Chapter X
A Sketch of the Theory of Functions of Several Variables
Up to this point, we have been concerned exclusively with functions of a single independent variable. We must now go on to consider functions of several independent variables. Even the applications of the calculus force us to take this step. In almost all the relationBhips which occur in nature, in fact, the functions in question do not depend on a single independent variable; on the contrary, the dependent variable is usually determined by two, three, or more independent variables. Thus, for example, the volume of an ideal gas is a function of a single variable, the pressure, if we keep the temperature constant, but not otherwise. As a rule, the temperature also varies and the volume depends upon a pair of values, namely, the value of the pressure and that of the temperature, whence it is a function of two independent variables.
Also from the point of view of pure mathematics, the need for a detailed study of functions of several independent variables is urgent. Here we shall be able to take advantage of what we have learned previously, so that in many cases we have only to make simple extensions of our arguments.
It is usually sufficient to consider the case of only two independent variables x and y, as long as no essentially new considerations are required for an extension to functions of three or more variables. Hence, in order to keep our statements and notation simple, we shall, as a rule, consider only two independent variables.
A systematic presentation of the differential and integral calculus for functions of several variables is impossible within the compass of this volume, but will be given in Volume II of this treatise. All which can be done here is to give the reader a preliminary view of some of the most important new concepts and operations. We shall frequently rely on intuitive plausibility, the full proofs to be developed subsequently in Volume II.
10.1 The Concept of Function in the Case of Several Variables
10.1.1 Functions and their Ranges of Definition: Equations of the form
assign a functional value u to each pair of values (x, y). In the first three of our example, this correspondence holds for every system of values (x, y), while in the last case the correspondence has a meaning only for those pairs of values (x, y) for which the inequality x² + y² £ 1 is true.
In these cases we say that u is a function of the independent variables x and y. In general, we use this expression whenever some law assigns a value of u as dependent variable, corresponding to each pair of values (x,y) belonging to a certain specified set. The relation between x, y and u may be stated in terms of a functional equation, as above, or by means of a verbal description such as u is the area of the rectangle with sides x and y, or it may follow from physical observations as for instance in the case of the magnetic declination at different latitudes and longitudes. The essential thing is that there exists a correspondence. Similarly, u is said to be a function of the three independent variables x, y, z, if for each triad of values (x, y, u) of a certain set there exists a corresponding value of u given by some definite law; it is similar for the general case of functions of n independent variables x1, x2, ···, xn.
The set of values which the pair (x, y) can assume is called the range of definition of the function u = f(x, y). For the purposes of this chapter, we shall restrict our attention to the simplest types of range of definition. We shall consider that (x, y) is limited either to a so-called rectangular region (domain)
or else to a circle, determined by an inequality of the form
In the case of functions of three variables x, y, z, we shall again consider only rectangular regions
and spherical regions
In dealing with more than three independent variables, geometrical intuition fails us, but it is often convenient to also extend geometrical terminology to this case. Thus, for functions of n variables x1, x2, ···, xn, we shall consider regions
and also regions
which we will call rectangular and spherical regions, respectively.
10.1.2 The Simplest Types of Functions. Just as in the case of functions of one variable, the simplest functions are the rational integral functions or polynomials. The most general polynomial of the first degree (linear function) has the form
where a, b and c are constants. The general polynomial of the second degree has the form
The general polynomial is a sum of terms of the form amnxmyn, where the quantities amn are arbitrary constants.
Rational fractional functions are quotients of polynomials; for example, there belongs to this class the linear fractional function
By extraction of roots, we pass on from the rational functions to certain algebraic functions, for example,
An accurate definition of the term algebraic function is given in at the end of 10.5.1.
In the construction of more complicated functions of several variables, we almost always fall back on the well-known functions of a single variable, for example (10.4.1),
10.1.3 Geometrical Representation of Functions: Just as we represent functions of one variable by means of curves, we seek to represent functions of two variables geometrically by means of surfaces; we shall consider hereafter only those functions which can actually be represented in this way. We achieve this representation very simply by considering a rectangular co-ordinate system in space with co-ordinates x, y and u, and marking off above each point (x, y) of the range (R) of definition of the function the point P with the third co-ordinate u = f(x, y). As the point (x, y) ranges over the region R, the point P describes a surface in space. We take this surface as the geometrical representation of the function.
Conversely, in analytical geometry, surfaces in space are represented by functions of two variables, so that there is between such surfaces and functions of two variables a reciprocal relationship.
For example, there corresponds to the function
the hemi-sphere above the x, y plane with unit radius and centre at the origin, to the function
the so-called parabaloid of revolution, obtained by rotating the parabola w=x² about the u-axis (Fig. 1), to the functions
the so-called hyperbolic paraboloids (Fig. 2). The linear function
has for its graph a plane in space.
If in the function u = f(x, y) one of the independent variables, say y, does not occur, so that u depends on x only, say u = g(x), the function is represented in the xyu-space by a cylindrical surface, obtained by erecting the perpendiculars to the ux-plane at the points of the curve u=g(x).
