BULETINUL INSTITUTULUI POLITEHNIC DIN IAŞI

10 Ion Crăciun
∂ ∂ ∂ ∂
∇ = e1 + e2 + e3
= ej
.
∂x1 ∂x2 ∂x3
∂x j
When applied to the scalar field f( x1, x2, x3) = f( x ), the vector operator
∇ yields a vector field or a tensor of rank one, who is known as the gradient of
that scalar field. Thus,
f = ∇ f = f e ,
grad
, i i
A comma in the front of an index i denotes partial differentiation into
respect the x
i
variable.
In a vector field, denoted for example by ux ( ), the components of the
vector are functions of spatial coordinates x , x , x denoted by
1 2
ui
( x).
Assuming that functions ui
( x1,
x2, x3)
are differentiable, the nine partial
∂ui
derivatives can be written in an index notation as u i , j
. It can be shown that
∂x
u i , j
j
are the components of a second-rank tensor.
3
u ( x , x , x )
i
1 2 3
When the vector operator ∇ operates on a vector ux ( ) = uie i
in a way
analogous to scalar multiplication, the result is a scalar field termed the
divergence of that vector field ux ( ) having the expression
div u= ∇ ⋅ u= u1,1 + u2,2 + u3,3 = u ii ,
.
By taking the cross product of the operator ∇ to the vector field
ux ( ) = uiei
we obtain a vector field termed the curl of ux ( ) and denoted by
curl u or ∇ × u, whose analytical expression is
where
ε ijk
∇ × u=
curl u= ε u ,
ijk
k , j
is a component of the Ricci's alternating tensor.
2
The Laplace operator ∇ is obtained by taking the divergence of a
gradient. The Laplace operator of a twice differentiable scalar field f is the
following scalar field
2
div grad f = ∇∇ f =∇ f = f ii ,
.
The operator ∇ can be applied to the divergence of an amplitude vector
ux ( ) = uiei
and the result can be written as
2
grad div u= ∇∇ ⋅ u=∇ u=u j , ji
e i
.
2
The Laplace operator ∇ of the vector field ux ( ) = uie i
is the vector
2
∇ u= ∇⋅ ∇u= u k , jj
e k
= ∇∇⋅u− ∇× ( ∇ × u ).
or