Developments and Applications of the Differential Calculus
3.1.1 General Remarks: In analytical geometry, it frequently happens that the equation of a curve is not given in the form y=f(x), but as F(x, y) = 0. Accordingly, a straight line may be represented by the equation ax + by + c = 0 or an ellipse by the equation x²/a² + y²/b² = 1. In order to obtain the equation of the curve in the form y =f(x), we must solve F(x, y) = 0 for y.
We have also considered in Volume 1 the problem of finding the inverse of a function y = f(x), in other words, the problem of solving the equation F(x, y) = y - f(x) = 0 for the variable x. These examples suggest the importance of studying the notion of solving an equation F(x, y)=0 for x or for y. We shall now study this problem and extend the results in 3.3.6 to functions of several variables.
In the simplest cases, such as the equations mentioned above, the solution can readily be found in terms of elementary functions. In other cases, the solution can be approximated to as closely as we please. However, for many purposes, it is preferable not to work with the solved form of the equation or with these approximations, but instead to draw conclusions regarding the solution by studying the function F(x, y), in which neither of the variables x, y is preferred.
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. For example, the equation x² + y² = 0 is only satisfied by the single pair of values x = 0, y = 0, while the equation x² + y² + 1 = 0 is not at all satisfied by real values. Hence, we must investigate the matter more closely, in order to find out whether an equation F(x, y) = 0 defines a function y = f(x) and what are the properties of this function.
3.1.2
Geometrical Interpretation (10.5.1): In order to clarify the
situation, we will think of the 
function u = F(x,
y) as being represented by a surface
in three- dimensional space. The solutions of
the equation F(x, y) = 0 are the
same as the simultaneous solutions of the two equations u
= F(x, y) and u = 0. Geometrically
speaking, our problem is to find whether curves y=f(x)
or x=f (y) exist
in which the surface u = F(x, y)
intersects the xy-plane. (How far
such a curve of intersection may extend does not concern us
here.)
A first possibility is that the surface and the plane may have no point in common. For example, the paraboloid
u = F(x, y) = x² + y² + 1
lies entirely above the xy-plane. In such a case, there is obviously no curve of intersection, whence we need only consider cases in which there is a point (x0, y0) at which F(x0, y0) = 0; the values x0, y0 are called an initial solution.
If an initial
solution exists, there remain two
possibilities. Either the tangent plane at the point (x0,y0)
is horizontal or it is not. If it is horizontal, we can readily
show by means of examples that 
the solution y = f(x)
or x = f(x) may
fail to exist. For example, the paraboloid
u = x² + y² has the initial solution x
= 0, y = 0, and has no other point in the xy-plane.
Again, the surface u = xy has the initial solution x
== 0, y = 0, and, in fact, intersects the xy-plane
along the lines x = 0 and y = 0 (Figs.
1/2). But in no neighbourhood of the origin can we represent the entire intersection by a
function y=f(x) or x=f(y). On the other hand,
it is quite possible for the equation F(x,y)=0
to have a solution, even when the tangent plane at the initial solution is
horizontal, as, for example, in the case (y—x)4
= 0. Hence, in the (exceptional)
case of a horizontal tangent plane,
no definite general statement can be made.
The remaining possibility is that at the initial solution the tangent plane is not horizontal. Intuition tells us, roughly speaking, that the surface u = F(x, y) cannot bend fast enough to avoid cutting the xy-plane near (x0, y0) along a single well-defined curve and that a portion of the curve near the initial solution can be represented by the equation y=f(x) or x =f (y). The statement that the tangent plane is not horizontal is the same as the statement that Fx(x0, y0) and Fy(x0, y0) are not both zero. This is the case which we shall discuss analytically in the next section.
3.1.3 The Theorem of Implicit Functions: The general theorem which states sufficient conditions for the existence of implicit functions and at the same time gives a rule for their differentiation is:
If F(x, y) has continuous Fx and Fy, and if at the point (x0, y0) within its region of definition the equation F(x0, y0) = 0 is satisfied, 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 exactly one value y = f(x) lying in the interval y1£y£y2. This function satisfies the equation. y0 = f(x0) and for every x in the interval the equation
is satisfied. The function f(x) is continuous and differentiable, and its derivative and differential are given by the equations
respectively.
For the presentm we shall assume that the first part of the theorem, relating to the existence and continuity of the implicitly defined function, has already been proved and shall confine ourselves to proving the differentiability of the function and the differentiation formulae; we shall postpone the proof of the existence and continuity of the solution to 3.1.6.
If we could differentiate the terms of the equation F(x, f(x)) = 0 by the chain rule, the above equation would follow at once (Vol. 1, 10.5.1). However, since the differentiability of f(x) must first be proved, we must consider the matter in somewhat greater detail
As the derivatives Fx and Fy have been assumed to be continuous, the function F(x, y) is differentiable. We can therefore write
where e1 and e2 are two quantities
tending to zero with h, k or 
. We will now confine
our attention to pairs of values (x, y) and (x
+ h, y + k) for which both x and x+
h lie in the interval x1
£ x £
x2,
and for which y = f(x) and y+k=f(x+h).
