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Mathematics for Physicists by Graham Shaw, Brian R. Martin

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7 Partial differentiation

In this chapter we generalise the discussion of differential calculus in Chapter 3 to functions of more than one variable. Many results will be taken over from Chapter 3 and will be dealt with rather briefly, so that we can focus on the differences between the two cases.

7.1 Partial derivatives

Given a function f(x1, x2, …, xn) of n independent variables, x1, x2, …, xn, the partial derivative of f with respect to x1 is defined by

(7.1) Unnumbered Display Equation

provided the limit exists. In other words, it is obtained by differentiating f with respect to x1, while treating the other variables x2, x3, …, xn as fixed parameters. Partial derivatives with respect to the other variables are defined in a similar way. For example, if

(7.2) Unnumbered Display Equation

then differentiating with respect to x keeping y fixed gives, using the product rule (3.20),

(7.3a) Unnumbered Display Equation

while differentiating with respect to y keeping x fixed gives

(7.3b) Unnumbered Display Equation

Higher derivatives are obtained by repeated partial differentiation, so that

(7.4)

for the second derivatives. Thus for the function (7.2), using (7.3) one obtains

and ...

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