However, this representation by means of rectangular co-ordinates has two disadvantages. Firstly, intuition fails us whenever we have to deal with three or more independent variables. Secondly, even in the case of two independent variables, it is often more convenient to confine the discussion to the xy-plane, since in the plane we can sketch and construct geometrically without difficulty. From this point of view, another geometrical representation of the function by means of contour lines is to be preferred. In the xy-plane, we take all the points for which u = f(x, y) has a constant value, say u = k. These points will usually lie on a curve or curves, the so-called contour line for the given constant value of the function. We can also obtain these curves by cutting the surface u = f(x, y) by the plane u = k parallel to the xy-plane and projecting the curves of intersection perpendicularly onto the xy-plane. The system of these contour lines, marked with the corresponding values k1, k2, ··· of the height k, gives us a representation of the function. As a rule, k is assigned values in arithmetic progression, say k = n h, where n = 1, 2, ···. The distance between the contour lines then gives us a measure of the steepness of the surface u=f(x,y); in fact, between every two neighbouring lines, the value of the function changes by the same amount. Where the contour lines are close together, the function rises or falls steeply, where they are far apart, the surface is flattish. This is the principle on which contour maps such as those of the Ordnance Surrey and the U.S. Geological Survey are constructed.
In this method, the linear function u = ax + by+ c is represented by a system of parallel straight lines ax + by+ c = k. The function u = x² + y² is represented by a system of concentric circles (Fig. 3). The function u = x² - y², the surface of which has a saddle point at the origin (Fig. 2) is represented by the system of hyperbolas shown in Fig. 4.
The method of representing the function u = f(x,y) by contour lines has the advantage of it being capable of extension to functions of three independent variables. Instead of contour lines, we then have level surfaces f(x,y,z) = k, where k is a constant to which we can assign any suitable sequence of values. For example, the level surfaces for the function u = x² + y² + z² are concentric spheres about the origin of the co-ordinate system.
1. For each of the following functions, sketch the contour lines corresponding to a z = -2, -1, 0, 1, 2, 3:
No Answers and no Hints
10.2.1 Definition:. As in the case of functions of a single variable, the basic requirement, by which functions should he capable of being represented geometrically leads to the analytic condition of continuity. Here again, the concept of continuity is given by the definition:
A function u = f(x, y), defined in a region R, is said to be continuous at a point (x,h) of R, if for all points (x,y) near (x,h) the value of the function f(x, y) differs but little from f(x,h), the difference being arbitrarily small if only (x, y) is near enough to (x,h).
More precisely: The function f(x, y), defined in the region R, is continuous at the point (x,h) of R, provided that it is possible for every positive number e to find a positive distance d = d(e) (in general, depending on e and tending to 0 with e) such that for all points of the region, the distance of which from (x,h) is less than d , that is, for which the inequality
holds, there is satisfied the relation
or, in other words, the relation
is to hold for all pairs of values (h, k) such that h² + k² £ d ² and (x+ h, h + k) belongs to the region R.
If a function is continuous at every point of a region B, we say that it is continuous in R.
In the definition of continuity, we can replace the distance condition h²+k²£d² by the equivalent condition:
There shall correspond to every e > 0 two positive numbers d1and d2 such that
whenever
The two conditions are equivalent. In fact, if the original condition is fulfilled, so is the second one, if we take d1= d2 = d/2; and conversely, if the second condition is fulfilled, so is the first, if we take for d the smaller one of the two numbers d1and d2.
The following facts are almost obvious:
The sum, difference and product of continuous functions are also continuous. The quotient of continuous functions is continuous except when the denominator vanishes. Continuous functions of continuous functions are themselves continuous (10.4.1). In particular, all polynomials are continuous and so are all rational fractional functions except when the denominator vanishes.
Another obvious fact, which, however, is worth stating, is:
If a function f(x, y) is continuous in a region R and differs from zero at an interior point P of the region, it is possible to mark off about P a neighbourhood, say a circle, belonging entirely to R, in which f(x,y) is nowhere equal to zero. In fact, if the value of the function at P is a, we can mark off about P a circle so small that the value of the function within the circle differs from a by less than a/2, and therefore it is certainly not zero.
10.2.2 Examples of Discontinuities: In the case of functions of one variable, we encounter three kinds of discontinuities: Infinite and jump discontinuities, and discontinuities at which no limit is approached from one or both sides. With functions of two or more variables, no such simple classification is possible. In particular, the situation is made more complicated by the fact that discontinuities may occur not merely at isolated points, but also along entire curves.
Thus, for the function u = 1/(x
- y), the line x = y is a line of
infinite discontinuity. As we approach the line from one side or
the other, the values of u increase numerically beyond
all bounds through positive or negative values. The function u
= 1/(x - y)² has the same line of
discontinuity, but tends to + ¥
as we approach the line from either side.
The function u = 1/(x² + y²) has the
single point of discontinuity x = 0, y = 0. The
function 
tends to no limit as we approach the origin; the
surface which it represents is obtained by rotating the graph of
the function u = sin 1/x about the u-axis.
Another instructive example of a discontinuous function is given by the rational function u = 2xy/(x² + y²). In the first instance, the function is undefined at x = 0, y = 0, and we supplement the definition by assuming that u(0,0) = 0. This function has a peculiar type of discontinuity at the origin. If we put x = 0, i.e., if we move along the y-axis, the function becomes u(0,y) = 0, which has the constant value 0 for all values of y. Along the x-axis, we likewise have u(x, 0) = 0. Thus, at the origin, the function u(x,y) is continuous in x, if we keep y at the constant value 0, and is continuous in y, if we keep x at the constant value 0. Nevertheless, the function is discontinuous when considered as a function of the two variables x and y. In fact, at every point of the line y=x, we find that u = 1, so that arbitrarily near to the origin we can find points at which u assumes the value 1. The function is therefore discontinuous at the origin and cannot be defined at the origin in such a way as to make it continuous.
