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

288 Multiple and Line Integrals [Ch. 7
Sec. 12. Fundamentals of Field Theory
1. Scalar and vector fields. A scalar field is defined by the scalar function
of the point = /(/>) = /(*, t/, z), where P (x, y, z) is a point of space. The
surfaces f (x, y, z) = C, where C = const, are called level surfaces of the scalar
field.
A vector field is defined by the vector function of the point a = a(P)~
a(r), where P is a point of space and r=xi-\-yj+zk is the radius vector
of the point P. In coordinate form, a ax i + a
vj-\-azk, where ax ~ax (x, y, z),
a
y ay(x, y, z), and a = z a z (x, //, z) are projections of the vector a on the
coordinate axes. The vector lines (force lines, flow lines) of a vector field are
found from the following system of differential equations
~~~'
dx__dy _dz
A scalar or vector field that does not depend on the time t is called
'ry; if it stationary; depends on t
Gradient. The vector
if it depends on the time, it is called - ~ "
nonstationary.
t. Th
where V = ^3-+y^- + ^y is the Hamiltonian operator (del, or nabla), is
called the gradient of the field U = f (P) at the given point P (ci. Ch. VI, Sec. 6).
The gradient is in the direction of the normal n to the level surface at the
point P and in the direction of increasing function U, and has length equal
to
dn~~ \dx
If the direction is given by the unit vector / {cos a, cos p, cos Y},
= -
cos a + - -
cos p + cosy
then
(the derivative of the function U in the direction /).
3. Divergence and rotation. The divergence of a vector field a (P) ~ a^i \~
+ a yj+az k is the scalar diva-^ +^ + ^^Va.
The rotation (curl) of a vector field a (P) = a x i + a y j+ a z k is the vector
da z
da y
4. Flux of a vector. The flux of a vector field a(P) through a surfaces
in a direction defined by the unit vector of the normal ujcosa, cos p, COSY}
to the surface S is the integral
\ \ an dS = \ \ a n dS \ \ (ax cos a -|- a y cos p + a z cos Y) dS.
S s S
If S is a closed surface bounding a volume V, and n is a unit vector of the
outer normal to the surface S, then the Ostrogradsky -Gauss formula holds,