Definition
The Laplace operator is a second order differential operator in the n-dimensional Euclidean space, defined as the divergence (∇·) of the gradient (∇ƒ). Thus if ƒ is a twice-differentiable real-valued function, then the Laplacian of ƒ is defined by
-
(1)
Equivalently, the Laplacian of ƒ is the sum of all the unmixed second partial derivatives in the Cartesian coordinates :
-
(2)
As a second-order differential operator, the Laplace operator maps Ck-functions to Ck−2-functions for k ≥ 2. The expression (1) (or equivalently (2)) defines an operator Δ : Ck(Rn) → Ck−2(Rn), or more generally an operator Δ : Ck(Ω) → Ck−2(Ω) for any open set Ω.
Read more about this topic: Laplace Operator
Famous quotes containing the word definition:
“No man, not even a doctor, ever gives any other definition of what a nurse should be than this—”devoted and obedient.” This definition would do just as well for a porter. It might even do for a horse. It would not do for a policeman.”
—Florence Nightingale (1820–1910)
“One definition of man is “an intelligence served by organs.””
—Ralph Waldo Emerson (1803–1882)
“It’s a rare parent who can see his or her child clearly and objectively. At a school board meeting I attended . . . the only definition of a gifted child on which everyone in the audience could agree was “mine.””
—Jane Adams (20th century)