For such pairs of values, we have F(x, y) =
0 and F(x + h, y + k) = 0, so
that the preceding equation reduces to
We assume here that f(x) has been proved to be continuous. Hence, as h tends to 0, so does k, and with them also e1 and e2 tend to 0. If we divide by hFy (which, by assumption, is non-zero), the last equation yields
and, on performing the passage to the limit h,
However,
this proves the differentiability of f(x) and gives the required rule for differentiation:
We can write this rule in the form
This last equation states that, by virtue of the equation F(x, y) = 0, the differentials dx and dy cannot be chosen independently of each other.
As a rule, an implicit function can be differentiated more easily by means of this rule than by first writing down its explicit form. The rule can be used whenever the explicit representation of the function is theoretically possible according to the theorem of implicit functions, even in cases where the practical solution in terms of the ordinary functions (rational, trigonometric functions, etc.) is extremely complicated or impossible.
Suppose that the second order partial derivatives of F(x, y) exist and are continuous. In the equation y' = - Fx/Fy, the right hand side of which is a compound function of x, we can differentiate according to the chain rule and then substitute for y' its value -Fx/Fy which yields
as the formula for the second derivative of y = f(x).
In the same way, we can. obtain, the higher derivatives of f(x) by repeated differentiation.
1. For the function y = f(x) obtained from the equation of the circle
we obtain the derivative
This is readily verified directly. If we solve for y,
the equation of the circle yields either the function 
or the
function 
representing the upper and lower semi-circles,
respectively. In the first case, differentiation gives
and in the second case
Thus, in both cases, y' = -x/y.
2. In the case of the lemniscate
,
it is not easy to solve for y. For x = 0 and y = 0, we obtain F = 0, Fx = 0, Fy = 0. Here, our theorem fails, as might be expected, from the fact that two different branches of the lemniscate pass through the origin. However, for all points of the curve for which y ¹ 0, our rule applies and the derivative of the function y = f(x) is given by
We can obtain important information about the curve from this
equation, without introducing the explicit expression for y. For
example, maxima or minima may occur where y'
= 0, i.e., for x = 0 or for x²
+ y² = a². From the equation of the
lemniscate, y=0 when x=0; however, at the origin,
there is no extreme value (Fig. 26), whence the two
equations give the four points (±a
,±a) as the maxima and minima.
3. In the case of the folium
of Descartes (Fig. 3)
,
the explicit solution would be exceedingly inconvenient. At the origin, where the curve intersects itself, our rule again fails, since at that point F=Fx=Fy=0. For all points at which y² ¹ ax, we have
Hence, there is a zero of the derivative when x²- ay = 0 or, if we use the equation of the curve, if
3.1.5 The Theorem of Implicit Functions for more than Two Independent Variables: The general theorem of implicit functions can be extended to the case of several independent variables as follows:
Let F(x, y, ··· , z, u) be a continuous function of the independent variables s, y, ··· , z, u, and let it possess continuous partial derivatives Fx, Fx, ··· , Fz , Fu. For the system of values x0, y0, ··· , z0, u0, corresponding to an interior point of the region of definition of F, let F(x0, y0, ··· , z0, u0) = 0 and
Then, we can mark off an internal interval u1 £ u £ u2 about u0 and a region R containing (x0, y0, ··· , z0) in its interior such that for every (x, y, ··· , z) in R the equation F(x,y,···,z,u)=0 is satisfied by exactly one value of u. in the interval u1 £ u £ u2. For this value of u, which we denote by u = f(x,y,···,z), the equation
holds identically in R; in addition,
The function f is a continuous function of the independent variables x, y, ··· , z and possesses continuous partial derivatives given by the equations
For the proof of the existence and continuity of f(x, y, ··· , z), we refer the reader to 3.1.6. The formulae of differentiation follow from those for the case of one independent variable, since we can, for example, let y, ··· , z remain constant and thus find the formula for fx.
If we desire so, we can combine our differentiation formulae in the single equation
In words:
If in a function F(x, y, ··· , z, u) the variables are not independent of one another, but are subject to the condition F = 0, then the linear parts of the increments of these variables are likewise not independent of one another, but are connected by the condition dF = 0, that is, by the linear equation
If we replace here du by the expression uxdx + uydy + ··· + uzdz and then equate the coefficient of each of the mutually independent differentials dx, dy, ··· , dz to zero, we again obtain the above differentiation formulae.
Incidentally, the concept of implicit functions enables us to give a general definition of the concept of an algebraic function. We say that u = f(x, y, ···) is an algebraic function of the independent variables x, y, ···, if it can be defined implicitly by an equation F(x, y,··· , u) = 0, where F is a polynomial in the arguments x, y, ··· , u; briefly speaking, if u satisfies an algebraic equation. All functions which do not satisfy an algebraic equation are said to be transcendental.
As an example of our differentiation formulae, consider the equation of the sphere
We obtain for the partial derivatives
and, by further differentiation,
3.1.6 Proof of the Existence and Continuity of Implicit Functions: Although, in many special cases, the existence and continuity of implicit functions follows from the fact that the equation f(x,y) = 0 can actually be solved in terms of the ordinary functions by means of some special device, yet it is still necessary to give a general analytical proof of the existence theorem stated above.
