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

From this, one sees that

In general

(7.5)

for any f such that both the derivatives in (7.4) are continuous in x_i and x_j at the point of evaluation.

It is very important when working with partial derivatives to keep track of which variables are kept constant. This can be made explicit by adopting a notation in which the partial derivatives are written in brackets with the fixed variables as subscripts, so that (7.1) becomes

(7.6)

and (7.3a) and (7.3b) are written

To emphasise the importance of keeping track of which variables are held constant, we note that if we define z = xy, then (7.2) can be written

so that

This notation is widely used in thermal physics, for example, where different choices of variables are often used within the same calculation. Thus, the energy E of a gas at equilibrium is often written both as a function of temperature T and volume V, and also as a function of temperature and pressure P, but

except in the case of a ‘perfect gas’. In this chapter, we shall generally use the simpler notation (7.1), resorting to (7.6) only where there is room for ambiguity.

7.2 Differentials

For functions f(x) of a single variable x, we are already familiar with the result [cf. (5.27)]

(7.7)

for small changes δx, provided the derivative exists. In the same way, the definition (7.1) implies

(7.8)

since x₂, x₃, …, x_n are treated as fixed parameters in defining the partial derivatives. Analogous results are obtained for small changes in the other variables x₂, x₃, …, x_n. From this, for a function of two variables f(x₁, x₂) one obtains

and substituting (7.8) into the first term of this equation gives

where the omitted terms are quadratic in δx₁, δx₂. On generalising to n variables, this becomes

(7.9)

where the omitted terms are again quadratic in δx_i.

At this point, we denote small changes by dx or dx_i, and define the differential df by

(7.10)

for the case of single variables, and

(7.11)

for the case of multi-variables. The important distinction between (7.10, 7.11) and (7.7, 7.9) is that the latter are approximations, with corrections of the order indicated, whereas the former, being definitions, are exact.

Differentials are used repeatedly throughout the rest of this chapter. Here we will show, by an example, how they can be used to obtain partial derivatives when the definition of the relevant function is implicit.

7.2.1 Two standard results

In this subsection we will consider a function of two variables f(x, y) and use differentials to derive the standard results

(7.12)

and

(7.13)

To do this, we use (7.11) to give

(7.14a)

and then consider the corresponding function x(y, f) that specifies x in terms of y and f, to obtain the corresponding differential

(7.14b)

Substituting (7.14b) into (7.14a) gives

Since any two of dx, dy, df are independent, the coefficient of df on the right-hand side must be unity, which gives (7.12), and the square bracket giving the coefficient of dy must vanish, which gives (7.13), as required.

Finally, we stress again that in using (7.12) and (7.13), it is important to pay attention to the variables being kept fixed in each derivative. In particular,

in general, and the equality only holds if, as in (7.12), the same variables are kept fixed in each partial derivative.

7.2.4 Homogeneous functions and Euler's theorem

A function f(x₁,  x₂, …, x_n) is said to be homogeneous of degree k if

(7.22)

where λ is an arbitrary parameter. For example, the functions

are both homogeneous, of degree −1 and 3, respectively. Euler's theorem states that if f(x₁,  x₂, …, x_n) is homogeneous of degree k, then

(7.23)

To derive (7.23) we make the substitutions x_i = λt_i and write

For any fixed set of t₁, t₂, …, t_n, this is a function of λ only, and differentiating using the chain rule (7.20) gives

Euler's theorem then follows on multiplying by λ.

7.3 Change of variables

In this section we address the problem of how to change variables in equations that contain partial derivatives. To do this, we firstly consider a function f ≡ f( x₁,  x₂, …, x_n) of n variables x₁,  x₂, …, x_n that are each functions of another n variables x_i ≡ x_i(t₁,  t₂, …, t_n). Using (7.11) twice then gives

where we remind the reader that partial differentiation with respect to x_i implies that all the other x_j(j ≠ i) are kept constant; and similarly differentiating with respect to t_j means that all the other t_i(i ≠ j) are kept constant. In the same notation, expressing f directly in terms of t_j, j = 1 ,  2 , …, n gives

and comparing these two results gives the relation

(7.24)

between partial derivatives with respect to x_i and t_j.

