GAlgebra.jl

import SymPy: symbols, sympy, Sym
using GAlgebra

# In console, uncomment to enable colored printing with ANSI escape sequences 
# galgebra.printer.Eprint()
# In Jupyter, uncomment to enable LaTeX printing with MathJax
# galgebra.printer.Format()

(x, y, z) = xyz = symbols("x,y,z",real=true)
o3d = galgebra.ga.Ga("e", g=[1, 1, 1], coords=xyz)

u = o3d.mv("u", "vector")
v = o3d.mv("v", "vector")
A = o3d.mv("A", "mv")
R = o3d.mv("R", "spinor")
# Volume element
I = o3d.I()

# Wedge product: ∧ \wedge
v ∧ A
# Hestenes' inner product: ⋅ \cdot
v ⋅ A
# Left contraction: ⨼ \intprod
v ⨼ A
# Right contraction: ⨽ \intprodr
v ⨽ A
# Scalar product: ⊛ \circledast
# A ⊛ B = <A B†>
v ⊛ A
# Commutator product: ⊠ \boxtimes
# A⊠B = (AB-BA)/2
v ⊠ A
# Anti-commutator product: ⊙ \odot
# A⊙B = (AB+BA)/2
v ⊙ A

# Norm: norm(A) = A.norm() := ||A||
norm(v)

# Inverse: postfix ⁻¹ \^-\^1
# (A)⁻¹ = A^-1 = inv(A) = A.inv()
(R)⁻¹
R^-1
inv(R)

# Reversion: ~A = rev(A) = A.rev()
# A^† is usually used in literature
~A
rev(A)

# Dual: postfix '
# orthogonal complement, Λ^p -> Λ^(n-p)
# note: Ga.dual_mode_value is default to "I+", so A' = A * I
# change Ga.dual_mode_value to get a different definition
A'
dual(A)

# Grade involution: postfix ˣ \^x
# (A)ˣ = A[:*] = involute(A) := A+ - A- = A.even() - A.odd()
# A^* is usually used in literature
(A)ˣ
involute(A)

# Clifford conjugate: postfix ǂ \doublepipe
# (A)ǂ = conj(A) := ((A)^*)^†
(A)ǂ
conj(A)

# Projection: proj(B, A) = A.project_in_blade(B)
proj(u, v)

# Reflection: refl(B, A) = A.reflect_in_blade(B)
refl(u, v)

# Rotation: rot(itheta, A) = A.rotate_multivector(itheta)
# rotate the multivector A by the 2-blade itheta
rot(u ∧ v, A)

# Natural base exponential of x: e^x
exp(u ∧ v)

# Grade-i part: A[i] = A.grade(i) := <A>_i
A[2]

# Scalar (grade-0) part: scalar(A) = A.scalar() := <A> = <A>_0
# note: it returns a SymPy expression unlike A[0] which returns a Mv object
scalar(A)

# Even-grade part: A[:+] = (A)₊ = even(A) = A.even() := A+
A[:+]
even(A)

# Odd-grade part: A[:-] = (A)₋ = odd(A) = A.odd() := A-
A[:-]
odd(A)