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

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

to a sum of n - 1 squares all of which have the same sign, while the same transformation does not reduce the B_i to 0. Also known as parabolic partial differential equation.