To illustrate the use of this result, consider a function f(x, y) of the Cartesian co-ordinates x, y. We will change variables to the plane polar co-ordinates r ,  θ of Figure 2.3, where [cf. (2.34) and (2.35)]

(7.25a)

and conversely

(7.25b)

From (7.24), setting (x₁, x₂) = (x, y) and (t₁, t₂) = (r, θ), we obtain

and

where

Using (7.25a), these equations imply

(7.26a)

and

(7.26b)

and conversely¹

(7.27a)

and

(7.27b)

Corresponding results involving higher order partial derivatives can be obtained by repeated use of (7.26) and (7.27). As an example, we will transform Laplace's equation in two dimensions,

(7.28)

into polar co-ordinates (r, θ). From (7.27a), we have

In a similar way one obtains

Adding these two results and substituting in (7.28) then gives

(7.29)

as Laplace's equation expressed in plane polar co-ordinates.

7.4 Taylor series

The generalisation of Taylor's theorem (5.21) to more than one variable is straightforward. For simplicity, we start by finding an expansion of a function f(x, y) of two variables about x = x₀, y = y₀ in powers of h = x − x₀, k = y − y₀. To do this, for any given values of h and k, we define a function

which reduces to f(x, y) when the new variable t → 0. Provided the first N + 1 derivatives of F(t) exist over the whole range 0 ≤ t ≤ 1, Taylor's theorem (5.21) gives

(7.32a)

where is the nth derivative of F with respect to t [cf. (3.40b)], and where the remainder term is

(7.32b)

for at least one θ in the range 0 ≤ θ ≤ 1. However, from the chain rule (7.20) we also have

Substituting this into (7.32) and setting t = 1 then gives

(7.33a)

where

and

means the derivatives of f(x, y) are evaluated at x = x₀,  y = y₀. The remainder term is

(7.33b)

for at least one θ in the range 0 ≤ θ ≤ 1. Assuming R_N → 0 as N → ∞, then leads to the Taylor series

(7.34)

The above results are easily generalised to more than two variables. For a function f(x₁, x₂, …, x_k) of k variables, (7.34), for example, becomes

(7.35)

on expanding about x_i = a_i(i = 1, 2, …, k), where the right-hand side is evaluated at x₁ = a₁, x₂ = a₂, …, x_k = a_k. However, expansions such as (7.35) for several variables rapidly become unwieldy, so we will restrict ourselves to explicitly expanding (7.34), when one obtains

(7.36)

where all the derivatives are evaluated at x = x₀, y = y₀, and we have assumed (7.5). In general, if one assumes that the order of the cross derivatives is unimportant, that is,

as is usually the case², (7.34) becomes

(7.37)

where we have used the binomial expansion (1.23) and where all the derivatives are again evaluated at x = x₀, y = y₀.

7.5 Stationary points

The necessary and sufficient conditions for the differential df(x₁, x₂, …, x_n) to vanish for arbitrary dx₁, dx₂, …, dx_n are, from (7.11),

(7.38)

Points at which (7.38) are satisfied are called stationary points, in analogy to those discussed for a function of a single variable in Section 3.4.1. However, determining whether such points are local minima, maxima or saddle points is more complicated than for functions of a single variable. For simplicity, we shall restrict ourselves to functions of two variables f(x, y), which can be regarded as two-dimensional surfaces as shown in Figures 7.2 and 7.3. Suppose that f(x, y) has a stationary point at x = x₀, y = y₀, where by (7.38),

Then making a Taylor expansion about (x₀, y₀) gives

(7.39)

where we have neglected higher-order terms and assumed for all values of h and k. Then if (x₀ , y₀) is a minimum (maximum), as opposed to a saddle point, we must have for all non-zero h, k values. For h ≠ 0, this implies that

has no real roots, where . Since the condition for a quadratic az² + bz + c = 0 to have no real roots is b² < 4ac, this implies

(7.40a)

and the same condition is obtained if instead we consider k ≠ 0. Hence (7.40a) is a necessary condition for f(x₀, y₀) to be either a maximum or a minimum, and since the left-hand side is positive definite, this implies that and are either both positive or both negative. Specifically, if (7.40a) is true and

(7.40b)

then and f(x₀, y₀) is a maximum; whereas if (7.40a) holds and

(7.40c)

then and f(x₀, y₀) is a minimum. Examples of a maximum and a minimum in two variables are shown in Figure 7.2.

