A3.1 Sufficient Conditions for Extreme Values
In the treatment of the heory of maxima and minima in the preceding chapter, we were content with finding necessary conditions for the occurrence of an extreme value. In many cases, occurring in practice, the nature of the stationary point thus found can be determined from the special nature of the problem, whence we can decide whether it is a maximum or a minimum. Yet, it is important to have general sufficient conditions for the occurrence of an extreme value. Such criteria will be developed here for the typical case of two independent variables.
If we consider a point (x0, y0) at which the function is stationary, i.e., a point at which both of its first partial derivatives vanish, the occurrence of an extreme value is connected with the question whether the expression
has or does not have the same sign for all sufficiently small values of h and k. If we expand this expression by Taylor's theorem with the remainder of third order, by virtue of the equations fx(x0, y0) = 0 and fy(x0, y0) = 0, we obtain at once
where r2 = h2 + k2 and e tends to zero with r.
Hence, we see that in a sufficiently small neighbourhood of the point (x0, y0) the behaviour of the functional difference
is essentially determined by the expression
where, for the sake of brevity, we have put
In order to study the problem of extreme values, we must investigate this homogeneous quadratic expression in h and k, or, as we say, the quadratic form Q. We will assume that not all the coefficients a, b, c vanish. In the exceptional case when all of them vanish, which we shall not consider, we must begin with a Taylor series extending to terms of higher order.
As regards the quadratic form Q, there are three different possible cases:
1. The form is definite, i.e., when h and k assume all values, Q only assumes values of one sign and only vanishes for h = 0, k = 0. We then say that the form is positive definite or negative definite, according to whether the sign is positive or negative. For example, the expression h2 + k2, which we obtain when a = c = l b = 0, is positive definite, while the expression -h2 + 2hk - 2k2 = -(h - k)2 - k2 is negative definite.
2. The form is indefinite, i.e., it can assume values of different sign, e.g., the form Q = 2hk, which has the value 2 for h = 1, k = 1 and the value -2 for h = -1, k = 1.
3. Finally, there is still the third possibility, when the form vanishes for certain values of h, k other than h = 0, k = 0, but otherwise assumes values of one sign only, e.g., the form (h + k)2, which vanishes for all sets of values h, k such that h = -k. Such forms are said to be semi-definite.
The quadratic form Q = ah + 2bhk + ck2 is definite if, and only if, the condition
is satisfied; it is then positive definite if a > 0 (so that also c > 0), otherwise it ia negative defmite.
In order that the form may be indefinite, it ia necessay and sufficient that
while the semi-definite case is characterized by the equation
Those conditions are easily obtained as follows. Either a = c = 0, when we must have b ¹ 0, and the form is, as has already been noted, indefinite, whence the criterion holds for this case; or we must have, say, a ¹ 0, when we can write
Obviousely , this form is definite, if ca - b2 > 0, and it then has the same sign as a. It is semi-definite, if ca - b2 = 0, for then it vanishes for all values of k, k which satisfy the equation h/k = -b/a, but has for all other values the same sign. It is indefinite if ca - b2 < 0, for it then assumes values with different signs when k vanishes and when h +-bk/a vanishes.
We shall now prove the following statements:
If the quadratic form Q(k,
k) is positive definite, the stationaiy
value assumed for h = 0, k = 0 is a minimum.
If the form is negative definite, the stationary
value is a maximum.
If the form is indefinite, we have neither a
maximum nor a minimum; the point is a saddle point.
Thus, a definite character of the form Q is a sufficient condition for an extreme value, while an indefinite character of Q excludes the possibility of an extreme value. We shall not consider the semi-definite case, which leads to involved discussions.
In order to prove the first statement above, we need only use the fact that, if Q is a positive definite form, there is a positive number m, independent of h and k, such that
Therefore
If we now choose r so small that the absolute value of e is less than m/2, we obviously have
Thus, for this neighbourhood of the point (x0, y0), the value of the function is everywhere larger than f(x0, y0), except, of course, at (x0, y0) itself. In the same way, when the form is negative definite, the point is a maximum.
In order to see this, consider the quotient Q(h, k)/(h² + k²) as a function of the two quantities
Then u² + v² and the form become a continuous function of u and v which must have a least value 2m on the circle u²+v²=1. Obviously, this value m satisfies our conditions. It is not zero, because on the circle u and v never vanish simultaneously.