More generally, on the straight line y = x tan a, inclined at the angle a to the x-axis, we have u = 2tana/(1+tan²a) = 2sinacosa = sin2a. The surface corresponding to the u = 2 xy/(x²+y²) is therefore formed by rotating a straight line at right angle to the u-axis about that axis until it coincides with the x-axis and simultaneously raising or lowering it so that the height sin2a is associated with the angle a. As a increases to 45º, the straight line rises to the height 1, and consequently falls to the level of the y-axis and below to the depth 1, thereafter rising again to the level of the x-axis. The surface enveloped by the moving straight line is known as the cylindroid and is important in mechanics.
The above example shows that a function can he continuous in x for every fixed value of y and continuous in y for every fixed value of x and yet be discontinuous when considered as a function of the two variables. The essential point in this definition of continuity is that the value of the function at a point P must be arbitrarily close to the value of the function at a point Q, provided only that Q is near enough to P; it is not permissible to restrict the position of Q relative to P in any other way.
1. Examine the continuity of the function 
Sketch
the level lines z=k (k = -4, -2, 0, 2, 4).
Exhibit (on the same graph) the behaviour of z as a
function of x alone for y = -2, -1, 0, 1, 2,
Similarly, exhibit the behaviour of z as a function of y
alone for x = 0, ± 1, ± 2. Finally, exhibit the
behaviour of z as a function of r alone when q is
constant (r, q being polar co-ordinates).
2. Show that the following functions are continuous:
3. Decide whether or not the following functions are continuous and, they are not, where they are discontinuous:
10.3 The Derivatives of a Function of Several Variables
10.3.1 Definition. Geometrical Representation: If we assign in a function of several variables definite numerical values to all but one of the variables and allow only that one variable, say x, to vary, the function becomes a function of one variable. For
example, consider a function u = f(x,y) of the two variables x and y and give y the definite fixed value y = y0 = c. The function u = f(x, y0) of the single variable x, which is thus formed, may be simply represented geometrically by letting the surface u = f(x, y) be cut by the plane y = y0 (Figs. 5 and 6). The curve of intersection thus formed in the plane is represented by the equation u=f(x,y0). If we differentiate this function in the usual way at the point x = x0 (assuming that the derivative exists), we obtain the partial derivative of f(x,y) with respect to x at the point (x0, y0). According to the ordinary definition of the derivative, this is the limit
If (x0, y0) is a point on the boundary of the region of definition, we make the restriction that in the passage to the limit the point (x + h, y0) must always remain in the region.
Geometrically speaking, this partial derivative denotes the tangent of the angle between a parallel to the x-axis and the tangent line to the curve u = (x,y0). It is therefore the slope of the surface u = f(x,y) in the direction of the x-axis.
In order to represent these partial derivatives, several different notations are in use of which we mention:
If we wish to emphasize -that the partial derivative is the limit of a difference quotient, we denote it by
We use here a special curved letter 
instead of the ordinary d
used in the differentiation of functions of one variable, in
order to show that we are dealing with a function of several
variables and differentiating with respect to one of them.
Sometimes. it is convenient to use Cauchy's symbol D and to write
however, we shall rarely use this symbol.
In exactly the same way, we define the partial derivative of f(x, y) with respect to y at the point (x0, y0) by the relation
It represents the slope of the curve of intersection of the surface u = f(x, y) with the plane x = x0, perpendicular to the x-axis.
Now, let us think of the point (x0, y0), hitherto considered to be fixed, as variable and accordingly omit the suffices. In other word, we think of the differentiation as being carried out at any point (x, y) of the region of definition of f(x, y). Then the two derivatives are themselves functions of x and y:
For example, the function 
has the partial
derivative 
(in differentiating with respect to x, the
term y² is considered to be a constant, whence is has
the derivative 0) and 
. The partial derivatives of u = x²y
are 
Similarly, we make the definition for any number (n) of independent variables:
it being assumed that the limit exists.
Of course, we can also form higher partial
derivatives of f(x, y) by again
differentiating the partial, first order derivatives 
with
respect to one of the variables, and repeating this process. We
indicate the order of differentiation by the order of the
suffixes or by the order of the symbols 
in the denominator from
the right to the left hand side, and use for the second partial
derivatives the:
On the other hand, in Continental
usage, 
becomes 
We likewise denote the third partial derivatives by
and, in general, the n-th derivatives
Finally, we shall study a few examples of the actual calculation of partial derivatives. According to the definition, all the independent variables are to be kept constant except the one with respect to which we are differentiating. We therefore have merely to regard the other variables as constants and carry out out the differentiation according to the rules by which we differentiate functions of a single independent variable.
For example, we have:
r
(Thus for the radius vector 
from the origin to the
point (x, y), the partial derivatives with
respect to x and y are given by the cosine, cosj=x/r,
and the sine, sin j = y/r of the angle j, which the radius
vector makes with the positive direction of the x-axis.)
