaternions are traditional quaternions. The type is named after W.R. Hamilton, who discovered quaternions.
This type can be used to study Hurwitz and Lipschitz integers.
The rational.Cockle type represents a rational Cockle quaternion. It corresponds to a hyperbolic Cayley-Dickson construct with rational.Complex values. The imaginary unit element is denoted i, and the split unit elements are denoted t and u. The multiplication rules are:
Mul(i, i) = -1
Mul(t, t) = Mul(u, u) = +1
Mul(i, t) = -Mul(t, i) = u
Mul(u, t) = -Mul(t, u) = i
Mul(u, i) = -Mul(i, u) = t
Cockle quaternions are more commonly known as split-quaternions. The type is named after J. Cockle, who discovered them.
Alternatively, you can obtain the Cockle quaternions from an elliptic Cayley-Dickson construct with rational.Perplex values; or from a hyperbolic Cayley-Dickson construct with rational.Perplex values.
The rational.Supra type represents a rational supra number. It corresponds to a parabolic Cayley-Dickson construct with rational.Infra values. The dual unit elements are denoted α, β, and γ. The multiplication rules are:
Mul(α, α) = Mul(β, β) = Mul(γ, γ) = 0
Mul(α, β) = -Mul(β, α) = γ
Mul(β, γ) = Mul(γ, β) = 0
Mul(γ, α) = Mul(α, γ) = 0
Note that supra numbers are very different from hyper-dual numbers: the multiplication operation for supra numbers is noncommutative. In some ways, supra numbers are the dual analog of quaternions.
The rational.InfraComplex type represents a rational infra complex number. It corresponds to a parabolic Cayley-Dickson construct with rational.Complex values. The imaginary unit element is denoted i, and the dual unit elements are denoted β and γ. The multiplication rules are:
Mul(i, i) = -1
Mul(β, β) = Mul(γ, γ) = 0
Mul(β, γ) = Mul(γ, β) = 0
Mul(i, β) = -Mul(β, i) = γ
Mul(γ, i) = -Mul(i, γ) = β
Note that i does not commute with either β or γ. This makes the infra complex numbers different from the more familiar dual complex numbers.
Alternatively, you can obtain the infra complex numbers from an elliptic Cayley-Dickson construct with rational.Infra values.
The rational.InfraPerplex type represents a rational infra perplex number. It corresponds to a parabolic Cayley-Dickson construct with rational.Perplex values. The split unit element is denoted s, and the dual unit elements are denoted τ and υ. The multiplication rules are:
Mul(s, s) = +1
Mul(τ, τ) = Mul(υ, υ) = 0
Mul(τ, υ) = Mul(υ, τ) = 0
Mul(s, τ) = -Mul(τ, s) = υ
Mul(s, υ) = -Mul(υ, s) = τ
Like i in the infra complex numbers, s does not commute with either τ or υ. This makes the infra perplex numbers different from the more familiar dual split-complex numbers.
Alternatively, you can obtain the infra perplex numbers from a hyperbolic Cayley-Dickson construct with rational.Infra values.
There are seven eight-dimensional types. The (binary) multiplication operation for all eight-dimensional types is noncommutative and nonassociative.
The rational.Cayley type represents a rational Cayley octonion. It corresponds to an elliptic Cayley-Dickson construct with rational.Hamilton values. The imaginary unit elements are denoted i, j, k, m, n, p, and q. The multiplication rules are:
Mul(i, i) = Mul(j, j) = Mul(k, k) = -1
Mul(m, m) = Mul(n, n) = Mul(p, p) = Mul(q, q) = -1
Mul(i, j) = -Mul(j, i) = +k
Mul(i, k) = -Mul(k, i) = -j
Mul(i, m) = -Mul(m, i) = +n
Mul(i, n) = -Mul(n, i) = -m
Mul(i, p) = -Mul(p, i) = -q
Mul(i, q) = -Mul(q, i) = +p
Mul(j, k) = -Mul(k, j) = +i
Mul(j, m) = -Mul(m, j) = +p
Mul(j, n) = -Mul(n, j) = +q
Mul(j, p) = -Mul(p, j) = -m
Mul(j, q) = -Mul(q, j) = -n
Mul(k, m) = -Mul(m, k) = +q
Mul(k, n) = -Mul(n, k) = -p
Mul(k, p) = -Mul(p, k) = +n
Mul(k, q) = -Mul(q, k) = -m
Mul(m, n) = -Mul(n, m) = +i
Mul(m, p) = -Mul(p, m) = +j
Mul(m, q) = -Mul(q, m) = +k
Mul(n, p) = -Mul(p, n) = -k
Mul(n, q) = -Mul(q, n) = +j
Mul(p, q) = -Mul(q, p) = -i
Cayley octonions are traditional octonions. The type is named after A. Cayley, who was not the first person to discover octonions. The first person to discover octonions was J.T. Graves.