Finally, if the form is indefinite, there is a pair of values (h1, k1) for which Q is negative and another pair (h2, k2) for which Q is positive, whence we can find a positive number m such that
If we now set h = th1, k = tk1, r² = h² + k² (t ¹ 0), i.e., if we consider a point (x + h, y +k) on the line joining (x0, y0) and (x0 + h1, ,y0 + k1), then it follows from Q(h, k) = t² Q(h1, k1) and r² = t²r1² that
Thus, we can make by choice of a sufficiently small t (and corresponding r) the expression f(x0+ h, y0 +k) - f(x0, y0) negative. We need only choose t so small that for h = th1, k = tk1 the absolute value of the quantity e is less than ½ m. For such a set of values, we have f(x0+ h, y0 +k) - f(x0, y0) < -mr²/2, so that the value f(x0+ h, y0 +k) is less than the stationary value f(x0, y0). In the same way, on carrying out the corresponding process for the system h = th2, k = tk2, we find that there are in an arbitrarily amall neighbourhood of (x0, y0) points at which the value of the function is larger than f(x0, y0). Thus, we have neither a maximum nor a minimum, but instead what we may call a saddle value.
If at the stationary point a = b = c = 0 , so that the quadratic form vanishes identically, as well in the semi-definite case, this argument does not apply. In order to obtain sufficient conditions for these cases would lead to involved manipulations. Thus, we have the rule for distinguishing maxima:
If at a point (x0, y0) the equations
hold as well as the inequality
then at that point the function has an extreme value. This is a maximum if fxx < 0 (and consequently fyy < 0), and a minimum if fxx > 0. On the other hand, if
the stationary value is neither a maximum nor a minimum. The case
remains undecided.
These conditions permit a simple geometrical mterpretation. The necessary conditions fx = fy = 0 state that the tangent plane to the surface z = f(x, y) is horizontal. If we really have an extreme value, then, in the neighbourhood of the point in question, the tangent plane does not intersect the surface. In contrast, in the case of a saddle point, the plane cuts the surface in a curve which has several branches at the point. This aspect will become clearer after the discussion of singular points.
As an example, we shall find the extreme values of the function
If we equate the first derivatives to zero, we obtain the equations
which have the solution x = (b - 2a)/3, y = (a - 2b)/3. The expression
is positive, and so is fxx = 2, whence the function has a minimum at the point in question.
The function
has a stationary point at the origin. The expression fxx fyy - fxy² vanishes there and our criterion fails. However, we readily see that the function does not have an extreme value there, because in the neighbourhood of the origin it assumes both positive and negative values.
On the other hand, the function
has a minimum at the point x = 1, y = 1, although the expression fxx fyy - fxy² vanishes there. In fact,
which is positive when r ¹ 0.
If f (a) = k ¹ 0 and x, y, z satisfy the relation
prove that the function
has a maximum when x = y = z = a, provided that
A3.2 Singular Points of Plane Curves
In 3.2.2, we have seen that, in general, a curve f(x, y) = 0 has a singular point at x = x0, y = y0, where the three equations
hold. In order to study systematically these singular points, we assume that in the neighbourhood of the point in question the function f(x, y) has continuous derivatives up to the second order and that at that point not all of the second derivatives vanish. By expanding in a Taylor series up to terms of the second order, we obtain the equation of the curve in the form
where we have put r² = (x - x0)² + (y — y0)² and e tends to zero with r.
Using a parameter t, we can write the equation of the general straight line through the point (x0, y0) in the form
where a and b are two arbitrary constants, which we may suppose to be chosen so that (a² + b²)= 1. In order to determine the point of intersection of this line with the curve f(x, y) = 0, we substitute these expressions in the above expansion for f(x,y); thus, we obtain for the point of intersection the equation
A first solution is t = 0, i.e., obviously, this is the point (x0, y0) itself. However, note that the left hand side of the equation is divisible by t², so that t is a double root of the equation. For this reason, the singular points are also sometimes called double points of the curve.
If we remove the factor t², we are left with the equation
We now ask whether it is possible for the line to intersect the curve in another point which tends to (x0, y0) as the line tends to some particular limiting position. Naturally, we call such a limiting position of a secant a tangent. In order to discuss this aspect, we observe that as a point tends to (x0, y0), the quantity t tends to zero, whence also c tends to zero. If the equation above is still to be satisfied, the expression a²fxx + 2abfxy + b²fyy must also tend to zero, i.e., we must have for the limiting position of the line
This equation yields a quadratic condition determining the ratio a/b which fixes the line.
If the discriminant of the equation is negative, i.e., if
we obtain two distinct real tangents. The curve has a double point or node, as is exhibited by the lemniscate
at the origin or the strophoid
at the point x0 = a, y0 = 0.