As a first step, we mark out a rectangle x1 £ x £ x2, y1 £ y £ y2 in which the equation F(x, y) = 0 determines a unique function y = f(x). We shall make no attempt to find the largest such rectangle; we only wish to show that such a rectangle exists.
Since Fy(x, y) is
continuous and Fy(x0, y0) ¹ 0, we can find a
rectangle R, with the point P(x0,y0) as centre, so small
that in all of R the function Fy
remains different from zero and thus has always the same sign.
Without loss of generality, we can assume that this sign is
positive, so that Fy is positive
everywhere in R; otherwise, we should merely have to
replace the function F by -F, which does not change
the equation F(x, y) = 0. Since Fy
> 0 on every line-segment x = const parallel to
the y-axis and lying in R, the function F(x,
y), as a function of y alone, is 
monotonic increasing.
But F(x0,y0)=0, whence, if A
is a point of R with co-ordinates x0 and y2 ( y2 > y0) lies on the
vertical line through P (Fig. 4), the value of the
function at A, F(x0, y1), is negative,
while at the point B with co-ordinates x0 and y2 (y2 > y0), the value of the
function F(x0,
y2) is positive. Owing to the
continuity of F(x, y), it follows
that it has negative values
along a certain horizontal line-segment y = y1 through A and
lying in R, and has positive
values along a line-segment y=y2 through B and
lying in R. We can therefore mark off an interval x1 £ x £ x2 about x2 so small that for
values of x in that interval the function F(x,
y) remains negative along the horizontal line through A
and positive along the horizontal line through B. In other
words, for x1£ x £ x2, the
inequalities F(x, y1) < 0 and F(x, y2) > 0 hold.
Assume now that x is fixed at any value in the interval x1 £ x £ x2 and let y increase from y1 to y2. The point (x, y) then remains in the rectangle
which we assume to lie completely within R. Since Fy(x, y) > 0, the value of the function F(x, y) increases monotonically and continuously from a negative to a positive value and can never have the same value for two points with the same abscissa. Hence, there is for each value of x in the interval x1 £ x £ x2 a uniquely determined value* of y for which the equation F(x,y) = 0 is satisfied. This value of y is thus a function of x; we have accordingly proved the existence and the uniqueness of the solution of the equation F(x, y) = 0. At the same time, the part played by the condition Fy ¹ 0 has been clearly shown. If this condition were not fulfilled, the values of the function at A and B might not have opposite signs, so that f(x, y) need not pass through zero on vertical line-segments. Or, if the signs at A and at B differed, the derivative Fy could change sign, so that, for a fixed value of x, the function f(x, y) would not increase monotonically with y and might have the value zero more than once, thus destroying the uniqueness of the solution.
* If the restriction y1 £ y £ y2 is omitted, this will not necessarily remain true. For example, let F = x² + y²-1 and x0=0, y0=1. Then, for -½£x£ ½, there is just a single solution y = f(x) in the interval 0 £ y £ 2; but if y is unrestricted, there are two solutions
This proof merely tells us that the function y =f(x) exists. It is a typical case of a pure existence theorem, in which the practical possibility of calculating the solution is not at all being considered.
The sacrifice of the statement of such practical methods in a general proof is sometimes an essential step towards the simplification of proofs.
The continuity of the function f(x) follows almost at once from the above considerations. Let R(x1' £ x £ x2'; y1' £ y £ y2') be a rectangle lying entirely within the rectangle x1 £ x £ x2, y1 £ y £ y2 found above. For this smaller rectangle, we can carry out exactly the same process as before, in order to obtain a solution y = f(x) of the equation F(x, y) = 0. However, in the larger rectangle, this solution was uniquely determined, whence the newly found function f(x) is the same as the old one. If we now wish, for example, to prove the continuity of the function f(x) at the point x = x0, we must show that for any small positive number e one has |f(x) - f(x0) < e , provided only that x lies sufficiently near to the point x0. For this purpose, we set
and we determine for these values y1' and y2' the corresponding interval x1' £ x £ x2'. Then, by the above construction, there lies for each x in this interval the corresponding f(x) between the bounds y1' ' and y2', and therefore it differs from y0 by less than e. This expresses the continuity of f(x) at the point x0. Since we can apply the above argument to any point x in the interval x1 £ x £ x2 , we have proved that the function is continuous at each point of this interval.
The proof of the general theorem for F(x, y, ... , z, u), a function with a larger number of independent variables, follows exactly along the same lines as the proof just completed and offers no further difficulties.
1. Prove that the following equations have unique solutions for y near the points indicated:
2. Find the first derivatives of the solutions in Example 1.
3. Find the second derivatives of the solutions in Example 1.
4. Find the maximum and minimum of the function y = f(x) = x² + xy + y² = 27.
5. Show that the equation x + y + z = sin xyz can he solved for z near (0, 0, 0). Find the partial derivatives of the solution.
