Chapter 11

Multivariable Calculus

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.

11.1 Partial Derivatives

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 image, while treating image as if they were constant quantities. The correct notation for such operations makes use of the “curly dee” symbol image, for example

image (11.1)

To begin with, we will consider functions of just two independent variables, such as image. Generalization to more than two variables is usually straightforward. The definitions of partial derivatives are closely analogous to that of the ordinary derivative:

image (11.2)


image (11.3)

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 image are image, and image. As shown in Figure 11.1, a partial derivative such as image can be interpreted geometrically as the instantaneous slope at the point (x, y) of the curve formed by the intersection of the surface image and the plane y = constant, and analogously for image. 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 image 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 image. The slopes of ab and cd equal image and image, 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,

image (11.4)


image (11.5)

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

image (11.6)

or, in more compact notation, image. Also possible are mixed second derivatives such as

image (11.7)

When a function and its first derivatives are single valued and continuous, the order of differentiation can be reversed, so that

image (11.8)

or, more compactly, image. Higher-order derivatives such as image and image can also be constructed.

Thus far we have considered changes in image 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 image cut in a direction not parallel to either the x- or y-axis. Consider the more general increment

image (11.9)

Adding and subtracting the quantity image and inserting the factors image and image, we find

image (11.10)

Passing to the limit image, the two bracketed quantities approach the partial derivatives (11.2) and (11.3). The remaining increments image approach the differential quantities image. The result is the total differential:

image (11.11)

Extension of the total differential to functions of more than two variables is straightforward. For a function of n variables, image, the total differential is given by

image (11.12)

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 image. We have then that

image (11.13)

But the ratio of dy to dx means, in this instance, image, since z was constrained to a constant value. Thus we obtain the important identity

image (11.14)

or in a cyclic symmetrical form

image (11.15)

As an illustration, suppose we need to evaluate image for one mole of a gas obeying Dieterici’s equation of state

image (11.16)

The equation cannot be solved in closed form for either V or T. However, using (11.14), we obtain, after some algebraic simplification:

image (11.17)

Problem 11.1.1

For the function image, evaluate


Problem 11.1.2

Show that the three partial derivatives from the Dieterici equation are consistent with the identity (11.15).

Problem 11.1.3

For the van der Waals equation of state


evaluate image.

11.2 Multiple Integration

A trivial case of a double integral can be obtained from the product of two ordinary integrals, say

image (11.18)

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 image. For well-behaved functions the integrations can be performed in either order. Thus

image (11.19)

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

image (11.20)

If the function image is continuous, either of the integrals above can be transformed into the other by inverting the functional relations for the limits from image to image. 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 image. On left, horizontal strips are integrated over x between image and image 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

image (11.21)


image (11.22)

This reduces to an integration over y:

image (11.23)

Since image, we obtain image. Thus we can express the hyperbolic functions in terms of the shaded area A:

image (11.24)

Problem 11.2.1

To test Fubini’s theorem, redo the integration over the shaded area in Figure 11.3 using vertical, rather than horizontal strips.


Figure 11.3 Integration over area A of the shaded crescent. This gives geometric representation of hyperbolic functions: image.

11.3 Polar Coordinates

Cartesian coordinates locate a point image 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, image, which specifies distance and direction from the origin. As shown in Figure 11.4, the direction is defined by an angle image, obtained by counterclockwise rotation from an eastward heading. Expressed in terms of Cartesian variables x and y, the polar coordinates are given by

image (11.25)

and conversely,

image (11.26)


Figure 11.4 Cartesian image and polar image coordinates of the point P.

Integration of a function over two-dimensional space is expressed by

image (11.27)

In Cartesian coordinates, the plane can be “tiled” by infinitesimal rectangles of width dx and height dy. Both x and y range over image. In polar coordinates, tiling of the plane can be accomplished by fan-shaped differential elements of area with sides dr and image, as shown in Figure 11.5. Since r and image have ranges image and image, respectively, an integral over two-dimensional space in polar coordinates is given by

image (11.28)

It is understood that, expressed in terms of their alternative variables, image.


Figure 11.5 Alternative tilings of a plane in Cartesian and polar coordinates.

A systematic transformation of differential elements of area between two coordinate systems can be carried out using

image (11.29)

In terms of the Jacobian determinant

image (11.30)

In this particular case, we find using (11.26)

image (11.31)

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

image (11.32)

The double integral can then be transformed to polar coordinates to give

image (11.33)

Therefore image.

Problem 11.3.1

Transform the Cartesian equation for a circle, image, into polar coordinates.

11.4 Cylindrical Coordinates

Cylindrical coordinates are a generalization of polar coordinates to three dimensions, obtained by augmenting r and image with the Cartesian z coordinate. (Alternative notations you might encounter are r or image for the radial coordinate and image or image for the azimuthal coordinate.) The image Jacobian determinant is given by

image (11.34)

the same value as for plane polar coordinates. An integral over three-dimensional space has the form

image (11.35)

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 image inside the desired volume and equal to zero everywhere else. The limits of r-integration are functions of z, such that image between image and image. (It is most convenient here to define the z-axis as pointing downward from the apex of the cone.) Thus,

image (11.36)

which is image the volume of a cylinder with the same base and altitude.

Problem 11.4.1

Find the volume of a paraboloidal bowl of height h and rim radius R.


