Mathematical Methods for Physics and Engineering - Matematica.NET

PARTIAL DIFFERENTIATIONcan be obtained. It will be noticed that the first bracket in (5.3) actually approximatesto ∂f(x, y +∆y)/∂x but that this has been replaced by ∂f(x, y)/∂x in (5.4).This approximation clearly has the same degree of validity as that which replacesthe bracket by the partial derivative.How valid an approximation (5.4) is to (5.3) depends not only on how small∆x and ∆y are but also on the magnitudes of higher partial derivatives; this isdiscussed further in section 5.7 in the context of Taylor series for functions ofmore than one variable. Nevertheless, letting the small changes ∆x and ∆y in(5.4) become infinitesimal, we can define the total differential df of the functionf(x, y), without any approximation, asdf = ∂f ∂fdx + dy. (5.5)∂x ∂yEquation (5.5) can be extended to the case of a function of n variables,f(x 1 ,x 2 ,...,x n );df = ∂f dx 1 + ∂f dx 2 + ···+ ∂f dx n . (5.6)∂x 1 ∂x 2 ∂x n◮Find the total differential of the function f(x, y) =y exp(x + y).Evaluating the first partial derivatives, we find∂f∂f= y exp(x + y),∂x=exp(x + y)+y exp(x + y).∂yApplying (5.5), we then find that the total differential is given bydf =[y exp(x + y)]dx +[(1+y)exp(x + y)]dy. ◭In some situations, despite the fact that several variables x i , i =1, 2,...,n,appear to be involved, effectively only one of them is. This occurs if there aresubsidiary relationships constraining all the x i to have values dependent on thevalue of one of them, say x 1 . These relationships may be represented by equationsthat are typically of the formx i = x i (x 1 ), i =2, 3,...,n. (5.7)In principle f can then be expressed as a function of x 1 alone by substitutingfrom (5.7) for x 2 ,x 3 ,...,x n , and then the total derivative (or simply the derivative)of f with respect to x 1 is obtained by ordinary differentiation.Alternatively, (5.6) can be used to givedfdx 1= ∂f∂x 1+( ∂f∂x 2) dx2dx 1+ ···+( ∂f∂x n) dxndx 1. (5.8)It should be noted that the LHS of this equation is the total derivative df/dx 1 ,whilst the partial derivative ∂f/∂x 1 forms only a part of the RHS. In evaluating154