Second derivatives
We see from this that there holds for the
function 
the equation
,
for all values of x, y, z except 0,0,0; as we may say, the equation
is satisfied identically in x,y,z by the function f(x,y,z) = l/r.
The equation
is therefore satisfied identically in x and y.
Just as in the case of one independent variable, the possession of derivatives is a special property of a function. (Note that the expression differentiable implies more than that the partial derivatives with respect to x and y exist (cf. Vol. II!) All the same, this property is possessed by all functions of practical importance except perhaps at isolated exceptional points.
In contrast to functions of one variable, the
possession of derivatives does not imply the continuity of the
function. This is clearly shown by the example 
already
considered. because the partial
derivatives exist for it everywhere and yet the function is
discontinuous at the origin. But, as is stated by the following
theorem, the possession of bounded
derivatives does imply continuity:
If a function f(x,y) has partial derivatives fx and fy everywhere in a region R, and these derivatives satisfy everywhere the in equalities
where M is independent of x and y, then f(x, y) is continuous everywhere in R.
In particular, if fx and fy are continuous, they are necessarily bounded, so that f(x,y) is also continuous.
We shall leave the proof of this theorem for Volume II. The reader will have noted that in all our examples the equation fxy = fyx is satisfied. In other words, it made no difference whether we differentiated first with respect to x and then with respect to y or vice versa. This is no accidental occurrence. In fact, we have the theorem:
If the mixed partial derivatives fxy and fyx of a function f(x,y) are continuous in a region R, then the equation
fyx = fxy
holds everywhere in the interior of this region, i.e., the order of differentiation with respect to x and y is immaterial.
Applying this theorem to fx and fy, then to fxx, fxy, fyy, etc., we find that
and, in general, we have the result:
In repeated differentiation of a function of two variables, the order of differentiation can be changed arbitrarily, provided only that the derivatives in question are continuous functions.
For the proof of this theorem, we again refer the reader to Volume II.
1. Find the first partial derivatives of the functions:
2.. Find all the first and second partial derivatives of the functions:
3.* Find a function f(x, y) which is a function of (x² + y²) and is also a product of the form y(x)y(y), i.e., solve the equations
10.4 The Chain Rule and the Differentiation of Inverse Functions
10.4.1 Functions of Functions (Compound Functions):. It often happens that a function u of the independent variables x, y is stated in the form
where the arguments x, h, ··· of the function f are themselves functions of x and y:
We then say that
is given as a compound function of x and y.
For example, the function
may be written as a compound function by means of the relations
Similarly, the function
can be expressed in the form
In order to make this concept more precise, we
assume to begin with that the functions 
are defined in a
certain region R of the independent variables x,
y. Then there corresponds to every point (x,y)
of R a point (x,h, ···). As the point (x,y) ranges
over R, the point (x,h, ···) will range over a certain set of values. We
assume that the point (x,h, ···) always lies within a region S in
which f(x,h, ···) is defined. The function
is then defined in the region R.
Referring to our examples, we find in the first
one of them that x and h are defined for every x, y and f(x,h) is
defined for every x,h, so that our region R can be taken to be the
entire xy-plane. However, in the second example, the
region S is restricted by the inequality |x| £ 1, since for |x| > 1 the function
arcos x is undefined. Secondly, the region R is
restricted by the inequalities x+1>0 and x²
+ y² £ 4, since for other values both x and h are not
defined. Thirdly, the region R must be further limited
by the inequality 3 £ x² + y² in order that the point with
co-ordinates x,h shall fall into S, i.e., the restriction |x|£1 implies that x²
+ y² ³ 3. Hence R consists of the part of the ring 3£ x²+y²£4 lying to the
right of the line x = -1.
The following theorem on compound functions is an immediate consequence of the definitions:
If the function u = f(x,h, ···) is continuous in S and the functions x=f (x, y), h=y (x, y)), ··· are continuous in R, then the compound function u = F(x, y) is continuous in R.x=f (x, y)
The reader should be able to prove this statement
10.4.2 The Chain Rule: We now turn our attention to compound functions of the type u = f(x,h, ···), where x,h, ··· depend on the single variable x:
For such functions, we have the important theorem known as the chain rule:
If the function u = f(x,h, ···) has continuous partial derivatives of the first order in S and the functions x = f(x), h = y(x), ··· have continuous first derivatives in the interval R, a £ x £ b, then u = f(f(x),y(x), ···) = Fxi) has a continuous derivative in R and
The right-hand side of this equation is an abbreviation for
In order to simplify the notation, we shall assume that f is a function of the three arguments x,h,z. We shall denote by x0 an arbitrary fixed point of the interval a £ x £ b, by x0,h0,z0 the corresponding values x0=f(x0), h0=y(x0), x0 =c(x0) and by x,h,z the values f(x), y(x), c(x), corresponding to a variable point x =x0 + h. We first write down the identity
In each bracket on the right hand side, we observe that only one of the independent variables changes its value. Hence we can apply to each bracket the mean value theorem for functions of a single variable and obtain
where 
lies between x and x0, 
between h0 and h, and 
between z0 and z. Moreover, by the mean value theorem, we have
where all x1, x2, and x3 lie between x0 and x. Substituting these values in the last equation and dividing by x - x0, we have
We now let x tend to x0.
