and astrophysics are a generalization of the Euler equations that account for the effects of special relativity. These equations reduce to the classical Euler equations if the fluid 3-space velocity is much less than the speed of light, the pressure is much less than the energy density, and the latter is dominated by the rest mass density. In flat spacetime and using Cartesian coordinates, if one combines this with the symmetry of the stress–energy tensor, one can show that angular momentum (relativistic angular momentum) is also conserved: where this zero is actually a (2,0)-tensor zero. As a Jacobian matrix for the SR Minkowski metric tensor The Jacobian matrix is the matrix of all first-order partial derivatives of a vector-valued function. The 4-gradient acting on the 4-position gives the SR Minkowski space metric : For the Minkowski metric, the components ( not summed), with non-diagonal components all zero. For the Cartesian Minkowski Metric, this gives . Generally, , where is the 4D Kronecker delta. As a way to define the Lorentz transformations The Lorentz transformation is written in tensor form as and since are just constants, then Thus, by definition of the 4-gradient This identity is fundamental. Components of the 4-gradient transform according to the inverse of the components of 4-vectors. So the 4-gradient is the "archetypal" one-form. As part of the total proper time derivative The scalar product of 4-velocity with the 4-gradient gives the total derivative with respect to proper time : The fact that is a Lorentz scalar invariant shows that the total derivative with respect to proper time is likewise a Lorentz scalar invariant. So, for example, the 4-velocity is the derivative of the 4-position with respect to proper time: or Another example, the 4-acceleration is the proper-time derivative of the 4-velocity : or As a way to define the Faraday electromagnetic tensor and derive the Maxwell equations The Faraday electromagnetic tensor is a mathematical object that describes the electromagnetic field in spacetime of a physical system. Applying the 4-gradient to make an antisymmetric tensor, one gets: where: Electromagnetic 4-potential , not to be confused with the 4-acceleration The electric scalar potential is The magnetic 3-space vector potential is By applying the 4-gradient again, and defining the 4-current density as one can derive the tensor form of the Maxwell equations: where the second line is a version of the Bianchi identity (Jacobi identity). As a way to