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:
“Mothers often are too easily intimidated by their children’s negative reactions...When the child cries or is unhappy, the mother reads this as meaning that she is a failure. This is why it is so important for a mother to know...that the process of growing up involves by definition things that her child is not going to like. Her job is not to create a bed of roses, but to help him learn how to pick his way through the thorns.”
—Elaine Heffner (20th century)
“Perhaps the best definition of progress would be the continuing efforts of men and women to narrow the gap between the convenience of the powers that be and the unwritten charter.”
—Nadine Gordimer (b. 1923)
“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)