Owing to the continuity of f(x), y(x), c(x), the quantities x,h,z tend
to x0,h0,z0,respectively, and, a fortiori, so
do ,,. Also x1,
x2 and x3 tend to x0.
Since all the functions on the right hand side are continuous, we
have
thus establishing the formula for F'(x).
The continuity of F'(x) follows immediately from the formula, since, by assumption, f ', y ', c ', are continuous fx, fh and fz are are continuous functions of continuous functions.
This theorem may be extended to compound functions of two or more variables, as follows:
If the funding u = f(x,h, ···) has continuous partial derivatives of the first order in the region S and the functions x = f(x,y), h = y(x,y), x,y have continuous partial derivatives of the first order in R, then u = F(x, y) = f{f(x,y), y(x,y), ··· has continuous partial derivatives of the first order in R and their derivatives are given by
These formulae are often written in the abbreviated form
In order to derive them, we temporarily introduce the notation g(x) = f(x,y0), h(x) = y(x,y0), ···, where y0 is a fixed value of y. By the definition of the partial derivatives, it follows that g'(x) = fx(x,y0), h'(x) = yx(x,y0), ···. Similarly, if we writs H(x) = F(x, y0), we have H'(x) = Fx(x, y0). We now apply the theorem just proved to the function u = H(x) =f(x,h, ···) = f{g(x), h(x), ···} and obtain
Returning to the original symbols, we have
The other formula is proved in a similar manner.
If we wish to calculate higher order derivatives, we need only differentiate again the right hand side of our formulae with respect to x and y, regarding fx, fh, ··· as compound functions. Thus, for u = f(x,h) = f{f(x,y), y(x,y)}, we have
We emphasize that the following differentiations can also be carried out directly without using the chain rule.
We have already dealt with a special case of this by rather artificial methods (cf. 2.3.3).
10.4.4 Change of the Independent Variables:. A particularly important type of compound function occurs in the process of changing the independent variables. For example, let u = f(x,h) be a function of x and h, which we interpret as rectangular co-ordinates in the xh-plane. If we rotate the axes in the xh-planes through an angle q, we obtain a new system of co-ordinates x,y, related to the co-ordinates (x,h) by the equations:
The function u = f(x,h) can then be expressed as a function of the new variables x, y:
The chain rule yields then immediately:
Thus, the partial derivatives are transformed by the same formulae as the independent variables. This is true for rotation of the axes in space also. Another important type of change of co-ordinates is the change from rectangular co-ordinates x, y to polar co-ordinates r, q. This is done by means of the equations
We then find that for an arbitrary function u = f(x,x) with continuous partial derivatives of the first order we have
Whence we obtain the often useful equation
In general, let us consider a pair of functions x = f(x, y), h = y(x, y) which are continuous and have continuous derivatives in a region R of the xy-plane. These equations assign to each point (x, y) in R a point x = f(x, y), h = y(x, y) in the xh-plane. As (x, y) ranges over R, the corresponding point (x,h) will range over some set of values S in the xh-plane. Naturally, it is possible that several distinct points (x, y) will give the same values for (x,h), so that there corre-spond to several points (x, y) only one point (x,h). We shall assume that this is not s, but instead that there corresponds to one point Q(x,h) in S exactly one point P(x,y) in R. We may therefore look at the correspondence from either point of view - saying that Q corresponds to P or that P corresponds to Q. The latter point of view can be expressed as follows: There corresponds to each point (x,h) in S one x and one y, namely, the co-ordinates of P, or, in , equations, there are two functions x = g(x,h), y = h(x,h), defined in S, which represent the corresponding inverse to x = f(x,y), h = y(x,y).
It happens often that the functions g(x,h), h(x,h) are by no means easy to calculate, even when they do exist, whence we shall now discover how to obtain the partial derivatives gx, gh, hx, hh directly from the partial derivatives fx, fy, yx, yy without at all calculating g and h themselves. For this purpose, we observe that, if we choose any point Q(x, h), find the corresponding point P{g(x,h),h(x,h)} in R and then find the point in S corresponding to P, which is f{g(x,h),h(x,h)}, y{g(x,h),h(x,h)}, we have simply returned to the point Q. In other words, the equations x = f{g(x,h),h(x,h)}, h = y{g(x,h),h(x,h)} are identities in x and h. We now differentiate both sides of both equations with respect to x and h and find
If an equation expresses an identical relationship, differentiation with respect to any independent variable in it yields an identity, as follows immediately from the definition.
By solving these equations, we find that
where by D is the determinant
which we assume to be non-zero.
This determinant D, called the functional determinant or Jacobian of (x,h) with respect to (x,y), occurs so frequently that one uses frequently the special symbol
1. Calculate the first order partial derivatives of
2. Caculate the derivatives of
3. Prove that, if f(x, y) satisfies the Laplace equation
4. Prove that the functions
satisfy the respective Laplace equations:
5. Given z = r² cos q, where r and q are polar coordinates, find zx and zy at the point q = p/4, r = 2. Express zr and zq in terms of zx and zy
6.By the transformation x = a + ax + b y, h = b - b x + ya, in which a,b,a, b, are constants and a² + b² = 1, the function u(x, y] is transformed mto a function U(x,h) of x and h. Prove that
7. Find the Jacobians of the transformations:
8. If x = (u,v), y = y(u,v) and u = u(x,h), v = v(x,h), prove that
9, As a corollary to 8., prove that
10. Using 9., find the Jacobian of the transformations which are the inverses of those in Example 7.
In our study of functions of several variables, we have as yet had no analogue to the inverse function. We can regard the inverse function of y = f(x) to be the function obtained by solving the equation y - f(x) = 0 for x. In this section, we shall seek more generally to solve equations F(x, y) = 0 for x or y and to discuss functions of several variables in a corresponding way.