3.2 CURVES AND SURFACES IN IMPLICIT FORM
3.2.1 Plane Curves in Implicit Form: We have previously expressed plane curves in the form y = f(x), which is unsymmetrical and gives preference to one of the co-ordinates. The tangent and the normal to the curve are given by the equations
respectively, where x and h are the current co-ordinates of the tangent and normal, and x and y are the co-ordinates of the point of the curve. We have also found an expression for the curvature, and criteria for points of inflection (Volume 1, Chap. 5). We shall now obtain the corresponding formulae for curves which are represented implicitly by equations of the type F(x,y)=0. We do this under the assumption that at the point in question Fx and Fy are not both zero so that F²x+F²y¹0.
If we assume, say, that Fy ¹ 0, we can substitute for y' in the equation of the tangent at the point (x, y) of the curve its value - Fx/Fy and obtain at once the equation of the tangent in the form
Similarly, we have for the normal
Without going out of the way to use the explicit form of the equation of the curve, we can also obtain the equation of the tangent directly in the following way. If a and b are any two constants, the equation
with current co-ordinates x
and h represents a straight
line passing through the point P(x, y).
If now P is any point on the curve, i.e., if F(x,
y) = 0, we wish to find the line through P
with the property that, if P1 is a point of
the curve with the co-ordinates x1 = x +
h and y1 = y + k,
the distance from the line to P1 tends to
zero to a higher order than 
By virtue of the differentiability of the
function F, we can write
where r tends to 0 with e . Since both the points P and P1 lie on the curve, this equation reduces to hFx + kFy = - er. As we have assumed that F²x + F²y ¹ 0, we can write this last equation in the form
where also 
tends to zero with r.
If we write
the left-hand side of this equation may be regarded as the expression obtained when we substitute the co-ordinates of the point (x1 = x+ h, y1 = y + k) for x and h in the canonical form of the equation
is the same as the equation of the tangent found earlier. Hence we can regard the tangent at P as the line * the distance of which from neighbouring points P1 of the curve vanishes to a higher order than the distance PP1.
* The reader will find it easy to prove that two such lines cannot exist, so that our condition determines the tangent uniquely.
The direction cosines of the normal to the curve are given by
f
which represent the components of a unit vector in the direction of the normal, i.e., of a vector with length 1 in the direction of the normal to the curve at the point P(x, y). The direction cosines of the tangent at the point P(x, y) are given by
More generally, if we consider instead of the curve F(x, y) = 0 the curve
where c is any constant, everything in the above discussion remains unchanged. We have only to replace the function F{x, y) by F(x, y) - c, which has the same derivatives as the original function. Thus, for these curves, the equations of the tangent and the normal have exactly the same forms as above.
The class of all the curves which we obtain when we allow c to range through all the values in an interval is called a family of curves. The plane vector with components Fx and Fy, which is the gradient of the function F(x, y), is at each point of the plane perpendicular to the curve of the family passing through that point, as we have already seen in 2.7.3. This again yields the equation of the tangent. For the vector with components (x — x) and (h - y) in the direction of the tangent must be perpendicular to the gradient so that the scalar product
must vanish.
While we have taken the positive sign for the square root in the above formulae, we could equally well have taken the negative sign. This arbitrariness corresponds to the fact that we can call the direction towards either side of the curve the positive direction. We shall continue to choose the positive square root and thereby fix a definite direction of the normal. However, it is to be observed that, if we replace the function F(x, y) by -F(x, y), this direction is reversed, although the geometrical nature of the curve is unaffected. (As regards the sign of the normal, cf. 1.1.35.2.2).
We have already seen in Volume 1, 3.5.1 that, for a curve explicitly represented in the form y = f(x), the condition f "(x) = 0 is necessary for the occurrence of a point of inflection. If we replace this expression by its equivalent expression
we obtain the equation
as a necessary condition for the occurrence of a point of inflection. In this condition, there is no longer any preference given to either of the two variables x, y. It has a completely symmetrical character and no longer depends on the assumption Fy ¹ 0.
If we substitute for y' and y" in the formula for the curvature (Volume l, 5.2.6)
we obtain
which is likewise perfectly symmetrical.(sign of the curvature). We find for the co-ordinates of the centre of curvature
where
If the two curves F(x, y) = 0 and G(x, y) = 0 intersect at the point with co-ordinates x, y, the angle between them is defined as the angle w formed by their tangents (or normals) at the point of intersection. If we recall the above expressions for the direction cosines of the normals and the formula for the scalar product (1.1.3 ), we obtain
for the cosine of this angle. Since we have taken the positive square roots here, the cosine is uniquely determined; this corresponds to the fact that we have thereby chosen definite directions for the normals and have thus determined the angle between them uniquely.
By setting in the last formula w = p/2, we obtain the orthogonality condition, i.e., the condition for the curves to intersect at right angles
If the curves are to touch, the ratio of the differentials, dy : dx, must be the same for the two curves, i.e., there must be fulfilled the condition
This condition may be rewritten in the form
As an example, consider the parabolas
(Fig. 9), all of which have the origin as focus (confocal parabolas). If p1>0 and p2<0, the two parabolas
intersect each other and are at the intersection at right angles to each other, because
since
As a second example, consider the ellipse
The equation of the tangent at the point (x, y) is, as we know from analytical geometry,
or,
We find for the curvature
If a > b, this has its maximum a/b² at the vertices y = 0, x = ± a, its minimum b/a² at the other vertices x = 0, y = ±b.
