A differential operator maps a function to an expression depending on finitely many of its derivatives. A linear differential operator has the form ; its order is the largest derivative order with nonzero coefficient. The Laplacian is a second-order example, and boundary conditions specify the domain when defining an operator on a function space.
The formal transpose is defined by the bilinear integration by parts identity for test functions. For , it is ; unlike the Hermitian adjoint operator, it does not conjugate coefficients.
A differential operator is in divergence form when its highest derivatives occur inside a divergence, for example or . This form makes integration by parts express the operator through a lower-order bilinear form and boundary flux.
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A differential operator is a mathematical operator used to denote the process of differentiation. In the context of a function, it takes a function as its input and produces the derivative of that function as output. Differential operators are commonly used in calculus, physics, engineering, and many other fields to analyze and describe rates of change and various physical phenomena.