If on the other hand

(7.41)

f(x₀, y₀) is a saddle point. There are several different types of saddle point depending on the behaviour of the second derivative, and one example is shown in Figure 7.3.

Finally, if

then for all h and k, contradicting our earlier assumption, and higher-order terms in the Taylor expansion must be inspected to determine the nature of the stationary point.

*7.6 Lagrange multipliers

In the preceding section, we discussed how to find the stationary points of a function of two or more variables. However sometimes one needs to find the stationary points of the function when the variables are subject to one or more additional conditions, called ‘constraints’. To take a very simple example, one could ask: “What is the maximum area of a rectangular field surrounded by a fence of fixed length, say 200 m?” In other words, if the length and breadth of the field are x and y metres respectively, what is the maximum value of the area A = xy subject to the constraint x + y = 100 m. In simple problems of this kind, one can use the constraint to eliminate one of the variables. In the above case eliminating y gives

which is easily shown to have a maximum value A = 2500 m² for x = 50 m, corresponding to a square field with x = y = 50 m. However, in cases where the function and/or the constraint is more complicated, or there are more than two variables and more than one constraint, solving the problem by using each of the constraints to eliminate a variable can become very clumsy and tedious, and it is often easier to use an alternative method due to Lagrange.

Suppose we need to find the stationary points of a function f(x₁, x₂, …, x_n), where the variables are restricted to a limited range of values by k constraints, that we shall assume can be written in the form

(7.42)

where k < n. In this case, the relation

(7.43)

no longer leads to the usual conditions

because the dx_i are no longer independent, but are related by conditions of the form

(7.44)

This problem can in principle be solved, as in the simple example discussed above, by using conditions (7.42) to eliminate k of the variables, and expressing f(x₁, x₂, …, x_n) as a function of the remaining independent variables, which can then be minimised in the usual way. However, following Lagrange, it is often more efficient to consider a new function,

(7.45)

where the λ_j are new variables called undetermined multipliers. One then determines the stationary points of F by treating x₁, x₂, …, x_n as independent variables to give n conditions

(7.46)

These determine the values of x₁, x₂, …, x_n as functions of the variables λ_j(j = 1, 2, …, k), whose values can then be determined by requiring the k conditions (7.42) to be satisfied. In other words, the n + k variables x₁, …, x_n, λ₁, …, λ_k are determined by the n + k equations (7.46) and (7.42); and since F → f when (7.42) are satisfied, the x_i values correspond to the stationary points of f subject to the constraints (7.42). This procedure is best illustrated by example.

*7.7 Differentiation of integrals

We conclude this chapter by using the properties of partial derivatives to deduce the rules for differentiating integrals with respect to a variable parameter, starting with the indefinite integral

(7.49)

where (4.1) together with the definition of partial derivatives implies

(7.50)

Then the partial derivative

(7.51)

provided that F satisfies (7.5), that is,

(7.52)

To see this, we note that (7.52) and (7.50) imply

Integrating this equation with respect to x then gives

(7.53)

which together with (7.49) gives (7.51). In other words, we may reverse the order of the differentiation and integration, as in (7.51), provided (7.52) is satisfied. As we saw in Section 7.1, this is so if the first- and second-order partial derivatives of F are continuous in x and t, as is usually the case.

We next consider the definite integral

(7.54)

where the limits of integration, as well as the integrand, may also depend on t. Then, using the chain rule (7.21), we have

provided a, b are differentiable functions of t. In addition, (7.53) implies

so that, using this together with (7.50), one finally obtains Leibnitz's rule,

(7.55)

which reduces to

(7.56)

for fixed limits a, b. Finally, one may allow b → ∞ and/or a → −∞, provided all the integrals converge.

As well as allowing given integrals to be differentiated, these results can be exploited by using known integrals to evaluate related, unknown integrals. For example, in thermal physics one frequently needs to evaluate integrals of the form

(7.57)

where n ≥ 0 and α > 0. There are no problems with convergence, and for n = 0 one easily obtains

while differentiating (7.57) with respect to α using (7.56) gives

and hence

Thus, and in general

(7.58)