Most physical systems are characterized by more than two quantitative variables. Experience has shown that it is not always possible to change such quantities at will, but that specification of some of them will determine definite values for others. Functional relations involving three or more variables lead us to branches of calculus which make use of partial derivatives and multiple integrals.
We have already snuck in the concept of partial differentiation in several instances by evaluating the derivative with respect to x of a function of the form , while treating as if they were constant quantities. The correct notation for such operations makes use of the “curly dee” symbol , for example
To begin with, we will consider functions of just two independent variables, such as . Generalization to more than two variables is usually straightforward. The definitions of partial derivatives are closely analogous to that of the ordinary derivative:
The subscript y or x denotes the variable which is held constant. If if there is no ambiguity, the subscript can be omitted, as in (11.1). Some alternative notations for are , and . As shown in Figure 11.1, a partial derivative such as can be interpreted geometrically as the instantaneous slope at the point (x, y) of the curve formed by the intersection of the surface and the plane y = constant, and analogously for . Partial derivatives can be evaluated by the same rules as for ordinary differentiation, treating all but one variable as constants.
Figure 11.1 Graphical representation of partial derivatives. The curved surface represents in the first quadrant. Vertically and horizontally cross-hatched planes are x = constant and y = constant, respectively. Lines ab and cd are drawn tangent to the surface at point . The slopes of ab and cd equal and , respectively.
Products of partial derivatives can be manipulated in the same way as products of ordinary derivatives provided that the same variables are held constant. For example,
Since partial derivatives are also functions of the independent variables, they can themselves be differentiated to give second and higher derivatives. These are written, for example, as
or, in more compact notation, . Also possible are mixed second derivatives such as
When a function and its first derivatives are single valued and continuous, the order of differentiation can be reversed, so that
or, more compactly, . Higher-order derivatives such as and can also be constructed.
Thus far we have considered changes in brought about by changing one variable at a time. The more general case involves simultaneous variation of x and y. This could be represented in Figure 11.1 by a slope of the surface cut in a direction not parallel to either the x- or y-axis. Consider the more general increment
Adding and subtracting the quantity and inserting the factors and , we find
Passing to the limit , the two bracketed quantities approach the partial derivatives (11.2) and (11.3). The remaining increments approach the differential quantities . The result is the total differential:
Extension of the total differential to functions of more than two variables is straightforward. For a function of n variables, , the total differential is given by
A neat relation among the three partial derivatives involving x, y, and z can be derived from Eq. (11.11). Consider the case when z = constant, so that . We have then that
But the ratio of dy to dx means, in this instance, , since z was constrained to a constant value. Thus we obtain the important identity
or in a cyclic symmetrical form
As an illustration, suppose we need to evaluate for one mole of a gas obeying Dieterici’s equation of state
The equation cannot be solved in closed form for either V or T. However, using (11.14), we obtain, after some algebraic simplification:
A trivial case of a double integral can be obtained from the product of two ordinary integrals, say
Since the variable in a definite integral is just a dummy variable, its name can be freely changed, say from x to y, in the first equality above. It is clearly necessary that the dummy variables have different names when they occur in a multiple integral. A double integral can also involve a nonseparable function . For well-behaved functions the integrations can be performed in either order. Thus
For well-behaved functions the integrations above can be performed in either order.
More challenging are cases in which the limits of integration are themselves functions of x and y, for example
If the function is continuous, either of the integrals above can be transformed into the other by inverting the functional relations for the limits from to . This is known as Fubini’s theorem. The alternative evaluations of the integral are represented in Figure 11.2.
Figure 11.2 Evaluation of double integral . On left, horizontal strips are integrated over x between and and then summed over y. On right, vertical strips are integrated over y first. By Fubini’s theorem, the alternative methods give the same result.
