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Calculus of variations
Infinite-dimensional vector spaces and functional analysis
Magmas (groupoids), semigroups, monoids and groups
Homomorphisms and isomorphisms to Z with addition
Finite cyclic groups (C_n aka Z_n aka Z/nZ) and generating sets
Symmetric groups (S_n)
Subgroups, cosets, normal subgroups
Cayley's theorem and Lagrange's theorem
Monomorphisms, epimorphisms
Group theory: Linear
Endomorphisms and automorphisms
Uncountable groups
Lie algebra
Rings
Fields
Vector spaces
Linear endomorphisms and automorphisms including the General Linear Group
Linear forms and the dual space, including the Orthogonal group, special orthogonal group, special linear group
Indefinite orthogonal groups: The indefinite/pseudo orthogonal group, the split orthogonal group, the Lorentz group
Linear forms on complex fields. sesquilinear forms, unitary groups, special unitary groups, Hermitian and Normal matrices
Linear maps
Cross products and the Hodge star operator
Exterior algebra
Tensors
A morphism which is injective. That is:
\(f(a)=f(b)\rightarrow a=b\)
The following are monomorphisms:
\(f(x)=x\)
\(f(x)=cx\)
Converting natural numbers to integers
The following are not monomorphisms:
\(f(x)=0\)
\(f(x)=x+1\)