This type can be used to study Gravesian and Kleinian integers, as well as other integral octonions.
The rational.Zorn type represents a rational Zorn octonion. It corresponds to a hyperbolic Cayley-Dickson construct with rational.Hamilton values. The imaginary unit elements are denoted i, j, and k, and the split unit elements are r, s, t, and u. The multiplication rules are:
Mul(i, i) = Mul(j, j) = Mul(k, k) = -1
Mul(r, r) = Mul(s, s) = Mul(t, t) = Mul(u, u) = +1
Mul(i, j) = -Mul(j, i) = +k
Mul(i, k) = -Mul(k, i) = -j
Mul(i, r) = -Mul(r, i) = +s
Mul(i, s) = -Mul(s, i) = -r
Mul(i, t) = -Mul(t, i) = -u
Mul(i, u) = -Mul(u, i) = +t
Mul(j, k) = -Mul(k, j) = +i
Mul(j, r) = -Mul(r, j) = +t
Mul(j, s) = -Mul(s, j) = +u
Mul(j, t) = -Mul(t, j) = -r
Mul(j, u) = -Mul(u, j) = -s
Mul(k, r) = -Mul(r, k) = +u
Mul(k, s) = -Mul(s, k) = -t
Mul(k, t) = -Mul(t, k) = +s
Mul(k, u) = -Mul(u, k) = -r
Mul(r, s) = -Mul(s, r) = -i
Mul(r, t) = -Mul(t, r) = -j
Mul(r, u) = -Mul(u, r) = -k
Mul(s, t) = -Mul(t, s) = +k
Mul(s, u) = -Mul(u, s) = -j
Mul(t, u) = -Mul(u, t) = +i
Zorn octonions are commonly known as split-octonions. The type is named after M.A. Zorn, who developed a vector-matrix algebra for working with split-octonions.
Alternatively, you can obtain the Zorn octonions from an elliptic Cayley-Dickson construct with rational.Cockle values; or from a hyperbolic Cayley-Dickson construct with rational.Cockle values.
The rational.Ultra type represents a rational ultra number. It corresponds to a parabolic Cayley-Dickson construct with rational.Supra values. The dual unit elements are denoted α, β, γ, δ, ε, ζ, and η. The multiplication rules are:
Mul(α, α) = Mul(β, β) = Mul(γ, γ) = 0
Mul(δ, δ) = Mul(ε, ε) = Mul(ζ, ζ) = Mul(η, η) = 0
Mul(α, β) = -Mul(β, α) = +γ
Mul(α, γ) = Mul(γ, α) = 0
Mul(α, δ) = -Mul(δ, α) = +ε
Mul(α, ε) = Mul(ε, α) = 0
Mul(α, ζ) = -Mul(ζ, α) = -η
Mul(α, η) = -Mul(η, α) = +ζ
Mul(β, γ) = Mul(γ, β) = 0
Mul(β, δ) = -Mul(δ, β) = +ζ
Mul(β, ε) = -Mul(ε, β) = +η
Mul(β, ζ) = Mul(ζ, β) = 0
Mul(β, η) = Mul(η, β) = 0
Mul(γ, δ) = -Mul(δ, γ) = +η
Mul(γ, ε) = Mul(ε, γ) = 0
Mul(γ, ζ) = Mul(ζ, γ) = 0
Mul(γ, η) = Mul(η, γ) = 0
Mul(δ, ε) = Mul(ε, δ) = 0
Mul(δ, ζ) = Mul(ζ, δ) = 0
Mul(δ, η) = Mul(η, δ) = 0
Mul(ε, ζ) = Mul(ζ, ε) = 0
Mul(ε, η) = Mul(η, ε) = 0
Mul(ζ, η) = Mul(η, ζ) = 0
In some ways, ultra numbers are the dual analog of octonions.
