be written in matrix form, with F, E, and v as column vectors and B as an antisymmetric matrix: If a pseudovector is explicitly given by a cross product of two vectors (as opposed to entering the cross product with another vector), then such pseudovectors can also be written as antisymmetric tensors of second order, with each entry a component of the cross product. The angular momentum of a classical pointlike particle orbiting about an axis, defined by , is another example of a pseudovector, with corresponding antisymmetric tensor: Although Cartesian tensors do not occur in the theory of relativity; the tensor form of orbital angular momentum J enters the spacelike part of the relativistic angular momentum tensor, and the above tensor form of the magnetic field B enters the spacelike part of the electromagnetic tensor. Vector and tensor calculus The following formulae are only so simple in Cartesian coordinates – in general curvilinear coordinates there are factors of the metric and its determinant – see tensors in curvilinear coordinates for more general analysis. Vector calculus Following are the differential operators of vector calculus. Throughout, let Φ(r, t) be a scalar field, and be vector fields, in which all scalar and vector fields are functions of the position vector r and time t. The gradient operator in Cartesian coordinates is given by: and in index notation, this is usually abbreviated in various ways: This operator acts on a scalar field Φ to obtain the vector field directed in the maximum rate of increase of Φ: The index notation for the dot and cross products carries over to the differential operators of vector calculus. The directional derivative of a scalar field Φ is the rate of change of Φ along some direction vector a (not necessarily a unit vector), formed out of the components of a and the gradient: The divergence of a vector field A is: Note the interchange of the components of the gradient and vector field yields a different differential operator which could act on scalar or vector fields. In fact, if A is replaced by the velocity field u(r, t) of a fluid, this is a term in the material derivative (with many other names) of continuum mechanics, with another term being the partial time derivative: which usually acts on the velocity field leading to the non-linearity in the Navier-Stokes equations. As for the curl of a