Gradient, divergence & curl operators

Gradient of a scalar field

Del operator (directional derivative):

Divergence of a vector field

Diverging fields have non-zero divergence,

0V∇⋅ ≠

0V∇⋅ =

Curl of a vector field

Fields with non-zero curl have rotational parts

.( ) 0.V∇ ∇× =

0V∇× ≠

but

Laplacian operator in Cartesian co-ordinates

where

is an eigenfunction of the operator

Some Identities

HW: Complete the proof!

Gradient TheoremIt is similar to the well known result:

Divergence Theorem

V S

The amount of a vector field F coming out (or going into) a volume V is equal to the flux of F emerging (or entering) the volume through the surface enclosing the volume.

Curl (Stokes) Theorem

S C

The curling of a vector field F in a area A is equal to the line integral of the field through the boundary enclosing the area A completely.

=