The rational.InfraHamilton type represents a rational infra Hamilton quaternion. It corresponds to a parabolic Cayley-Dickson construct with rational.Hamilton values. The imaginary unit elements are denoted i, j and k, and the dual unit elements are denoted α, β, γ, and δ. The multiplication rules are:
Mul(i, i) = Mul(j, j) = Mul(k, k) = -1
Mul(α, α) = Mul(β, β) = Mul(γ, γ) = Mul(δ, δ) = 0
Mul(i, j) = -Mul(j, i) = +k
Mul(i, k) = -Mul(k, i) = -j
Mul(i, α) = -Mul(α, i) = +β
Mul(i, β) = -Mul(β, i) = -α
Mul(i, γ) = -Mul(γ, i) = -δ
Mul(i, δ) = -Mul(δ, i) = +γ
Mul(j, k) = -Mul(k, j) = +i
Mul(j, α) = -Mul(α, j) = +γ
Mul(j, β) = -Mul(β, j) = +δ
Mul(j, γ) = -Mul(γ, j) = -α
Mul(j, δ) = -Mul(δ, j) = -β
Mul(k, α) = -Mul(α, k) = +δ
Mul(k, β) = -Mul(β, k) = -γ
Mul(k, γ) = -Mul(γ, k) = +β
Mul(k, δ) = -Mul(δ, k) = -α
Mul(α, β) = Mul(β, α) = 0
Mul(α, γ) = Mul(γ, α) = 0
Mul(α, δ) = Mul(δ, α) = 0
Mul(β, γ) = Mul(γ, β) = 0
Mul(β, δ) = Mul(δ, β) = 0
Mul(γ, δ) = Mul(δ, γ) = 0
The infra Hamilton quaternions are different from the more familiar dual quaternions.
Alternatively, you can obtain the infra Hamilton quaternions from an elliptic Cayley-Dickson construct with rational.InfraComplex values.
The rational.InfraCockle type represents a rational infra Cockle quaternion. It corresponds to a parabolic Cayley-Dickson construct with rational.Cockle values. The imaginary unit element is denoted i, the split unit elements are denoted t and u, and the dual unit elements are denoted ρ, σ, τ, and υ. The multiplication rules are:
Mul(i, i) = -1
Mul(t, t) = Mul(u, u) = +1
Mul(ρ, ρ) = Mul(σ, σ) = Mul(τ, τ) = Mul(υ, υ) = 0
Mul(i, t) = -Mul(t, i) = +u
Mul(i, u) = -Mul(u, i) = -t
Mul(i, ρ) = -Mul(ρ, i) = +σ
Mul(i, σ) = -Mul(σ, i) = -ρ
Mul(i, τ) = -Mul(τ, i) = -υ
Mul(i, υ) = -Mul(υ, i) = +τ
Mul(t, u) = -Mul(u, t) = -i
Mul(t, ρ) = -Mul(ρ, t) = +τ
Mul(t, σ) = -Mul(σ, t) = +υ
Mul(t, τ) = -Mul(τ, t) = +ρ
Mul(t, υ) = -Mul(υ, t) = +σ
Mul(u, ρ) = -Mul(ρ, u) = +υ
Mul(u, σ) = -Mul(σ, u) = -τ
Mul(u, τ) = -Mul(τ, u) = -σ
Mul(u, υ) = -Mul(υ, u) = +ρ
Mul(ρ, σ) = Mul(σ, ρ) = 0
Mul(ρ, τ) = Mul(τ, ρ) = 0
Mul(ρ, υ) = Mul(υ, ρ) = 0
Mul(σ, τ) = Mul(τ, σ) = 0
Mul(σ, υ) = Mul(υ, σ) = 0
Mul(τ, υ) = Mul(υ, τ) = 0
The infra Cockle quaternions are different from the more familiar dual split-quaternions.
Alternatively, you can obtain the infra Cockle quaternions from a hyperbolic Cayley-Dickson construct with rational.InfraComplex values; an elliptic Cayley-Dickson construct with rational.InfraPerplex values; or a hyperbolic Cayley-Dickson construct with rational.InfraPerplex values.