3.2.2 Singular Points of Curves: We now add a few
remarks on singular points of a curve.
We shall just give a number of typical
examples; for a more thorough investigation, we
refer the reader to the Appendix.
In the above formulae, there occurs frequently the expression 
in
the denominator. Accordingly, we may expect something unusual to
happen when this quantity vanishes, i.e., when Fx
= 0 as well as Fy = 0 at a point
of the curve. This is especially brought out by the fact that at
such a point y' = -Fx/Fy,
the slope of the tangent to the curve, loses its meaning.
We say that a point of a curve is a regular point, if in the neighbourhood of this point either the co-ordinate y can be represented either as a continuously differentiable function of x or x as a continuously differentiable function of y. In either case, the curve has a tangent and, in the neighbourhood of the point in question, the curve differs but little from that tangent. All other points of a curve are called singular points (or singularities).
We know from the theory of implicit functions that a point of the curve F(x, y) = 0 is regular if at that point Fy ¹ 0, since we can. then solve the equation in order to obtain a unique differentiable solution y = f(x). Similarly, a point is regular if Fx ¹ 0. Hence, the singular points of a curve are found among those points of the curve at which the equations
are satisfied in addition to the equation of the curve.
An important type of singularity is a multiple point, i.e., a point through which two or more branches of the curve pass. For example, the origin is a multiple point of the lemniscate
In the neighbourhood of such a point, it is impossible to express the equation of the curve uniquely in the form y = f(x) or x= f (y).
The truth of the relations Fx
= 0 and Fy = 0 is a necessary, but by no
means a sufficient
condition for a multiple point;
on the contrary, quite a different type of singularity
may occur such as a cusp.
As an example, consider the function
with a cusp at the origin. At that point, both the first partial derivatives of F vanish.
Moreover, cases may occur in which both Fx and Fy vanish, and yet there is no striking peculiarity of the curve at the point, the curve being regular there.
An example is the curve
or, in explicit form,
We see at once from the equations (-x)4/3 = x4/3, y' = (4/3)x1/3 that the curve is symmetrical with respect to the y-axis and touches the x-axis at the origin, like a parabola. Yet the origin is a somewhat special point on the curve, since the second derivative is infinite there, whence the curvature is infinite, while the direction of the tangent exhibits no peculiarity. Another example is the curve (y - x)² = 0, which is a straight line and therefore regular throughout, even though Fx=0 and Fy=0 for every point of the line.
As a result of this discussion, we see that in an investigation and discussion of singular points of a curve it is not enough to verify that the two equations Fx=0 and Fy=0 are satisfied; on the contrary, each case must be studied especially .
3.2.3 Implicit Representation of Surfaces: Hitherto, we have
usually represented a function z = f(x, y)
(we write here z instead of u, employed above) by
means of a surface
in xyz-space. However, if we are not given
initially the function, but a surface in space, the preference
which this form of expression gives to the co-ordinate z
may turn out to be inconvenient, just as in the case of the
representation of plane curves in the form y = f(x).
It is more natural, and more general, to represent surfaces in
space by equations of the form F(x, y, z) =
0 or F(x, y, z) = const; for
example, it is better to represent the sphere
by the equation x ² + y² + z² -
r²= 0, and not by 
The form z—f(x,
y) = 0 can then be treated as a special case.
In order to establish the equation of the tangent plane to the surface F(x, y, z) = 0 at the point (x,y,z), we first make the assumption * that at that point Fx² + Fy² +Fz² ¹ 0, i.e., that at least one of the partial derivatives, say Fx, is non-zero. Then we can determine from the equation of the surface z = f(x, y) explicitly as a function of x and y. If we substitute in the equation of the tangent plane
for the derivatives zx and zy their values
we obtain the equation of the tangent plane in the form
where x, h, z are the current co-ordinates.
* The vanishing of this expression indicates the possibility that certain singularities may occur; however, we shall not discuss this.
As in the case of the tangent to a plane curve, we can derive
this equation directly from the implicit representation of
the surface by considering the problem of finding a plane through
the point (x, y, z) of the surface with
the property that the distance from the plane to the point (x+h,y+k,z+1)
of the surface vanishes as 
to a higher order than
r.
Elementary theorems of analytical geometry ( 1.1.4 ) show that the direction cosines of the normal to a surface, that is, of the normal to the tangent plane, are given by
In taking the positive square root in the denominator, we assign a definite sense of direction to the normal (3.2.1).
