Mathematics for Physicists

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, …, x_n) of n independent variables, x₁, x₂, …, x_n, the partial derivative of f with respect to x₁ is defined by

(7.1)

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

(7.2)

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

(7.3a)

while differentiating with respect to y keeping x fixed gives

(7.3b)

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 ...