Let the function computing the derivative of be . f has no dependence on time. The Jacobian of f, is
where is the 3N-square zero matrix, and I is the 3N identity matrix. If all particles move, then the entire Jacobian is 6N square. If only M particles move, then there is no meaning to the entries in the Jacobian owned by particles that do not move, and so the Jacobian is only 6M square. The only difficult part of this Jacobian is , which is a 3N-square matrix, and where is the acceleration of particle i, given by equation A.1, and is its position in 3-space. Thus,
Finally, if we let represent one of , it can be shown that
where, for example, is the component of . If softening is employed, substitute for and for .