Editing Differential operator (section)

==Examples==
*The differential operator <math> P </math> is [[elliptic differential operator|elliptic]] if its symbol is invertible; that is for each nonzero <math> \theta \in T^*X </math> the bundle map <math> \sigma_P (\theta, \dots, \theta)</math> is invertible. On a [[compact manifold]], it follows from the elliptic theory that ''P'' is a [[Fredholm operator]]: it has finite-dimensional [[kernel (algebra)|kernel]] and cokernel.
*In the study of [[hyperbolic partial differential equation|hyperbolic]] and [[parabolic partial differential equation]]s, zeros of the principal symbol correspond to the [[method of characteristics|characteristics]] of the partial differential equation.
* In applications to the physical sciences, operators such as the [[Laplace operator]] play a major role in setting up and solving [[partial differential equation]]s.
* In [[differential topology]], the [[exterior derivative]] and [[Lie derivative]] operators have intrinsic meaning.
* In [[abstract algebra]], the concept of a [[derivation (abstract algebra)|derivation]] allows for generalizations of differential operators, which do not require the use of calculus. Frequently such generalizations are employed in [[algebraic geometry]] and [[commutative algebra]].  See also [[Jet (mathematics)]].
* In the development of [[holomorphic function]]s of a [[complex variable]] ''z'' = ''x'' + ''i''&thinsp;''y'', sometimes a complex function is considered to be a function of two real variables ''x'' and ''y''. Use is made of the [[Wirtinger derivative]]s, which are partial differential operators: <math display="block"> \frac{\partial}{\partial z} = \frac{1}{2} \left( \frac{\partial}{\partial x} - i \frac{\partial}{\partial y} \right) \ ,\quad \frac{\partial}{\partial\bar{z}}= \frac{1}{2} \left( \frac{\partial}{\partial x} + i \frac{\partial}{\partial y} \right) \ .</math> This approach is also used to study functions of [[several complex variables]] and functions of a [[motor variable]].
*The differential operator [[del]], also called ''nabla'', is an important [[Euclidean vector|vector]] differential operator. It appears frequently in [[physics]] in places like the differential form of [[Maxwell's equations]]. In three-dimensional [[Cartesian coordinates]], del is defined as
:<math display="block">\nabla = \mathbf{\hat{x}} {\partial \over \partial x}  + \mathbf{\hat{y}} {\partial \over \partial y} + \mathbf{\hat{z}} {\partial \over \partial z}.</math>
:Del defines the [[gradient]], and is used to calculate the [[Curl (mathematics)|curl]], [[divergence]], and [[Laplacian]] of various objects.
*A [[chiral differential operator]]. For now, see [https://ncatlab.org/nlab/show/chiral+differential+operator]