Laplace operator (Laplacian)

Originally created for Electromagnetic Fields and Waves course, this slide discusses what is Laplacian, why and how it is used and related mathematical equations.

1. —

WELCOME

2. Electromagnetic Fields & Waves
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Groößp. four

3. Topic: Laplacian .

hi. dl
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4. Be“ Y
SO, WHAT |S. -APLACIAN?
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6. In mathematics, the Laplace operator or Laplacian is a
differential operator given by the divergence of the gradient
of a function on Euclidean space. It is usually denoted by the

symbols v EXAMPLE

LAPLACIAN

KHANACADEMY
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Laplace operator - Wikipedia
https://en.wikipedia.org/wiki/Laplace_operator

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7. vi

Del Operator is the vector
differential operator. '
Denoted by

9. d'à

Gradient of a scalar V= VV.
| a?
Divergence of a VectorA = V-V
Ax ay
Curl of a Vector A=V XA

10. Second order differential
operator in the n-dimensional
Euclidean space

à En y E
Tr

11. | be defined as

14. In Cartesian Coordinates

DE up Gr

Aj RE nd à
Í de. Ay? W022

15. In Cylindrical Coordinates
( a) eas DO

p? Oy? 622 |

16. In Spherical Coordinates

Af= ee (PZ) ++ Le 6 6) Lu rome Yad

Or r2 sin 9 00 00 r2 sin? 0 Op? :

21. Laplacian can be
defined over a
vector field.

22. Scalar Laplacian applies to a
scalar field and returns a scalar
quantity. The vector Laplacian
applies to a vector field, returns
a vector quantity

23. The vector Laplacian of
Cartesian Coordinates

VA ww (7? A,, Va V’" A.),