the principal part of a PDE

When you want to know how a PDE will behave — whether it spreads things out like heat, propagates them like waves, or settles into equilibrium like Laplace's equation — you do not look at the whole equation. You look at one piece of it: the principal part, the collection of all the highest-order derivative terms. It is the part that 'leads', and it largely dictates the equation's personality, the way the leading coefficient of a polynomial controls its behaviour for large inputs.

Concretely, the principal part of a PDE is what is left after you keep only the terms containing derivatives of the maximal order and throw away everything of lower order (lower derivatives, u itself, source terms, and so on). For the heat equation u_t - k u_xx = 0 the highest order is two, and the second-order part is -k u_xx (the first-order u_t is lower order). For a general second-order equation A u_xx + B u_xy + C u_yy + (lower-order terms) = 0, the principal part is A u_xx + B u_xy + C u_yy, and the coefficients A, B, C — through the combination B^2 - 4 A C — decide whether the equation is elliptic, parabolic, or hyperbolic. The lower-order terms shift solutions around but do not change the basic type.

This is the deep reason the linear/semilinear/quasilinear/fully-nonlinear hierarchy is organised around the highest derivatives, and why classification ignores forcing terms and lower-order coefficients: they affect the details of a solution but not the fundamental way information travels. Knowing the principal part tells you what data makes the problem well-posed, whether disturbances travel at finite or infinite speed, and whether singularities get smoothed away or propagated — the structural questions that everything else hangs on.