Even in elementary analytical geometry, curves are frequently represented not by equations y =f(x) or x = f(y), but by an equation involving x and y in the form F(x, y) = 0. For example, we have the circle x² + y² - 1 = 0, the ellipse x²/a² + y²/b² - 1 = 0 and. the lemniscate (x² + y²)² - 2a²(x² - y²) = 0. In order to obtain y as a function of x or x as a function of y, we must solve the equation for y or for x. We then say that the function y = f(x) or x =f (y) found in this way is defined implicitly by the equation F(x, y) = 0, and that the solution of this equation yields the function explicitly. In the examples cited and in many others, the solution can be carried out and the solutions stated explicitly in terms of the elementary functions. In other cases, the solution can be obtained in terms of an infinite series or other limiting processes, i.e., we can approximate to the solution y = f(x) or x = f(y) as closely as we please.
However, for many purposes, it is more convenient to base the discussion on the implicit definition F(x, y) = 0, instead of resorting to an exact or approximate solution of the equation.
The idea that every function F(x, y) yields a function y = f(x) or x = f(y) given implicitly by means of the equation F(x, y) = 0 is erroneous. On the contrary, it is easy to give examples of functions F(x, y) which, when equated to zero, permit no solution in terms of functions of one variable. Thus, for example, the equation x² + y² = 0 is satisfied by the single pair of values x = 0, y = 0 only, while the equation x² + y² + 1= 0 is not satisfied by any real values. It is therefore necessary to investigate this matter more closely, in order to find out whether an equation F(x, y) = 0 can actually be solved and what properties its solution has. We cannot undertake in detail such an investigation here, but will content ourselves with a geometrical interpretation which suggests the required results, the rigorous proofs being left to Volume II.
10.5.1 Geometrical Interpretation of Implicit Functions:. In order to discuss this problem geometrically, we represent the function u = F(x, y) by a surface in three-dimensional space. Finding values (x, y) which satisfy the equation F(x, y) = 0 is the same thing as finding values (x, y) which satisfy two equations F(x, y) = u, u = 0; in other words, we wish to find the intersection of the surface u = F(x, y) and the plane u = 0 - the xy-plane. We then suppose that we have a definite point (x0,y0) which satisfies the equation F(x0, y0) = 0), i.e., at (x0, y0), the surface u = F(x, y) has a point in common with the plane u = 0. (If there does not exist such a point, there is no intersection and the equation F(x, y) = 0 cannot be solved.) If the tangent plane to the surface u = F(x, y) at the point (x0,y0) is not horizontal, it cuts the plane u = 0 in a single straight line. Intuition then tells us that the surface u = F(x, y), lying near the tangent plane, likewise cuts the plane u = 0 in a single well defined curve. How far this curve extends does not at present concern us. The tangent plane will be horizontal, if both the curves u = F(x0, y) and u = F(x, y0) have horizontal tangent lines at (x0, y0), i.e., if Fx(x0, y0) = 0 and Fy(x0, y0)) = 0. Thus, if either Fx(x0, y0) ¹ 0 or Fy(x0, y0) ¹ 0, the tangent plane is not horizontal, and, as we ave just seen, we may expect that a solution in the form y = f(x) or x = f (y) hwill exist.
On the other hand, if both Fx(x0, y0) and Fy(x0, y0) have the value 0, we readily see that there is no guarantee that a solution is possible.
For example, for 
, the corresponding
spherical surface is 
, has the point (0, 0) in connection with
the xy-plane. The partial derivatives Fx(0,0)
and Fy(0,0) are both zero; we find
that no point other than (0,0) satisfies the equation F
= 0. For the function F(x,y)=xy,
we find that F(0, 0) = 0, while Fx(0,
0) = Fy(0, 0) = 0. Here all the
points on the x-axis and all the points on the y-axis
satisfy the equation F(x,y)=0; in the
neighbourhood of the origin, we have no unique solution x=f(y)
or y =f(x). Thus, we see that when Fx(x0,y0)
= Fy(x0,y0)
= 0, we cannot be sure that a solution exists.
Accordingly, we return to the case in which one of the partial derivatives—say, Fy(x0,y0) to be specific—is not zero, the graphical suggestion that a smooth surface should be cut by a non-tangent plane in a smooth curve leads us to expect that one has the theorem:
If the function F(x,y) has continuous derivatives Fx and Fy, and if at the point (x0,y0) holds the equation F(x0,y0) = 0, while Fy(x0,y0) is not zero, then we can mark off about the point (x0,y0) a rectangle x1£ x £ x2,, y1£ y £ y2 such that for every x in the interval x1£ x £ x2 the equation F(x, y) = 0 determines just value y=f(x), lying in the interval y1£ y £ y2. This function y = f(x) satisfies the equation y = f(x0), and the equation
is satisfied for every x in the interval. Moreover, the function y = f(x) is continuous and has a continuous derivative.
