vector field A, this can be defined as a pseudovector field by means of the ε symbol: which is only valid in three dimensions, or an antisymmetric tensor field of second order via antisymmetrization of indices, indicated by delimiting the antisymmetrized indices by square brackets (see Ricci calculus): which is valid in any number of dimensions. In each case, the order of the gradient and vector field components should not be interchanged as this would result in a different differential operator: which could act on scalar or vector fields. Finally, the Laplacian operator is defined in two ways, the divergence of the gradient of a scalar field Φ: or the square of the gradient operator, which acts on a scalar field Φ or a vector field A: In physics and engineering, the gradient, divergence, curl, and Laplacian operator arise inevitably in fluid mechanics, Newtonian gravitation, electromagnetism, heat conduction, and even quantum mechanics. Vector calculus identities can be derived in a similar way to those of vector dot and cross products and combinations. For example, in three dimensions, the curl of a cross product of two vector fields A and B: where the product rule was used, and throughout the differential operator was not interchanged with A or B. Thus: Tensor calculus One can continue the operations on tensors of higher order. Let T = T(r, t) denote a second order tensor field, again dependent on the position vector r and time t. For instance, the gradient of a vector field in two equivalent notations ("dyadic" and "tensor", respectively) is: which is a tensor field of second order. The divergence of a tensor is: which is a vector field. This arises in continuum mechanics in Cauchy's laws of motion – the divergence of the Cauchy stress tensor σ is a vector field, related to body forces acting on the fluid. Difference from the standard tensor calculus Cartesian tensors are as in tensor algebra, but Euclidean structure of and restriction of the basis brings some simplifications compared to the general theory. The general tensor algebra consists of general mixed tensors of type (p, q): with basis elements: the components transform according to: as for the bases: For Cartesian tensors, only the order of the tensor matters in a Euclidean space with an orthonormal basis, and all indices can be lowered. A Cartesian basis does not exist unless the vector space has a