Mathematical Methods for Physics and Engineering : A Comprehensive Guide

11.5 SURFACE INTEGRALS

independent of the path taken. Sinceais conservative, we can writea=∇φ. Therefore,φ
must satisfy
∂φ
∂x

=xy^2 +z,

which implies thatφ=^12 x^2 y^2 +zx+f(y, z) for some functionf. Secondly, we require

∂φ

∂y

=x^2 y+

∂f

∂y

=x^2 y+2,

which impliesf=2y+g(z). Finally, since

∂φ

∂z

=x+

∂g

∂z

=x,

we haveg=constant=k. It can be seen that we have explicitly constructed the function
φ=^12 x^2 y^2 +zx+2y+k.

The quantityφthat figures so prominently in this section is called thescalar

potential functionof the conservative vector fielda(which satisfies∇×a= 0 ), and

is unique up to an arbitrary additive constant. Scalar potentials that are multi-

valued functions of position (but in simple ways) are also of value in describing

some physical situations, the most obvious example being the scalar magnetic

potential associated with a current-carrying wire. When the integral of a field

quantity around a closed loop is considered, provided the loop does not enclose

a net current, the potential is single-valued and all the above results still hold. If

the loop does enclose a net current, however, our analysis is no longer valid and

extra care must be taken.

If, instead of being conservative, a vector fieldbsatisfies∇·b=0(i.e.b

is solenoidal) then it is both possible and useful, for example in the theory of

electromagnetism, to define avector fieldasuch thatb=∇×a.Itmaybeshown

that such a vector fieldaalways exists. Further, ifais one such vector field then

a′=a+∇ψ+c,whereψis any scalar function andcis any constant vector, also

satisfies the above relationship, i.e.b=∇×a′. This was discussed more fully in

subsection 10.8.2.

11.5 Surface integrals

As with line integrals, integrals over surfaces can involve vector and scalar fields

and, equally, can result in either a vector or a scalar. The simplest case involves

entirely scalars and is of the form
∫

S

φdS. (11.8)

As analogues of the line integrals listed in (11.1), we may also encounter surface

integrals involving vectors, namely
∫

S

φdS,

∫

S

a·dS,

∫

S

a×dS. (11.9)