If two surfaces F(x, y, z) = 0 and G(x, y,z) = 0 intersect at a point, the angle w between the surfaces is defined as the angle between their tangent planes, or, what is the same thing, the angle between their normals. This is given by
In particular, the orthogonality condition is
Instead of the single surface F(x, y, z) = 0, we may consider the entire family of surfaces f(x,y,z) = c, where c is a different constant for each surface of the family. We assume here that there passes through each point of space, or at least through every point of a certain region of space, one and only one surface of the family; or, as we say, that the family covers the region simply. The individual surfaces are then called lead surfaces of the function F(x, y, z). In 2.7.3., we have considered the gradient of this function, that is, the vector with the components Fx,Fy,Fz. We see that these components have the same ratios as the direction cosines of the normal, whence we conclude that the gradient at the point with the co-ordinates (x, y, z) is perpendicular to the level surface passing through that point. (If we accept this fact as already proved in 2.7.3., we at once have a new and simple method for deriving the equation of the tangent plane, just like that given 3.2.1 for the equation of the tangent.)
As an example, consider the sphere
At the point (x, y, z), the tangent plane is
The direction cosines of the normal are proportional to x, y, z, i.e., the normal coincides with the radius vector drawn from the origin to the point (x, y, z).
For the most general ellipsoid with the principal axes as co-ordinate axes
the equation of the tangent plane is
1. Find the tangent plane
(a) of the surface
at the point (1, 1, 1);
(b) of the surface
at the point (1, 1, 1);
(c) of the surface
at the point (p/3, p/3, 0).
2. Calculate the curvature of the curve
at the origin.
3*. Find the curvature at the origin of each of the two branches of the curve
4. Find the curvature of a curve which is given in polar co-ordinates by
5. Prove that the three surfaces of the family of surfaces
which pass through a single point are orthogonal to each other.
6. The points A and B moves uniformly with the same velocity, A starting from the origin and moving along the x-axis, B starting from the point (a, 0,0) and moving parallel to the y-axis. Find the surface enveloped by the straight lines joining them.
7. Prove that the intersection of the curve
with the line x + y = a are inflections of the curve.
8. Discuss the singular points of the curves:
9. Let (x, y) be a double point of the curve F(x, y) = 0. Calculate the angle j between the two tangents at (x, y), assuming that not all the second derivatives of F vanish at (x, y).
Find the angle between the tangents at the double point (a) of the lemniscate, (b) of the folium of Descartes (3.1.4).
10. Determine a and b so that the conics
intersect orthogonally at the point (1,1) and have the same curvature at this point.
11. If F(x, y, z) = 1 in the equation of a surface, F being a homogeneous function of degree h, then the tangent plane at the point (x, y, z) is given by
12. Let K' and K" be two circles with two common points A and B. If a circle K is orthogonal to K' and K", then it is also orthogonal to every circle passing through A and B.
13. Let z be defined as a function of x and y by the equation
Express zx and zy as functions of x, y, z.
3.3 SYSTEMS OF FUNCTIONS, TRANSFORMATlONS AND MAPPINGS
3.3.1 General Remarks:. The results obtained above for implicit functions now allows us to consider systems of functions, that is, to discuss several functions simultaneously. We shall consider here the particularly important case of systems, where the number of functions is the same as the number of independent variables. We begin by investigating the meaning of such systems in the case of two independent variables. If the two functions
are both differentiable in a region R of the xy-plane, we can interpret this system of functions in two different ways. The first interpretation (the second will be given in 3.3.2) is by means of a mapping or transformation. There corresponds to the point P with co-ordinates (x, y) in the xy-plane the image point P with the co-ordinates (x, h) in the xh-plane.
An example of such a mapping is the affine mapping or transformation
of 1.4.1, where a, b, c, d are constants.
Frequently, (x, y) and (x, h) are interpreted as points of one and the same plane. In this case, we speak of a mapping of the xy-plane onto itself or a transformation of the xy-plane onto itself.
It is also possible to interpret a single function x = f(x) of a single variable as a mapping, if we think of a point with co-ordinate x on an x-axis as being brought by means of the function into correspondence with a point x on a x-axis. By this point-to-point correspondence, the whole or a part of the x-axis is mapped onto the whole or a part of the x-axis. A uniform scale of equidistant x-values on the x-axis will, in general, be expanded or contracted into a non-uniform scale of x-values on the x-axis. The x-scale may he regarded as a representation of the function x = f(x). Such a point of view is frequently found useful in applications (e.g., in nomography).
The fundamental 'problem connected with a mapping is that of its inversion, i.e., the question whether and how, by virtue of the equations
x and y can be regarded as functions of x and h, and how these inverse functions are to be differentiated.
If, as the point (x, y) ranges over the region R, its image point (x ,h) ranges over a region B of the (x,h)-plane, we call B image region of R. If two different points of R always correspond to two different points of B, we can always find for each point of B a single point of R - its image. Thus, we can assign to each point of B the point of R of which it is the image. (This point of R is sometimes called the model as opposed to the image.) Hence, we can invert the mapping uniquely, or determine x and y uniquely as functions
of x and h, which are defined in B. We then say that the original mapping can be inverted uniquely , or has a unique inverse, or is a one-to-one * mapping (often written (1,1))and we call x = g(x, h), y = h(x, h) the transformation inverse to the original transformation or mapping.