As an illustration, let us do the double integration over area involved in the geometric representation of hyperbolic functions (see Figure 4.17). Referring to Figure 11.3, it is clearly easier to first do the x-integration over horizontal strips between the straight line and the rectangular hyperbola. The area is then given by
This reduces to an integration over y:
Since , we obtain . Thus we can express the hyperbolic functions in terms of the shaded area A:
Cartesian coordinates locate a point in a plane by specifying how far east (x-coordinate) and how far north (y-coordinate) it lies from the origin (0, 0). A second popular way to locate a point in two dimensions makes use of plane polar coordinates, , which specifies distance and direction from the origin. As shown in Figure 11.4, the direction is defined by an angle , obtained by counterclockwise rotation from an eastward heading. Expressed in terms of Cartesian variables x and y, the polar coordinates are given by
Integration of a function over two-dimensional space is expressed by
In Cartesian coordinates, the plane can be “tiled” by infinitesimal rectangles of width dx and height dy. Both x and y range over . In polar coordinates, tiling of the plane can be accomplished by fan-shaped differential elements of area with sides dr and , as shown in Figure 11.5. Since r and have ranges and , respectively, an integral over two-dimensional space in polar coordinates is given by
It is understood that, expressed in terms of their alternative variables, .
A systematic transformation of differential elements of area between two coordinate systems can be carried out using
In terms of the Jacobian determinant
In this particular case, we find using (11.26)
in agreement with the tiling construction.
A transformation from Cartesian to polar coordinates is applied to evaluation of the famous definite integral
Taking the square and introducing a new dummy variable, we obtain
The double integral can then be transformed to polar coordinates to give
Cylindrical coordinates are a generalization of polar coordinates to three dimensions, obtained by augmenting r and with the Cartesian z coordinate. (Alternative notations you might encounter are r or for the radial coordinate and or for the azimuthal coordinate.) The Jacobian determinant is given by
the same value as for plane polar coordinates. An integral over three-dimensional space has the form
As an application of cylindrical coordinates, let us derive the volume of a right circular cone of base radius R and altitude h, shown in Figure 11.6. This is obtained, in principle, by setting the function inside the desired volume and equal to zero everywhere else. The limits of r-integration are functions of z, such that between and . (It is most convenient here to define the z-axis as pointing downward from the apex of the cone.) Thus,
which is the volume of a cylinder with the same base and altitude.
Spherical polar coordinates provide the most convenient description for problems involving exact or approximate spherical symmetry. The position of an arbitrary point P is described by three coordinates , as shown in Figure 11.7. The radial variable r gives the distance OP from the origin to the point P. The azimuthal angle, now designated as , specifies the rotational orientation of OP about the z-axis. The third coordinate, now called , is the polar angle between OP and the Cartesian z-axis. Polar and Cartesian coordinates are connected by the relations:
with the reciprocal relations
The coordinate is analogous to latitude in geography, in which and correspond to the North and South Poles, respectively. Similarly, the angle is analogous to geographic longitude, which specifies the east or west angle with respect to the Greenwich meridian. The ranges of the spherical polar coordinates are given by:
The volume element in spherical polar coordinates can be determined from the Jacobian:
Therefore, a three-dimensional integral can be written
A wedge-shaped differential element of volume in spherical polar coordinates is shown in Figure 11.8.
Integration over the two polar angles gives
This represents the steradians of solid angle which radiate from every point in three-dimensional space. For integration over a spherical symmetrical function , independent of and , (11.40) can be simplified to
This is equivalent to integration over a series of spherical shells of area and thickness dr.
Differential quantities of the type
known as Pfaff differential expressions are of central importance in thermodynamics. Two cases are to be distinguished. Eq. (11.43) is an exact differential if there exists some function for which it is the total differential; an inexact differential if there exists no function which gives (11.43) upon differentiation. If dq is exact, we can write
Comparing with the total differential of ,
we can identify
Note further that
As discussed earlier, mixed second derivatives of well-behaved functions are independent of the order of differentiation. This leads to Euler’s reciprocity relation
which is a necessary and sufficient condition for exactness of a differential expression. Note that the reciprocity relation neither requires nor identifies the function .
A simple example of an exact differential expression is
Here and , so that and Euler’s condition is satisfied. It is easy to identify the function in this case as , since . A differential expression for which the reciprocity test fails is
Here , so that dq is inexact and no function exists whose total differential equals (11.50).
However, an inexact differential can be cured! An inexact differential expression with can be converted into an exact differential expression by use of an integrating factor. This means that becomes exact with
For example, (11.50) can be converted into an exact differential by choosing so that
Alternatively, converts the differential to , while converts it to . In fact, times any function of is also an integrating factor. An integrating factor exists for every differential expression in two variables, such as (11.43). For differential expressions in three or more variables, such as
an integrating factor does not always exist.