Figure 11.6 Volume of a cone. Integration in cylindrical coordinates gives image.

11.5 Spherical Polar Coordinates

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 image, 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 image, specifies the rotational orientation of OP about the z-axis. The third coordinate, now called image, is the polar angle between OP and the Cartesian z-axis. Polar and Cartesian coordinates are connected by the relations:

image (11.37)

with the reciprocal relations

image (11.38)

The coordinate image is analogous to latitude in geography, in which image and image correspond to the North and South Poles, respectively. Similarly, the angle image 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:



Figure 11.7 Spherical polar coordinates.

The volume element in spherical polar coordinates can be determined from the Jacobian:

image (11.39)

Therefore, a three-dimensional integral can be written

image (11.40)

A wedge-shaped differential element of volume in spherical polar coordinates is shown in Figure 11.8.


Figure 11.8 Volume element in spherical polar coordinates.

Integration over the two polar angles gives

image (11.41)

This represents the image steradians of solid angle which radiate from every point in three-dimensional space. For integration over a spherical symmetrical function image, independent of image and image, (11.40) can be simplified to

image (11.42)

This is equivalent to integration over a series of spherical shells of area image and thickness dr.

Problem 11.5.1

The 1s orbital of the hydrogen atom is described by the wavefunction image, where image is the Bohr radius. Calculate the integral of image over all space.

11.6 Differential Expressions

Differential quantities of the type

image (11.43)

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 image 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

image (11.44)

Comparing with the total differential of image,

image (11.45)

we can identify

image (11.46)

Note further that

image (11.47)

As discussed earlier, mixed second derivatives of well-behaved functions are independent of the order of differentiation. This leads to Euler’s reciprocity relation

image (11.48)

which is a necessary and sufficient condition for exactness of a differential expression. Note that the reciprocity relation neither requires nor identifies the function image.

A simple example of an exact differential expression is

image (11.49)

Here image and image, so that image and Euler’s condition is satisfied. It is easy to identify the function in this case as image, since image. A differential expression for which the reciprocity test fails is

image (11.50)

Here image, 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 image with image can be converted into an exact differential expression by use of an integrating factorimage. This means that image becomes exact with

image (11.51)

For example, (11.50) can be converted into an exact differential by choosing image so that

image (11.52)

Alternatively, image converts the differential to image, while image converts it to image. In fact, image times any function of image 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

image (11.53)

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:

image (11.54)

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

image (11.55)

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 image, where image is the pressure and V, the volume. Using (11.55), the differential of heat equals image. Therefore the First Law (11.54) reduces to

image (11.56)

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 image, and identifies the coefficients

image (11.57)

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 enthalpyimage. This satisfies the differential relation

image (11.58)

which must be the differential of the function image. Analogously we can define the Helmholtz free energyimage, such that

image (11.59)

and the Gibbs free energyimage, which satisfies

image (11.60)

A Legendre transformation also connects the Lagrangian and Hamiltonian functions in classical mechanics. For a particle moving in one dimension, the Lagrangian image can be written

image (11.61)

with the differential form

image (11.62)

Note that

image (11.63)

which is recognized as the momentum of the particle. The Hamiltonian is defined by the Legendre transformation

image (11.64)

This leads to

image (11.65)

which represents the total energy of the system.

Problem 11.6.1

Derive the reciprocity relations for Eqs. (11.58), (11.59) and (11.60) analogous to Eq. (11.57).

Problem 11.6.2

Generalize the transformation from the Lagrangian image to the Hamiltonian image in three dimensions. The Lagrangian is given by


where image, and image, the velocity vector.

11.7 Line Integrals

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:

image (11.66)

where image and image. All the points image lie on a continuous curve C connecting image to image, 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 image along path C between image and image. The value of the integral along path image 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 image, we obtain

image (11.67)

More generally, when the curve C can be represented by a functional relation image can be eliminated from (11.66) to give

image (11.68)

In general, the value of a line integral depends on the path of integration. Thus the integrals along paths C and image 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 image is an exact differential equal to the total differential of the function image. We can therefore substitute in Eq. (11.66)

image (11.69)


image (11.70)

neglecting terms of higher order in image and image. Accordingly,


noting that all the shaded intermediate terms cancel out. In the limit image, we find therefore

image (11.72)

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 image. The closed curve is by convention traversed in the counterclockwise direction. If image is an exact differential, then

image (11.73)

for an arbitrary closed path C. (There is the additional requirement that image must be analytic within the path C, with no singularities.) When image 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 image [Eqs. (11.49) and (11.50)] we find

image (11.74)


Figure 11.10 Rectangular path for integration of image.

11.8 Green’s Theorem

A line integral around a curve C can be related to a double Riemann integral over the enclosed area S by Green’s theorem:

image (11.75)

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

image (11.76)


image (11.77)


image (11.78)

which establishes Green’s theorem for the rectangle:

image (11.79)

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 image, 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.


Figure 11.11 Steps in proof of Green’s theorem.

By virtue of Green’s theorem, the interrelationship between differential expressions and their line integrals can be succinctly summarized. For a differential expression image, any of the following statements implies the validity of the other two:

1. There exists a function image whose total differential equals image (dq is an exact differential).

2. image (Euler’s reciprocity relation).

3. image around an arbitrary closed path C.

Problem 11.8.1

Show that the area bounded by a smooth closed curve is given by


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