Mathematical Methods for Physics and Engineering - Matematica.NET

5Partial differentiationIn chapter 2, we discussed functions f of only one variable x, which were usuallywritten f(x). Certain constants and parameters may also have appeared in thedefinition of f, e.g.f(x) =ax+2 contains the constant 2 and the parameter a, butonly x was considered as a variable and only the derivatives f (n) (x) =d n f/dx nwere defined.However, we may equally well consider functions that depend on more than onevariable, e.g. the function f(x, y) =x 2 +3xy, which depends on the two variablesx and y. For any pair of values x, y, the function f(x, y) has a well-defined value,e.g. f(2, 3) = 22. This notion can clearly be extended to functions dependent onmore than two variables. For the n-variable case, we write f(x 1 ,x 2 ,...,x n )fora function that depends on the variables x 1 ,x 2 ,...,x n .Whenn =2,x 1 and x 2correspond to the variables x and y used above.Functions of one variable, like f(x), can be represented by a graph on aplane sheet of paper, and it is apparent that functions of two variables can,with little effort, be represented by a surface in three-dimensional space. Thus,we may also picture f(x, y) as describing the variation of height with positionin a mountainous landscape. Functions of many variables, however, are usuallyvery difficult to visualise and so the preliminary discussion in this chapter willconcentrate on functions of just two variables.5.1 Definition of the partial derivativeIt is clear that a function f(x, y) of two variables will have a gradient in alldirections in the xy-plane. A general expression for this rate of change can befound and will be discussed in the next section. However, we first consider thesimpler case of finding the rate of change of f(x, y) in the positive x- andydirections.These rates of change are called the partial derivatives with respect151