If in this mapping the point P with co-ordinates (x, y) describes a curve in the region R, its image point will likewise describe a curve in the region B, called the image curve of the first. For example, the curve x = c, which is parallel to the y-axis, corresponds to a curve in the xh-plane which is given in parametric form by the equations
where y is the parameter. Again, there corresponds to the curve y = k the curve
If we assign to c and k sequences of neighbouring values c1, c2, c3, ··· and k1, k2, k3, ··· , then the rectangular co-ordinate net consisting of the lines x = const and y = const (e.g., the network of lines on ordinary graph paper) usually gives rise to a corresponding curvi-linear net of curves in the xh-plane (Figs. 6,7). The two families of curves composing this net of curves can be written in implicit form. If we represent the inverse mapping by the equations
the equations of the curves are simply
respectively.
In the same way, the two families of lines x == g and h = k correspond in the xh-plane to the two families of curves
in the xy-plane.
As an example, consider inversion or the mapping by reciprocal radii or reflection in the unit circle. This transformation is given by
There corresponds to the point P with co-ordinates (x, y) the point P with co-ordinates (x, h), lying on the same line OP and satisfying the
that the radius vector to F is the reciprocal of the radius vector to P. Points inside the unit circle are mapped onto points outside the circle and vice versa.
We find from the relation
that the inverse transformation is
which is again an inversion.
For the region R, we may take the entire xy-plane with the exception of the origin and for the region B the entire xh-plane with the exception of the origin. The lines 5 = c and h = k in the xh-plane correspond to the circles
in the xy-plane, respectively; at the origin, these circles touch the y-axis and the x-axis, respectively. In the same way, the rectilinear co-ordinate net in the xy-plane corresponds to the two families of circles touching the x-axis and the h-axis, respectively at the origin.
As another example, consider the mapping
The curves x = const yield in the xy-plane the rectangular hyperbolae x² - y² = counts., the asymptotes of which are the lines x = y and x = -y; the curves h = const. also correspond to a family of rectangular hyperbolae with the co-ordinate axes as asymptotes. The hyperbola of each family intersect those of the other family at right angles (Fig. 8). The lines parallel to the axes in the xy-plane correspond to two families of parabolaes in the xh-plane, the parabolas h² = 4c²(c² - x) corresponding to the lines x = c and the parabolas h² = 4c²(c² + x) to the lines y = c. All these parabolas have the origin as focus and the x-axis as axis (a family of confocal and coaxial parabolas; Fig. 9). Systems of confocal ellipses and hyperbolas are treated in Example 5 p. 158.
(1,1)-transformations have an important interpretation and application in the representation of deformations or motions of continuously distributed substances such as fluids. If we think of such a substance as being spread out at a given time over a region R and then deformed by a .motion, the substance originally spread over R will, in general, cover a region B, different from R. Each particle of the substance can be distinguished at the beginning of the motion by its co-ordinates (x, y) in B and at the end of the motion by its co-ordinates (x ,h) in B. The (1.1) character of the transformation obtained by bringing (x, y) into correspondence with (x, h) is simply the mathematical expression of the physically obvious fact that the separate particles must remain recognizable after the motion, i.e., that separate particles remain separate.
3.3.2 Introduction of New Curvilinear Co-ordinates: Closely connected with the first interpretation (as a mapping), which we can give to a system of equations x = f(x, y), h = y(x, y) is the second interpretation, as a transformation of co-ordinates in the plane. If the functions f and y are not linear, this is no longer an affine transformation, but one to general curvilinear co-ordinates.
Again, we assume that when (x, y) ranges over a region R of the xy-plane and the corresponding point (x, h) over a region B of the xh-plane, and also that for each point of B the corresponding (x, y) in R can be uniquely determined, in other words, that the transformation is (1,1). The inverse transformation will again be denote by x = g(x, h), y = h(x, h).
We can mean by the co-ordinates of a point P in a region R any number-pair, which serves to specify uniquely the position of the point P in R . Rectangular co-ordinates are the simplest case of co-ordinates which extend over the entire plane. Another typical case is the system of polar co-ordinates in the xy-plane, introduced by the equations
When given, as above, a system of functions x = f(x, y), h = y(x, y), we can, in general, assign to each point P(x, y) the corresponding values (x, h) as new co-ordinates. In fact, each pair of values (x, h) belonging to the region B uniquely determines the pair (x, y) and thus uniquely the position of the point P in R; this entitles us to call x, h the co-ordinates of the point P. The co-ordinate lines x = const and h = const are then represented in the xy-plane by two families of curves, which are defined implicitly by the equations f(x, y) = const and y(x, y) = const, respectively. These co-ordinate curves cover the region R with a co-ordinate net (as a rule, curved), for which reason the co-ordinates (x, h) are also called curvilinear co-ordinates in R.
