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Compendium of vector analysis with applications to continuum mechanics
compiled by Valery P. Dmitriyev
Lomonosov University P.O.Box 160, Moscow 117574, Russia
e-mail: dmitr@cc.nifhi.ac.ru
1. Connection between integration and differentiation
Gauss-Ostrogradsky theorem
We transform the volume integral into a surface one:
+
| ( ) ∫∂iPdV
( ) V
= ∫∂iPdxidx jdxk V
=
∫ dx jdxk
S(V )
x
i
x
i
x j ,xk x j ,xk
P
=
( ( ) ) ( ( ) ) =
∫ dx jdxk
S(V )
 
P
xi+
x j ,xk
,x j ,xk
P xi
x j ,xk
,x j ,xk
 
=
= ∫cos θe+xtdSP ∫cosθintdSP = ∫cos θextdSP = ∫n⋅eiPdS
S+
S
S
Here the following denotations and relations were used:
( ) P is a multivariate function P xi,x j ,xk , ∂i = ∂ / ∂xi , V volume,
S surface, ei a basis vector, ei ⋅e j = /ij , n the external normal to the element dS of closed surface with
dx jdxk = n⋅ei dS , n⋅ei = cosθ .
Thus
∫ ∂ i PdV
V
= S(∫V )Pn⋅eidS
(1.1)
Using formula (1.1), the definitions below can be transformed into coordinate representation.
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Gradient
( ) S(∫VP) ndS
=
∫ n⋅ei
S(V )
ei PdS
= ∫∂iPeidV V
where summation over recurrent index is implied throughout. By definition
gradP = ∇P = ∂iPei
Divergence
( ) S(∫VA)
ndS
=
S
(V
n⋅ei
)
AidS = ∫∂i AidV V
(1.2)
By definition
di#A = ∇⋅A = ∂i Ai
Curl
( ) ∫n× AdS = ∫ n⋅ei ei × Aje jdS = ∫∂i Ajei×e jdV (1.3)
S(V )
S(V )
V
By definition
curlA = ∇× A = ∂i Ajei ×e j
Stokes theorem follows from (1.3) if we take for the volume a right cylinder
with the height h → 0 . Then the surface integrals over the top and bottom areas
mutually compensate each other. Next we consider the triad of orthogonal unit
vectors
m, n, 2
where m is the normal to the top base and n the normal to the lateral face
2 =m×n
Multiplying the left-hand side of (1.3) by m gives
∫m⋅n×AdS = ∫(m×n)⋅AdS = ∫ 2 ⋅AdS = h∫ 2⋅Adl
lateral
lateral
lateral
l
where 2 is the tangent to the line. Multiplying the right-hand side of (1.3) by m gives
h∫m⋅ curlAdS
S
where m is the normal to the surface. Now, equating both sides, we come to the
formula sought for
∫ 2⋅Adl = ∫m⋅ curlAdS
l
S
The Stokes theorem is easily generalized to a nonplanar surface (applying to it
Ampere's theorem). In this event, the surface is approximated by a polytope.
Then mutual compensation of the line integrals on common borders is used.
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2. Elements of continuum mechanics
( ) A medium is characterized by the volume density ρ x,t and the flow ( ) velocity u x,t .
Continuity equation
The mass balance in a closed volume is given by
∂t ∫ ρdV + ∫ ρu⋅ndS = 0
V
S(V )
where ∂t = ∂ / ∂t . We get from (1.2)
ρu⋅ndS = ∫∂i(ρui )dV
Thereof the continuity equations follows
( ) ∂t ρ + ∂i ρui = 0
Stress tensor
We consider the force df on the element dS of surface in the medium and are interested in its dependence on normal n to the surface
df (n)
where
df(n)= df(n)
With this purpose the total force on a closed surface is calculated. We have for the force equilibrium at the coordinate tetrahedron
df (n)+ df (n1)+ df(n2)+ df (n3)= 0
where the normals are taken to be external to the surface
( ) ( ) ( ) n1 = sign n⋅e1 e1 , n2 = sign n⋅e2 e2 , n3 = sign n⋅e3 e3
Thence
df (n)= sign(n⋅e j )df (e j )
(2.1)
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( ) The force density 1 n is defined by
Insofar as
df = 1dS
we have for (2.1)
dS j= n⋅e j dS
df (n)= sign(n⋅e j )1(e j )dS j = sign(n⋅e j ) n⋅e j 1(e j )dS = n⋅e j1(e j )dS
i.e.
1(n)= n⋅e j1(e j )
( ) = n⋅e jei1i e j
( ) The latter means that 1 n possesses the tensor property. The elements of the stress
tensor are defined by
( ) σij =σi e j
Now, using (1.2), the force on a closed surface can be computed as a volume integral
( ) ∫1ndS = ∫1e j e j ⋅ndS = ∫∂ j1 e j dV
(2.2)
V
Euler equation
The momentum balance is given by the relation
∂t ∫ ρudV + ∫(ρu)u⋅ndS = ∫1dS
(2.3)
V
S(V )
S(V )
We have for the second term by (1.2)
( ) ∫(ρu)u⋅ndS = ∫(ρu)u je j ⋅ndS = ∫∂ j ρu ju dV
Using also (2.2) gives for (2.3)
∂t (ρu)+ ∂ j (ρu ju)= ∂ j1(e j )
or
( ) ρ∂tu + ρu j∂ ju = ∂ j1 e j
(2.4)
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Hydrodynamics
The stress tensor in a fluid is defined from the pressure as
That gives for (2.4)
σ ij = pδij ρ∂tui + ρu j∂ jui + ∂ j p = 0
Elasticity
( ) The solid-like medium is characterized by the displacement s x,t . For small
displacements
u=∂ts
and the quadratic terms in the left-hand part of (2.4) can be dropped. For an isotropic homogeneous medium the stress tensor is determined from the Hooke's law as
( ) ( ) σ i e j = λδij∂k sk + µ ∂is j +∂ jsi
where λ and µ are the elastic constants. That gives
( ) ( ) ( ) ∂ jσi e j
= λ∂i∂k sk + µ ∂i∂ js j +∂2j si
=
λ+µ
∂i∂
j
s
j
+
µ∂
2 j
si
and
∂ j1(e j )= (λ+µ)graddi# s + µ∇2s
= (λ+2µ)∇2s + (λ+µ)curlcurls
= λ graddi# s µ curlcurls
where graddi# = ∇2 + curlcurl was used. Substituting it to (2.4) we get finally
Lame equation
( ) ρ∂t2s = λ+µ graddi# s + µ∇2s
where ρ is constant.
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