Advanced Calculus fi..

defined analogously:Chapter 2 Differential Calculus of Functions of Several Variables 137The origin of the v2 symbol lies in the interpretation of V as a "vector differentialoperator":One then has symbolicallyIThis point of view will be discussed further in Chapter 3.If z = f (x, y) has continuous second derivatives in a domain D andv2z = 0 (2.126)in D, then z is said to be harmonic in D. The same term is used for a function ofthree variables that has continuous second derivatives in a domain D in space andwhose Laplacian is 0 in D. The two equations for harmonic functions:are known as the Laplace equations in two and three dimensions, respectively.Another important combination of derivatives occurs in the biharmonicequation:which arises in the theory of elasticity. The combination that appears here can beexpressed in terms of the Laplacian, for one hasIf we write v4z = v2(v2z), then the biharmonic equation can be written:Its solutions are termed bihamzonic functions. This can again be generalized tofunctions of three variables, (2.129) suggesting the definition to be used.Harmonic functions arise in the theory of electromagnetic fields, in fluid dynamics,in the theory of heat conduction, and in many other parts of physics; applicationswill be discussed in Chapters 5, 8, and 10. Biharmonic functions are used mainly inelasticity. (See Section 8.25.)