This can actually be rigorously proved and will be proved in Volume 2. Assuming it to be true, we can add the following:
The derivative of the function y = f(x) is given by the equation
This follows immediately by using the chain rule. In fact,
However, since F{x,f(x)}identically zero, its derivative is also zero, whence Fx+ Fyf '= 0, and the formula is established.
If we regard the right hand side of the formula as a compound function of x and differentiate according to the chain rule, replacing y' by -Fx/Fy, we have
Continuing the process, we may calculate yiii, iv, etc.
By using this formula, we can usually find the derivative of a function given in implicit form much more easily than by solving first and then differentiating.
For example, for the circle
we have
This is easily verified. In fact, solving the
equation of the circle for y, we obtain two solutions,
namely, 
giving the upper and lower semi-circles,
respectively. For the upper part, we have
so that in either case
Another example is F(x,y) = ex+y + y - x = 0. We find Fx(½,-½) = 0, while Fy(½,-½) = 2. Thus, the equation has a solution y = f(x); but its actual explicit calculation is not simple. Nevertheless, we have
In order that the function f(x) may
have a maximum or minimum, we must have y' = 0, i.e., 
whence y
= -x. Substitution of y = -x into the
equation F(x, y) = 0 yields 1 - 2x
= 0, whence x = ½, y = -½. If we calculate
f "(x) for x = ½, we find it to
be negative so -½ is the maximum of y.
An extension of this theorem for implicit functions to functions of a larger number of independent variables readily suggests itself:
Let F(x, y, ··· , z, u) be a continuous function of the independent variables x, y, ··· , z, u with continuous partial derivatives Fx, Fy, ··· , Fz, Fu. For the system of values (x0, y0, ··· , z0, u0), let F(x0,y0,···,z0,u0)=0 and Fu(x0,y0,···,z0,u0)¹0. Then we can mark off an interval u1£u£ u2 about u0 and a region R containing (x0,y0,···,z0) such that for every (x, y, ··· , z) in R the equation F(x,y,···,z,u)= 0 is satisfied by just one value of a in the interval u1 £ u £ u2. This value of n, which u = we denote by u = f(x, y, ··· , z) is a continuous function of x, y, ··· , z and possesses continuous partial derivatives fx, fy, ··· , fz and
The derivatives of f are given by the equations
For the proof of the existence and continuity of u, we again refer the reader to Volume II. The formulae for fxy, etc., follow immediately from the chain rule.
Incidentally, the concept of an implicit function enables us to give a general definition of the term algebraic function.
We say that u = f(x, y, ··· , z) is an algebraic function of the independent variables x,y,···,z if u can be defined implicitly by an equation F(x,y,···,z,u)=0, where F is a polynomial in x,y,···,z,u, that is, if u satisfies an algebraic equation. Functions which do not satisfy any algebraic equation are called transcendental equations.
As an example of our differentiation formula, consider the ellipsoid
We have for the partial derivatives
and by redifferentiation
1. Prove that the following equations have unique solutions for y near the points indicated:
2. Find the first derivatives of the solutions in 1.
3. Find the second derivatives of the sulutions in 1.
4. Find the maxima and minima of the function y
= f(x) defined by the equation 
6. Show that the equation x + y + z = sin xyz can be solved for z near the point (0,0,0). Find the partial derivatives of the solution.
10.6 Multiple and Repeated Integrals
10.6.1 Multiple Integrals: Consider a function u = f(x,
y) which is defined and continuous in the rectangle R(a
£ x
£ b,
c £ y £ d) and which assumes only positive values. We
wish to assign a volume to the portion of three-dimensional space
bounded by the rectangle R, the surface u = f(x,
y, z), and the four planes x = a,
x = b, y = c, y = d
perpendicular to the xy-plane. Moreover, the volume
should be defined so as to satisfy certain 
elementary conditions:
(1), if the three-dimensional region is a prism, i.e., if the
function u is a constant k, the volume
should be the product of the base by the height, V=(b-a)(d-c)k
;
(2) if we divide the rectangle R into smaller rectangles
R1 and R2 by drawing
straight lines, then the volume over R1
should be equal to the volume over R1 plus
the volume over R2;
(3) if the three-dimensional region R1
completely includes R2, the volume of R1
should be at least as large as that of R2.
These considerations lead us to a method of defining V which is an immediate extension of the method of defining area in 2.1.1. By constructing lines parallel to the sides, we subdivide the rectangle R into smaller rectangles R1,R2, ···,Rn , the areas of which will be denoted by DR1, DR2, ··· , DRn. In each rectangle Rj, the fonction has a least value mj and a larger value Mj, whence a prism with base R and height Mj completely includes the portion of the region over Rj, while this portion of the region contains the prism with base Rj and height mj (Fig. 7). Hence, we see that the vohume of the portion in question lies between mjRj and MjRj. Thus, the total volume V should be such that
Now let the number n of rectangles
increase beyond all bounds in such a way that the length of the
longest diagonal tends to zero. Intuition leads us to expect that
both the sums 
will converge and tend to the same limit, whence we
call this limit the volume V.