If the discriminant vanishes, that is, if
we obtain two coincident tangents; for example, it is then possible that two branches of the curve touch or that the curve has a cusp.
Finally, if
there is not at all a real tangent . For example, this happens in the case so-called isolated or conjugate points of an algebraic curve. These are points at which the equation of the curve is satisfied, but in the neighbourhood of which there lies no other point of the curve.
The curve (x2 - a2)2 +
(y2 - b2)2 =
a4 + b4 demonstrates this case. The values
in x = 0, y = 0 satisfy
the equation, but for all other values in the region 
the
left hand side is less than the right hand side.
We have omitted the case in which all the derivatives of the second order vanish. This case leads to involved investigations and we shall not consider it. Through such a point, there may pass several branches of the curve or there may occur singularities of other types.
Finally, we shall briefly mention the link between these topics and the theory of maxima and minima. Owing to the vanishing of the first derivatives, the equation of the tangent plane to the surface z = f(x, y) at a stationary point (x0, y0) is simply
Hence the equation
yields the projection onto the xy-plane of the curve of intersection of the tangent plane with the surface, and we see that the point (x0, y0) is a singular point of this curve. If this is an isolated point, the tangent plane in a certain neighbourhood has no other point in common with the surface and the function f(x, y) has a maximum or a minimum at the point (x0, y0) However, if the singular point is a multiple point, the tangent plane cuts the surface in a curve with two branches and the point corresponds to a saddle value. These remarks lead us precisely to the sufficient conditions which we have found already.
A3.3 Singular Points of Surfaces
In a similar manner, we can discuss a singular point of a surface f(x, y, z) =, i.e., a point for which
Without loss of generality, we may take the point as the origin O. If we let
at this point, we obtain the equation
for a point (x, y, z) which lies on a tangent to the surface at O
This equation represents a quadratic cone touching the surface at the singular point - instead of the tangent plane at an ordinary point of the surface - if we assume that not all of the quantities a, b, ··· , n vanish and that the above equation has real solutions other than x = y = z = 0.
A3.4 Link between Euler's and Lagrange's Representations of the Motion of a Fluid
Let (a, b, c) be the co-ordinates of a particle at the time t = 0 in a moving continuum (liquid or gas). Then the motion can be represented by three functions
or in terms of a position vector x =
x(a, b, c, t).
The velocity and acceleration are given by the derivatives with
respect to the time t. Thus, the velocity vector is 
with
the components 
and the acceleration vector
is 
with the components 
all of which appear as
functions of the position (a, b, c)
and the parameter t. For each value of t,
we have a transformation of the co-ordinates (a, b,
c), belonging to the different points of the moving continuum into the
co-ordinates (x, y, z) at the time t.
This is the so-called Lagrange representation of
the motion. Another representation, introduced by Euler, is based on the
knowledge of three functions
representing the components of the velocity of
the motion at the point (x, y, z) at the
time t.
In order to pass from the first to the second representation,
we use the first representation to calculate a, b, c
as functions of x, y, z, t and
substitute these expressions in the expressions for 
We then get the components of the acceleration from
as follows:
or
In continuum mechanics, the following equation, linking Euler's and Lagrange's representations, is fundamental:
where
is the Jacobian which characterizes the motion.
The reader may complete the proof of this and the correponding theorem in two dimensions by means of the various rules for the differentiation of implicit functions.
A3.5 Tangential Representation of a Closed Curve
A family of straight lines with parameter a may be given by
where p(a) denotes a function which is twice continuously differentiable and periodic with period 2p (a so-called tangential function). The envelope C of these lines is a closed curve satisfying (1) and the additional equation
Hence
is the parametric representation of C (with the parameter a). Equation (1) is the equation of the tangents of C and is referred to as the tangential equation of C.
Since
we have at once the following expressions for the length L and the area A of C:
since p'(a) is also a function of period 2p.
Since obviously p(a) + c is the tangential function of the parallel curve at a distance c from C, the formulae for the area and the length of a parallel curve (cf. Vol. I, Ex. 22, 9.5) are readily derived from these expressions.
We deduce from this the isoperimetric inequality
where the equality sign holds only for the circle . This may also be expressed by the statement: Among all closed curves of given length, the circle has the largest area.
For the proof, we employ the Fourier expansion of p(a),
when
so that (using the orthogonality relations ) we have
Thus,
in particular, A = L²/4p only if an, = bn = 0 for n ³ 2, i.e., p(a) = a0/2 + a1cos a + b1 sin a; the last equation defines a circle, as is readily proved from (2).