16-02-2013, 04:12 PM
Vector analysis and Maxwell equations
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Maxwell equations
In 1860’s Maxwell has discovered the equations that describe the dynamics of electromagnetic
fields. Feinmann called this discovery the most important historical event of the XIXth
century.
These equations describe the dynamics of two vector fields E and B, electric and magnetic
ones, in the three space. The equations are written in local terms, but they express
global physical laws. In the vacuum, in the domain free of currents and charges they have
the form:
Local statements Global statements
I. divE = 0 A current of E through any closed surface equals
zero provided that there are no charges inside.
II curl E = ∂B
∂t The circulation of E over a closed curve
equals to the time derivative of the current
of the magnetic field through this curve.
III div B = 0 The current of B through any closed surface
equals zero.
IV c2 curl B = ∂E
∂t The circulation of B over a closed curve,
multiplied by c2, equals to the time derivative
of the current of the field E through a closed curve
in a domain free from currents (moving charges)
These laws are written in the language of the vector analysis. They have also formed
this language.
Our goal is to answer the following questions:
How to deduce local statements from the global ones?
What is “a current of a vector field through a closed curve”?
How to solve Maxwell equations?
2 Work and curle, current and divergence
When a k-form is given in a domain of Rn, with no coordinates chosen, and an oriented
compact k-surface is specified, an integral of this form over this surface immediately occurs.
A vector field is also a coordinate-free notion. When a vector field and a closed curve
are given, are the following quantities well defined:
• The work of the vector field over this curve?
• A current of the vector field through this curve?
We will now turn to these questions.
Consider the space Rn with an Eucledean structure on it
Definition 1 The circulation of a vector field v over a parameterized curve γ : [0, 1] → Rn
is defined by an integral
w(v, γ) =
∫ 1
0
(v(γ(t)), γ˙ (t))dt,
where (ξ, η) is a scalar product of vectors ξ and η.
Definition 2 A current of a vector field v in a Eucledean space Rn through a hypersurface
S with or without boundary and with a direction of the normal chosen, is an integral of a
scalar product of the vector of the field to a unit normal to the surface S:
F(v, S) =
∫
S
(v, n)(x)dsx,
2
where dsx is an element of the surface area.
The following definitions use coordinates.
Definition 3 Let v has coordinates v1, ..., vn. Then
.
A vector field with vanishing divergence is called divergence free.
Definition 4 In R3, let the vector field v have the components X, Y,Z. Then
curl v = (Zy − YZ,Xz − Zx, Yx − Xy).
To remember this definition, the following formula serves:
curl v = det
Here Dj is the operator of the derivative along the vector field ej .
Definition 5 The gradient of a function f at a point x of a Eucledean space is a vector
grad f(x) ∈ TxRn, whose scalar product equals to the differential of the function at x:
dfx(ξ) = (grad f(x), ξ).
The gradient vector field is called a a potential field, and a function itself, a potential.
3 Main formulas of vector analysis
Gauss–Ostrogradski formula
Let S be a closed surface in Rn, Ω is a domain bounded by S, and v be a smooth vector
field. Then the current of this vector field through the surface equals to the integral of the
divergence of the field over the domain bounded by the surface:
Stokes formula
Let γ be a closed curve in R3, S be a surface bounded by γ. Then the work (circulation)
of the field over γ equals to the integral of the curle of the field through the surface S:
w(v, γ) = F(S, curl v). (2)
Note that the Gausse formula holds in any dimension, and the Stokes formula for the
current holds in R3 only.
Note also that there was no definition of a current of a vector field through a closed
curve. No wonder! This notion may be well defined not for all the vector fields , but fot the
divergence free ones only. Indeed, the current of a vector field through a film bounded by
the curve, does not depend on the film, only in the case when the integral of the divergence
of the field over a domain bounded by these two films equals zero.
The coordinate definitions eqsily imply that
curl grad = 0, (3)
div curl = 0. (4)
Prove it!
More tricky formula: for the divergence free vector fields :
curl curl v = −Δv (5)
Prove it!
The Laplace operator is applied to the vector field component-wise.
The modern proofs of these formulas are given in the language of the differential forms.
4 Work and current forms
Definition 6 For any vector field in a Eucledean space the following 1-form is well defined:
ωv(x, ξ) = (v(x), ξ).
It is called a work form.
Problem 1 The integral of the work form over an oriented curve equals to circulation of
the field over the curve.
Therefore, to define the work of a vector field along a curve, we need not only the field
and the curve, but a Eucledean structure as well.
Consider now the Rn space with no Eucledean structure specified, but with a volume
n-form ωn given.
Definition 7 The flow form of the vector field v in Rn with the volume form ωn, specified,
is the interior derivative of the volume form along this field:
ϕv(x; ξ1, . . . , ξn¡1) = ωn(v(x), ξ1, . . . , ξn¡1).
Lemma 1 The flow of a vector field through an oriented surface S is the integral over S
of the correspondent flow form.