The First and Second Laws of Thermodynamics can be formulated mathematically in terms of exact differentials. Individually, dq, an increment of heat gained by a system and dw, an increment of work done on a system are represented by inexact differentials. The First Law postulates that their sum is an exact differential:
which is identified with the internal energy U of the system. A mathematical statement of the Second Law is that 1/T, the reciprocal of the absolute temperature, is an integrating factor for dq in a reversible process. The exact differential
defines the entropy S of the system. These powerful generalizations hold true no matter how many independent variables are necessary to specify the thermodynamic system.
Consider the special case of reversible processes on a single-component thermodynamic system. A differential element of work in expansion or compression is given by , where is the pressure and V, the volume. Using (11.55), the differential of heat equals . Therefore the First Law (11.54) reduces to
sometimes known as the fundamental equation of thermodynamics. Remarkably, since this relation contains only functions of state, U, T, S, P, and V, it applies very generally to all thermodynamic processes—reversible and irreversible. The structure of this differential expression implies that the energy U is a natural function of S and , and identifies the coefficients
The independent variables in a differential expression can be changed by a Legendre tranformation. For example, to reexpress the fundamental equation in terms of S and p, rather than S and V, we define the enthalpy. This satisfies the differential relation
which must be the differential of the function . Analogously we can define the Helmholtz free energy, such that
and the Gibbs free energy, which satisfies
A Legendre transformation also connects the Lagrangian and Hamiltonian functions in classical mechanics. For a particle moving in one dimension, the Lagrangian can be written
with the differential form
which is recognized as the momentum of the particle. The Hamiltonian is defined by the Legendre transformation
This leads to
which represents the total energy of the system.
The extension of the concept of integration considered in this section involves continuous summation of a differential expression along a specified path C. For the case of two independent variables, the line integral can be defined as follows:
where and . All the points lie on a continuous curve C connecting to , as shown in Figure 11.9. In mechanics, the work done on a particle is equal to the line integral of the applied force along the particle’s trajectory.
Figure 11.9 Line integral as limit of summation at points along path C between and . The value of the integral along path will, in general, be different.
The line integral (11.66) reduces to a Riemann integral when the path of integration is parallel to either coordinate axis. For example along the linear path , we obtain
More generally, when the curve C can be represented by a functional relation can be eliminated from (11.66) to give
In general, the value of a line integral depends on the path of integration. Thus the integrals along paths C and in Figure 11.9 can give different results. However, for the special case of a line integral over an exact differential, the line integral is independent of path, its value being determined by the initial and final points. To prove this, suppose that is an exact differential equal to the total differential of the function . We can therefore substitute in Eq. (11.66)
neglecting terms of higher order in and . Accordingly,
noting that all the shaded intermediate terms cancel out. In the limit , we find therefore
independent of the path C, just like a Riemann integral.
Of particular significance are line integrals around closed paths, in which the initial and final points coincide. For such cyclic paths, the integral sign is written . The closed curve is by convention traversed in the counterclockwise direction. If is an exact differential, then
for an arbitrary closed path C. (There is the additional requirement that must be analytic within the path C, with no singularities.) When is inexact, the cyclic integral is, in general, different from zero. As an example, consider the integral around the rectagular closed path shown in Figure 11.10. For the two prototype examples of differential expressions [Eqs. (11.49) and (11.50)] we find
A line integral around a curve C can be related to a double Riemann integral over the enclosed area S by Green’s theorem:
Green’s theorem can be most easily proved by following the successive steps shown in Figure 11.11. The line integral around the small rectangle in (i) gives four contributions, which can be written
which establishes Green’s theorem for the rectangle:
This is applicable as well for the composite figure formed by two adjacent rectangles, as in (ii). Since the line integral is taken counterclockwise in both rectangles, the common side is transversed in opposite directions along paths C and , and the two contributions cancel. More rectangles can be added to build up the shaded figure in (iii). Green’s theorem remains valid when S corresponds to the shaded area and C to its zigzag perimeter. In the final step (iv), the elements of the rectangular grid are shrunken to infinitesimal size to approach the area and perimeter of an arbitrary curved figure.
By virtue of Green’s theorem, the interrelationship between differential expressions and their line integrals can be succinctly summarized. For a differential expression , any of the following statements implies the validity of the other two: