Generalizations of The Derivative - Derivatives in Analysis - Higher-order Derivatives and Differential Operators

Higher-order Derivatives and Differential Operators

One can iterate the differentiation process, that is, apply derivatives more than once, obtaining derivatives of second and higher order. A more sophisticated idea is to combine several derivatives, possibly of different orders, in one algebraic expression, a differential operator. This is especially useful in considering ordinary linear differential equations with constant coefficients. For example, if f(x) is a twice differentiable function of one variable, the differential equation

may be rewritten in the form

   where   

is a second order linear constant coefficient differential operator acting on functions of x. The key idea here is that we consider a particular linear combination of zeroth, first and second order derivatives "all at once". This allows us to think of the set of solutions of this differential equation as a "generalized antiderivative" of its right hand side 4x − 1, by analogy with ordinary integration, and formally write

Higher derivatives can also be defined for functions of several variables, studied in in multivariable calculus. In this case, instead of repeatedly applying the derivative, one repeatedly applies partial derivatives with respect to different variables. For example, the second order partial derivatives of a scalar function of n variables can be organized into an n by n matrix, the Hessian matrix. One of the subtle points is that the higher derivatives are not intrinsically defined, and depend on the choice of the coordinates in a complicated fashion (in particular, the Hessian matrix of a function is not a tensor). Nevertheless, higher derivatives have important applications to analysis of local extrema of a function at its critical points. For an advanced application of this analysis to topology of manifolds, see Morse theory.

As in the case of functions of one variable, we can combine first and higher order partial derivatives to arrive at a notion of a partial differential operator. Some of these operators are so important that they have their own names:

• The Laplace operator or Laplacian on R3 is a second-order partial differential operator Δ given by the divergence of the gradient of a scalar function of three variables, or explicitly as

Analogous operators can be defined for functions of any number of variables.

• The d'Alembertian or wave operator is similar to the Laplacian, but acts on functions of four variables. Its definition uses the indefinite metric tensor of Minkowski space, instead of the Euclidean dot product of R3: