the consequences of special relativity can be deduced. More generally this show how transformations in Euclidean space and spacetime can all be described using the same kind of algebra. Maxwell's equations (Note: in this section traditional 3-vectors are indicated by lines over the symbols and spacetime vector and bivectors by bold symbols, with the vectors J and A exceptionally in uppercase) Maxwell's equations are used in physics to describe the relationship between electric and magnetic fields. Normally given as four differential equations they have a particularly compact form when the fields are expressed as a spacetime bivector from ⋀2R3,1. If the electric and magnetic fields in R3 are and then the electromagnetic bivector is where e4 is again the basis vector for the time-like dimension and c is the speed of light. The product e123 yields the bivector that is Hodge dual to in three dimensions, as discussed above, while e4 as a product of orthogonal vectors is also bivector-valued. As a whole it is the electromagnetic tensor expressed more compactly as a bivector, and is used as follows. First it is related to the 4-current J, a vector quantity given by where is current density and ρ is charge density. They are related by a differential operator ∂, which is The operator ∇ is a differential operator in geometric algebra, acting on the space dimensions and given by . When applied to vectors ∇·M is the divergence and ∇∧M is the curl but with a bivector rather than vector result, that is dual in three dimensions to the curl. For general quantity M they act as grade lowering and raising differential operators. In particular if M is a scalar then this operator is just the gradient, and it can be thought of as a geometric algebraic del operator. Together these can be used to give a particularly compact form for Maxwell's equations with sources: This equation, when decomposed according to geometric algebra, using geometric products which have both grade raising and grade lowering effects, is equivalent to Maxwell's four equations. It is also related to the electromagnetic four-potential, a vector A given by where is the vector magnetic potential and V is the electric potential. It is related to the electromagnetic bivector as follows using the same differential operator ∂. Higher dimensions As has been suggested in earlier sections much of geometric algebra generalises well into higher dimensions. The