Problems in Mathematical Analysis.pdf - pwp.net.ipl.pt

Sec. 12] Fundamentals of Field Theory 289
which in vector form is
ff (\ r\ r*
div a dx dy dz.
5. Circulation of a vector, the work of a Held. The line integral of the
vector a along the curve C is defined by the formula
f a dr= \ a s ds V a x dx -f- a ydy -f a z dz
C C C
and represents the work done by the field a along the curve C (as
is the
projection of the vector a on the tangent to C).
If C is closed, then the line integral (1) is called the circulation of the
vector field a around the contour C.
If the closed curve C bounds a two-sided surfaces, then Stokes' formula
holds, which in vector form has the form
adr= f f /i rotadS,
where n is the vector of the normal to the surface S; the direction of the
vector should be chosen so that for an observer looking in the direction of n
the circulation of the contour C should be counterclockwise in a right-handed
coordinate system.
6. Potential and solenoidal fields. The vector iield a(r) is called poten-
tial if
where Uf(r) is a scalar function (the potential of the field).
For the potentiality of a field a, given in a simply-connected domain,
it is necessary and sufficient that it be non rotational, that is, rota = 0. In
that case there exists a potential U defined by the equation
U,
dU ~a x dx-}- a v dy -f- a 2 dz.
If the potential U is a single-valued function, then \ a dr U (B) U (A);
AB
in particular, the circulation of the vector a is equal to zero: m adr=Q.
A vector field a (r) is called solenoidal if at each point of the field div
a = 0; in this case the flux of the vector through any closed surface is zero.
If the field is at the same time potential and solenoidal, then div (grad U)=.Q
and the potential function U is harmonic; that is, it satisfies the Laplace
or AU=0 where A= isthe
*'=;+>+>
"Er++5-. '
Laplacian operator
2371. Determine the level surfaces of the scalar field
where r \fx*+y*-\-z*. What will the level surfaces be of a field
U = F(Q), where
10-1900
(0