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