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 − ∫cosθi−ntdSP = ∫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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