A general type of second-order partial differential equation which includes Laplace's equation and has the form

where A_ij, B_i, C, and F are suitably differentiable real functions of x₁, x₂, ..., x_n , and there exists at each point (x₁, x₂, ..., x_n ) a real linear elliptic partial differential equation transformation on the x_i which reduces the quadratic form

to a sum of n squares, all of the same sign. Also known as elliptic partial differential equation.