The rational.SupraComplex type represents a rational supra complex number. It corresponds to a parabolic Cayley-Dickson construct with rational.InfraComplex values. The imaginary unit element is denoted i, and the dual unit elements are denoted α, β, γ, δ, ε, and ζ. The multiplication rules are:
Mul(i, i) = -1
Mul(α, α) = Mul(β, β) = Mul(γ, γ) = 0
Mul(δ, δ) = Mul(ε, ε) = Mul(ζ, ζ) = 0
Mul(i, α) = -Mul(α, i) = +β
Mul(i, β) = -Mul(β, i) = -α
Mul(i, γ) = -Mul(γ, i) = +δ
Mul(i, δ) = -Mul(δ, i) = -γ
Mul(i, ε) = -Mul(ε, i) = -ζ
Mul(i, ζ) = -Mul(ζ, i) = +ε
Mul(α, β) = Mul(β, α) = 0
Mul(α, γ) = -Mul(γ, α) = +ε
Mul(α, δ) = -Mul(δ, α) = +ζ
Mul(α, ε) = Mul(ε, α) = 0
Mul(α, ζ) = Mul(ζ, α) = 0
Mul(β, γ) = -Mul(γ, β) = +ζ
Mul(β, δ) = -Mul(δ, β) = -ε
Mul(β, ε) = Mul(ε, β) = 0
Mul(β, ζ) = Mul(ζ, β) = 0
Mul(γ, δ) = Mul(δ, γ) = 0
Mul(γ, ε) = Mul(ε, γ) = 0
Mul(γ, ζ) = Mul(ζ, γ) = 0
Mul(δ, ε) = Mul(ε, δ) = 0
Mul(δ, ζ) = Mul(ζ, δ) = 0
Mul(ε, ζ) = Mul(ζ, ε) = 0
Alternatively, you can obtain the supra complex numbers from an elliptic Cayley-Dickson construct with rational.Supra values.
The rational.SupraPerplex type represents a rational supra perplex number. It corresponds to a parabolic Cayley-Dickson construct with rational.InfraPerplex values. The split unit element is denoted s, and the dual unit elements are denoted ρ, σ, τ, υ, φ, and ψ. The multiplication rules are:
Mul(s, s) = +1
Mul(ρ, ρ) = Mul(σ, σ) = Mul(τ, τ) = 0
Mul(υ, υ) = Mul(φ, φ) = Mul(ψ, ψ) = 0
Mul(s, ρ) = -Mul(ρ, s) = +σ
Mul(s, σ) = -Mul(σ, s) = +ρ
Mul(s, τ) = -Mul(τ, s) = +υ
Mul(s, υ) = -Mul(υ, s) = +τ
Mul(s, φ) = -Mul(φ, s) = -ψ
Mul(s, ψ) = -Mul(ψ, s) = -φ
Mul(ρ, σ) = Mul(σ, ρ) = 0
Mul(ρ, τ) = -Mul(τ, ρ) = +φ
Mul(ρ, υ) = -Mul(υ, ρ) = +ψ
Mul(ρ, φ) = Mul(φ, ρ) = 0
Mul(ρ, ψ) = Mul(ψ, ρ) = 0
Mul(σ, τ) = -Mul(τ, σ) = +ψ
Mul(σ, υ) = -Mul(υ, σ) = +φ
Mul(σ, φ) = Mul(φ, σ) = 0
Mul(σ, ψ) = Mul(ψ, σ) = 0
Mul(τ, υ) = Mul(υ, τ) = 0
Mul(τ, φ) = Mul(φ, τ) = 0
Mul(τ, ψ) = Mul(ψ, τ) = 0
Mul(υ, φ) = Mul(φ, υ) = 0
Mul(υ, ψ) = Mul(ψ, υ) = 0
Mul(φ, ψ) = Mul(ψ, φ) = 0
Alternatively, you can obtain the supra perplex numbers from a hyperbolic Cayley-Dickson construct with rational.Supra values.
Using any of the Cayley-Dickson constructs on any of the eight-dimensional types would produce one of nine sixteen-dimensional types. These types include the sedenions, which are infamous for containing zero divisors.
Besides all the types mentioned above, other types are included. These types have multiplication operations that do not involve conjugation.
The rational.BiComplex type represents a bicomplex number. It corresponds to a non-sesquilinear elliptic construct with rational.Complex values. The imaginary units are denoted i and J. The multiplication rules are:
Mul(i, i) = Mul(J, J) = -1
Mul(i, J) = Mul(J, i)
Note that this multiplication operation is commutative and associative.