The reader will have observed that we have
carried out an immediate generalization of the discussion in 2.1.2. As in Chapter II, we
call the common limit of the sums the integral of the
function n = f(x, y) over the
rectangle R and denote it by the symbol
It is at once clear that, if we choose in each rectangle Rj a point (xj,hj) and find the corresponding value of the function f(xj,hj), then there must hold the limiting relation
in fact, the sum 
lies
between , both of which approach the integral as a limit.
As a particular method of subdividing r into smaller rectangles, we may subdivide the side a £ x £ b into n intervals of length Dx = (b - a)/n and the side c £ y £ d into m intervals of length Dy = (d - c)/m, and then draw parallels to the axes through the points of division thus marked. The area of each rectangle Rj is then DRj = DxDy. Choosing a point (xj,hj) arbitrarily in each rectangle Rj, we form the sum
As both n and m increase without limit, this sum approaches the integral as a limit. This type of subdivision suggests a second notation for the integral, which has been in common use since the time of Leibnitz, namely
The proof that such a limit exists if u = f(x, y) is continuous can be carried out as in the A2.1). However. we shall assume without proof the even stronger statement:
The function f(x, y) is continuous except along a finite number of smooth curves (curves with continuous derivatives) y = f(x) or x = f(y) along which f(x,y) has jump discontinuities, then there exists double integral
We leave its proof to Volume II. It depends
essentially on the fact that, as the number of rectangles
increases, the total area of the rectangles having points in
common with the curves of discontinmty tends to zero. Thus, even
though Mj and mj
may differ considerably for such rectangles, they give rise to a
little difference between the sums .
With this assumption, we can find the area
under surfaces u = f(x, y)
for which (x, y) ranges over quite complicated
regions R. In fact, let the region R be bounded
by a finite number of curves x = f(y) or y
= y(x) with continuous derivatives and f(x,
y) be continuous in R. We enclose R in
a rectangle R' and assign the points of R',
which 
taken over the region R', as the volume
under the surface u = f(x, y),
where (x, y) is in R. This integral is
usually denoted by .
Certain simple, but important theorems relating to these double integrals follow directly from the definition. We shall simply state the theorems, as the reader will be able to prove them without any trouble.
If f(x, y) and g (x, y) are integrable over a rectangle, then so are f ± g and cf, where c is a constant:
lf f(x, y) ³ g(x, y) in R, then
If IjR is the sum of two regions R1 and R2, then
10.6.2 Reduction of Double Integrals to Repeated Single Integrals:
We now have a definition of the double integral with its interpretation as a volume and with the many possibilities of usefulness which our experience with the single integral suggests; however, as yet, we do not possess a method for evaluating such integrals. In this section, we shall see how the calculation of a double integral can be reduced to that of two single integrals.
Let u = f(x, y)
be a function which is defined and continuous in a rectangle R,
a £ x £ b, c £ y £ d. If we
fix upon any value x0 in the interval a £ x £ b, the
function f(x0, y)
is a continuous function of the remaining variable y,
whence there exists the integral
and it can be evaluated by the methods of the earlier chapters. This integral has a definite value for each value of x0 which we may select; in other words, the integral is a function f(x0) of the quantity x0:
For example, let u = f(x, y)
= x²y³, 0 £ x £ 1, 0 £ y £ 3. For each fixed
x in the interval 0 £ x £ 1, the integral 
can be
evaluated and, in fact, is 81x²/4, i.e., it is a
function of x. Or if f(x, x)
= exy, 1 £ x £ 2, 1 £ y £ 4, we
have 
Having thus found the function f (x), we can prove that it is continuous; this is a simple consequence of the uniform continuity of f(x, y).It is therefore possible to integrate f (x) between the limits a and b, thus obtaining the repeated integral
By reversing the order of the process, first
calculating the function of y defined by 
and
then integrating from c to d, we obtain the
other repeated integral
These integrals, as we have seen, are obtained by a double application of the ordinaly simple integration which we have studied before. Their importance lies in the following fact:
For continuous functions f(x, y) and for functions f(x, y) with at most jump discontinuities on a finite number of smooth curves, the repeated integrals are equal to the double integral:
We shall content ourselves with an intuitive
discussion of the case where f(x,y) is
continuous. In our original discussion of the double integral
regarded as the volume lying above the rectangle a £ x £ b, c
£ y
£ d
and below the surface
u = f(x, y), we obtained this
volume by subdividing the solid into vertical colunms and then
letting the diagonals of the bases of these columns approach 
zero. Instead of this, we can divide the solid into
slices of thickness k=(d-c)/n by
drawing the lines y = c + nk (n = 0, 1,
··· , n) parallel to the x-axis and then
constructing a plane perpendicular to the xy-plane
through each line (Fig. 8). These planes cut the solid into n
slices which grow thinner as n increases and the total
volume of which is equal to the double integral. We now see that
the volume of each slice is approximately (but, of course, not as
a rule exactly) equal to the product of the thickness k
by the area of the left-hand face, i.e., equal to
Hence, if we write
the desired volume is represented approximately by
As n ® ¥, these sums tend to
It is therefore reasonable to expect that the volume or doube integral is exactly equal to
which is the statement made ahove. A similar discussioa makes it equally plausible that the statement
is also true.