convert the ε symbols into δ tensors: thusly: Note this is antisymmetric in b and c, as expected from the left hand side. Similarly, via index notation or even just cyclically relabelling a, b, and c in the previous result and taking the negative: and the difference in results show that the cross product is not associative. More complex identities, like quadruple products; and so on, can be derived in a similar manner. Transformations of Cartesian tensors (any number of dimensions) Tensors are defined as quantities which transform in a certain way under linear transformations of coordinates. Second order Let a = aiei and b = biei be two vectors, so that they transform according to j = aiLij, j = biLij. Taking the tensor product gives: then applying the transformation to the components and to the bases gives the transformation law of an order-2 tensor. The tensor a⊗b is invariant under this transformation: More generally, for any order-2 tensor the components transform according to; , and the basis transforms by: If R does not transform according to this rule – whatever quantity R may be – it is not an order-2 tensor. Any order More generally, for any order p tensor the components transform according to; and the basis transforms by: For a pseudotensor S of order p, the components transform according to; Pseudovectors as antisymmetric second order tensors The antisymmetric nature of the cross product can be recast into a tensorial form as follows. Let c be a vector, a be a pseudovector, b be another vector, and T be a second order tensor such that: As the cross product is linear in a and b, the components of T can be found by inspection, and they are: so the pseudovector a can be written as an antisymmetric tensor. This transforms as a tensor, not a pseudotensor. For the mechanical example above for the tangential velocity of a rigid body, given by , this can be rewritten as where Ω is the tensor corresponding to the pseudovector ω: For an example in electromagnetism, while the electric field E is a vector field, the magnetic field B is a pseudovector field. These fields are defined from the Lorentz force for a particle of electric charge q traveling at velocity v: and considering the second term containing the cross product of a pseudovector B and velocity vector v, it can