The rational.BiPerplex type represents a biperplex number. It corresponds to a non-sesquilinear hyperbolic construct with rational.Perplex values. The split units are denoted s and T. The multiplication rules are:
Mul(s, s) = Mul(T, T) = +1
Mul(s, T) = Mul(T, s)
Note that this multiplication operation is commutative and associative.
The rational.Hyper type represents a hyper-dual number. It corresponds to a non-sesquilinear parabolic construct with rational.Infra values. The dual units are denoted α and Γ. The multiplication rules are:
Mul(α, α) = Mul(Γ, Γ) = 0
Mul(α, Γ) = Mul(Γ, α)
Note that this multiplication operation is commutative and associative.
Hyper-dual numbers are useful for computing second-order derivatives. If z = a + 1α + 1Γ + 0αΓ, and f is a function, then
f(z) = f(a) + f'(a)α + f'(a)Γ + f''(a)αΓ
Another name for the dual numbers could be the nilplex numbers. Thus, in analogy with the bicomplex numbers, another name for the hyper-dual numbers could be the binilplex numbers.
The rational.DualComplex type represents a dual-complex number. It corresponds to a non-sesquilinear parabolic construct with rational.Complex values. The imaginary unit is denoted i, and the dual unit is denoted Γ. The multiplication rules are:
Mul(i, i) = -1
Mul(Γ, Γ) = 0
Mul(i, Γ) = Mul(Γ, i)
Note that this multiplication operation is commutative and associative.
Dual-complex numbers are useful for evaluating two-dimensional vector derivatives. Let (x, y) be the coordinate vector in a two-dimensional space. Also, let f = (g, h) be a function vector. Then, the divergence and the curl of f are:
Div(f) = (∂g / ∂x) + (∂h / ∂y)
Curl(f) = (∂h / ∂x) - (∂g / ∂y)
A two-dimensional vector (x, y) can be associated with a complex number z = x + yi. Introduce the Wirtinger derivative
(∂ / ∂z) = (1 / 2) * ((∂ / ∂x) - (∂ / ∂y)i)
The function vector (g, h) becomes the complex number f = g + hi. Then:
2 * (∂f / ∂z) = Div(f) + (Curl(f))i
Thus, in analogy with the dual numbers, given a dual-complex number p = a + bi + 2Γ + 0iΓ, and a function f, then
f(p) = f(a + bi) + (∂f / ∂z) * (2Γ + 0iΓ)
= f(a + bi) + (Div(f))Γ + (Curl(f))iΓ
In this manner, f, Div(f), and Curl(f) can be calculated at the point a + bi by just evaluating f(a + bi + 2Γ + 0iΓ). Note that this derivation assumes that f is holomorphic (i.e. that it only depends on a and b in the combination a + bi).
The rational.DualPerplex type represents a dual-perplex number. It corresponds to a non-sesquilinear parabolic construct with rational.Perplex values. The split unit is denoted s, and the dual unit is denoted Γ. The multiplication rules are:
Mul(s, s) = +1
Mul(Γ, Γ) = 0
Mul(s, Γ) = Mul(Γ, s)
Note that this multiplication operation is commutative and associative.
Just like dual-complex numbers, the dual-perplex numbers can be used to evaluate certain vector derivatives. Let w = x + ys and f = g + hs. The Wirtinger derivative is now
(∂ / ∂w) = (1 / 2) * ((∂ / ∂x) + (∂ / ∂y)s)
Then:
2 * (∂f / ∂w) = ((∂g / ∂x) + (∂h / ∂y)) + ((∂h / ∂x) + (∂g / ∂y))s
This suggests the introduction of the hyperbolic version of the curl:
Hurl(f) = (∂h / ∂x) + (∂g / ∂y)
Now let p = a + bs + 2Γ + 0sΓ. Thus,
f(p) = f(a + bs) + (Div(f))Γ + (Hurl(f))sΓ
Again, f, Div(f), and now Hurl(f) can be calculated at the point a + bs by just evaluating f(a + bs + 2Γ + 0sΓ).
- Improve documentation
- Tests
- DualHamilton type
- DualCockle type
- Elementary and special functions via Padé approximants
- Simplify symbols for constructs from plexification
- Complexification, Nilplexification, Perplexification