Notation in Variational Calculus

Variational calculus is concerned with finding extrema for example, what is the shortest distance between two points on the surface of a parabolic cylinder In ordinary, garden-variety calculus, we deal wiihfunctions, which are objects whose values depend on the values of numerical quantities. But in the variational calculus, the focus of attention is on functionals, which are objects whose values depend on functions. For example, we may interpret the entropy as a functional because its value depends on other thermodynamic functions, such as temperature, pressure, and composition. Since the functionals differ from functions, we sometimes find it convenient to use a notation for operators on functionals that differs somewhat from the notation for operators on functions. For our purposes, the most important notational distinction occurs for differential operators. 

Let/ be a functional that depends on C functions a , i = 1,2. C. When/ is at a stationary point (a maximum or minimum), the functions x have values x,. The variation of any x,- about its stationary value can be represented by 

Then the first-order variation of/merely means the total differential, evaluated at the stationary point. 

Since the quantities /of interest to us form exact differentials, the second-order variation in (G.0.5) is invariant under an exchange of the indices i and j. We need not proceed further into the variational calculus here because in this book we use the variations 8/and 8 /merely as a notational convenience relations (G.0.4) and (G.0.5) define the notation we use. 

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