Mathematical Methods for Physics and Engineering - Matematica.NET

11.5 SURFACE INTEGRALSindependent of the path taken. Since a is conservative, we can write a = ∇φ. Therefore, φmust satisfy∂φ∂x = xy2 + z,which implies that φ = 1 2 x2 y 2 + zx + f(y, z) for some function f. Secondly, we require∂φ∂y = x2 y + ∂f∂y = x2 y +2,which implies f =2y + g(z). Finally, since∂φ∂z = x + ∂g∂z = x,we have g =constant=k. It can be seen that we have explicitly constructed the functionφ = 1 2 x2 y 2 + zx +2y + k. ◭The quantity φ that figures so prominently in this section is called the scalarpotential function of the conservative vector field a (which satisfies ∇×a = 0), andis unique up to an arbitrary additive constant. Scalar potentials that are multivaluedfunctions of position (but in simple ways) are also of value in describingsome physical situations, the most obvious example being the scalar magneticpotential associated with a current-carrying wire. When the integral of a fieldquantity around a closed loop is considered, provided the loop does not enclosea net current, the potential is single-valued and all the above results still hold. Ifthe loop does enclose a net current, however, our analysis is no longer valid andextra care must be taken.If, instead of being conservative, a vector field b satisfies ∇ · b =0(i.e.bis solenoidal) then it is both possible and useful, for example in the theory ofelectromagnetism, to define a vector field a such that b = ∇×a. Itmaybeshownthat such a vector field a always exists. Further, if a is one such vector field thena ′ = a + ∇ψ + c, whereψ is any scalar function and c is any constant vector, alsosatisfies the above relationship, i.e. b = ∇×a ′ . This was discussed more fully insubsection 10.8.2.11.5 Surface integralsAs with line integrals, integrals over surfaces can involve vector and scalar fieldsand, equally, can result in either a vector or a scalar. The simplest case involvesentirely scalars and is of the form∫φdS. (11.8)SAs analogues of the line integrals listed in (11.1), we may also encounter surfaceintegrals involving vectors, namely∫ ∫∫φdS, a · dS, a × dS. (11.9)SS389S