Once again, we point out how closely these two interpretations of our system of equations are interrelated. The curves in the xh-plane, which in the mapping correspond to straight lines parallel to the axes in the xy-plane can be directly regarded as the co-ordinate curves for the curvilinear co-ordinates x = g(x, h), y = h(x, h) in the xh-plane; conversely, the coordinate curves of the curvilinear co-ordinate system x = f(x, y), h = y(x, y) in the xy-plane in the mapping are the images of the straight lines parallel to the axes in the xh-plane. Even in the interpretation of (x, h) as curvilinear co-ordinates in the zy-plane we must consider a xh-plane and a region B of that plane in which the point with the co-ordmatea (x; h) can vary, if we wish to keep the situation clear. The difference depends mainly on the point of view.* If we are chiefly interested in the region R of the xy-plane, we regard x, h simply as a new means of locating points in the region R, the region B of the xh-plane then being merely subsidiary; if we are equally interested in the two regions R and B in the xy-plane and the xh-plane, respectively, it is preferable to regard the system of equations as specifying a correspondence between the two regions, i.e., a mapping of one onto the other. However, it always desirable to keep both the interpretations - mapping and transformation of co-ordinates - in mind.
* However, there is a real difference in that the equations always define a mapping, no matter how many points (x, y) correspond to one point (x, h), while they define a co-ordinate transformation only when the correspondence is (1,1).
If, for example, we introduce polar co-ordinates (r, q) and interpret r and q as rectangular co-ordinates in an rq-plane, the circles r = const and the lines q = const. are mapped onto straight lines parallel to the axes in the rq-plane. If the region R of the xy-plane is the circle x²+y²£1, the point (r, q) of the rq -plane will range over a rectangle 0 £ r £ 1, 0 £ q < 2p, where corresponding points of the sides q = 0 and q = 2p are associated with one and the same point of R and the whole side r = 0 is the image of the origin x = 0, y = 0.
Another example of a curvilinear co-ordinate system is the system of parabolic co-ordinates. We arrive at these by considering the family of confocal parabolas in the xy-plane (3.2.1, Fig.9)
all of which have the origin as focus and the x-axis as axis. There pass through each point of the plane two parabolas of the family, one corresponding to a positive parameter value p = x and the other to a negative parameter value p = h. We obtain these two values by solving for p the quadratic equation which results when we substitute in the above equation the values of x and y, corresponding to the point, whence
These two quantities may be introduced. as curvilinear co-ordinates in the xy-plane, the confocal parabolas then becoming the co-ordinate curves. These are indicated in Fig. 9 above, if we imagine the symbols (x, y) and (x, h) to have been interchanged.
In introducing parabolic co-ordinates (x, h), we must bear in mind that the one pair of values (x,h) corresponds to the two points (x, y) and (x, -y) which are the two intersections of the corresponding parabolas. Hence, in order to obtain a (1,1) correspondence between the pair (x,y) and the pair (x, h), we must restrict ourselves to, say, the half-plane y ³ 0. Then every region R in this half-plane is in a (1,1) correspondence with a region B of the xh-plane and the rectangular co-ordinates (x, h) of each point in this region B are exactly the same as the parabolic co-ordinates of the corresponding point in the region R.
3.3.3 Extension to More than Two Independent Variables: In the case of three or more independent variables, the state of affairs is analogous. Thus, a system of three continuously-differentiable functions
defined in a region R of xyz-space, may be regarded as the mapping of the region R onto a region B of xhz-space. If we assume that this mapping of R onto B is (1,1), so that for each image point (x, h, z) of B the co-ordinates (x, y, z) of the corresponding point (model-point) in R can be uniquely calculated by means of functions
then (x, h, z) may also be regarded as general co-ordinates of the point P in the region R. The surfaces x=const, h=const, z=const or, in other symbols,
then form a system of three families of surfaces which cover the region R and may be called curvilinear co-ordinate surfaces.
Just as in the case of two independent variables, we can
interpret (1,1) transformations in three 
dimensions as
deformations of a substance spread continuously throughout a
region of. space. A very important case of transformation of
co-ordinates is given by polar co-ordinates in space.
These specify the position of a point P in space by three
numbers: (1) the distance from the
origin 
, (2) the geographical
longitudef
, i.e., the angle
between the xz-plane and the plane, determined by
P and the z-axis, and (3) the polar
distance q,
i.e., the angle
between the radius vector OP and the positive z-axis.
As we see from Fig. 10, the three polar co-ordinates r, f, q
are related to the rectangular co-ordinates by the transformation
equations
from which we obtain the inverse relations
For polar co-ordinates in the plane,
the origin is an exceptional point, at which the (1,1)
correspondence fails,
since the angle is there indeterminate. In the same way, for
polar co-ordinates in space, all of the z-axis is an
exception, since the longitude f
is there indeterminate. At the origin itself, the polar distance q is also indeterminate.
The co-ordinate surfaces for three-dimensional polar co-ordinates are: (1) for constant values of r, the concentric spheres about the origin; (2) for constant values of f, the family of half-planes through the a-axis; (3) for constant values of q, the circular cones with the z-axis as axis and the origin as vertex (Fig. 11).
Another frequently employed co-ordinate system is the system of cylindrical co-ordinates. They are obtained by introduction of polar co-ordinates r, f in the xy-plane and retaining z as the third co-ordinate. Then the transformation formulae from rectangular co-ordinates to cylindrical co-ordinates are
and the inverse transformation is
The co-ordinate surfaces r = const are the vertical circular cylinders which intersect, the xy-plane in concentric circles with the origin as centre; the surfaces f = const are the half-planes through the z-axis; the surfaces z = const are the planes parallel to the xy-plane.