3.3.4 Differentiation Formulae for the Inverse Functions: In many cases of practical importance, it is possible to solve, as in the above examples, the given system of equations directly and thus to recognize that the inverse functions are continuous and have continuous derivatives. Hence, for the time being, we will assume the differentiability of the inverse functions. Then, without actually solving the equations explicitly, we can calculate the derivatives of the inverse functions as follows: Substitute the inverse functions x = g(x, h), y = h(x, h) in the given equations x = f(x, y). We obtain then on the right hand side the compound functions f(g(x,h), h(x, h)) and y(g(x,h), h(x, h)) of x and h; however, these functions must be equal to x and h, respectively. We now differentiate each of these equations with respect to x and h, considering them to be independent variable.* If we apply on the right hand side the chain rule for the differentiation of compound functions, we obtain the system of equations

* These equations hold for all values of x and h under consideration, as we say, they hold identically, in contrast to equations between variables which are satisfied only for certain of the values of these variables. Such identical equations or identities, when differentiated with respect to any of the variables occurring in them, again yield identities, as follows immediately from the definition.

Solving these equations, we obtain

i.e., the partial derivatives of the inverse functions x = g(x, h) and y = h(x, h) with respect to x and h, expressed in terms of the derivatives of the original functions f(x, y) and y(x, y) with respect to x and y. For the sake of brevity, we have introduced

This expression. D, which we assume is not zero at the point in question, is called the Jacobian or functional determinant of the functions x = f(x, y) and h = y(x, y) with respect to the variables x and y.

In the above work, as occasionally elsewhere, we have used the shorter notation x (x,y) instead of the more detailed notation x = f(x, y), which distinguishes between the quantity x and its functional expression f(x, y). In the future, we shall often use similar abbreviations when there is no risk of confusion.

For example, in the case of polar co-ordinates in the plane, expressed in terms of rectangular co-ordinates,

the partial derivatives are

Hence the Jacobian has the value

and the partial derivatives of the inverse functions - rectangular co-ordinates expressed in terms of polar co-ordinates - are

which we could have found more easily by direct differentiation of the inverse formulae x = rcosq, y=rsinq.

The Jacobian occurs so frequently that one uses for it the special symbol

The appropriateness of this abbreviation will soon become obvious. The formulae

for the derivatives of the inverse functions yield that the Jacobian of the functions x = x(x, h) and y = y(x, h) with respect to x and h is given by

Thus, the Jacobian of the inverse system of functions is the reciprocal of the Jacoian of the original system.

In the same way, we can also express the second derivatives of the inverse functions in terms of the first and second derivatives of the given functions. We have only to differentiate the linear equations above with respect to x and h by means of the chain rule. (We assume, of course, that the given function possess continuous derivatives of the second order.) We then obtain linear equations from which the required derivatives are readily calculated.

For example, in order to calculate the derivatives

we again differentiate the two equations

once with respect to x and obtain by the chain rule

If we solve this system of linear equations, using the quantities x_xx and y_xx as unknowns (the determinant of the system is again D, and therefore, by assumption, not zero) and then replace x_x and y_x by the values already known, brief manipulations yield

The third and higher derivatives can be obtained in the same manner by repeated differentiation of the linear system of equations; we obtain at each stage a system of linear equations with the non-vanishing) determinant D.

3.3.5 Resolution and Combination of Mappings and Transformations: In Chapter I, we saw that every affine transformation can be analyzed into simple or, as we say, primitive transformations, the first of which deforms the plane in one direction only and the second deforms the already deformed plane again in another direction. In each of these transformations, really only one new variable is introduced.

We can now do exactly the same for transformations in general. We begin with some remarks on the combination of transformations. If the transformation

yields a (1,1) mapping of the point (x, y), which ranges over a region R onto the point (x, y) of the region B in the xh-plane and if the equations

yield a (1,1) mapping of the region B onto a region R' in the w-plane, then there simultaneously occurs a (1,1) mapping of R onto R'. Of course, we call this mapping the resultant mapping or the resultant transformation, and say that it is obtained by combining the two given mappings. The resultant transformation is given by the equations

it follows from the definition at once that this mapping is (1,1). From the rules for differentiating compound functions, we obtain

On comparing these equations with the law for the multiplication of determinants, we find that the Jacobian of u and v with respect to x and y is

The same result can, of course, be obtained by straightforward multiplication.

In words: The Jacobian of the resultant transformation is equal to the product of the Jacobians of the individual transformations.

In symbols:

This equation brings out the appropriateness of the symbol for the Jacobian. When transformations are combined, the Jacobians behave in the same way as the derivatives behave when functions of one variable are combined. The Jacobian of the resultant transformation differs from zero, provided the same is true for the individual (or component) transformations.

In particular, if the second transformation

is the inverse of the first transformation,

and if both transformations are differentiable, the resultant transformation will simply be the identical transformation, i.e., u=x, v=y. The Jacobian of this last transformation is obviously 1, whence we obtain again the earlier obtained relation

Incidentally, it follows from this that neither of the two Jacobians can vanish.

Before we take up the question of the resolution of an arbitrary transformation into primitive transformations, we shall consider the primitive transformation

We assume that the Jacobian D = f_x of this transformation differs from zero throughout the region R, i.e., we assume, say, that f_x > 0 in the region. The transformation deforms the region R into a region B; we may imagine that the effect of the transformation is to move each point in the direction of the x-axis, since the ordinate is unchanged. After deformation, the point (x, y) has a new abscissa which depends on both x and y. The condition f_x > 0 means that, when y is fixed, x varies monotonically with x;. This ensures the (1.) correspondence of the points on a line y = const before and after the

transformation; in fact, two points P(x₁, y) and Q(x₂, y) with the same ordinate y and x₂ > x₁ are transformed into two points P' and Q' which again have the same ordinate and the abscisae of which satisfy the inequality x ₂ > x₁ (Fig.12). This fact also shows that after the transformation the sense of rotation is the same as that in the xy-plane.

If f_x were negative, the two points P and Q would correspond to points with the same ordinate and with abscisae x₁ and x₂, but this time we should havex₁ > x₂, (Fig. 13). The sense of rotation would therefore be reversed, as we have already seen in 1.4.3 for the simple case of affine transformations.

If the primitive transformation

is continuously differentiable and its Jacobian f_x differs from zero at a point P(x₀, y₀), then in a neighbourhood of P the transformation has a unique inverse, and this inverse is also a primitive transformation of the same type. By virtue of the assumption f_x ¹ 0, we can apply the theorem on implicit functions and thus find that, in a neighbourhood of (x₀, y₀), the equation x = f(x, y) determines x uniquely as a continuously differentiable function x = g(x , y) of x and y. (We use here the fact that a function with two continuous derivatives is differentiable.) Whence the two formulae

yield the inverse transformation with the determinant g_x = 1/f_x ¹ 0.

If we now think of the region B in the xh-pane as itself mapped onto a region R in the uv-plane by means of a primitive transformation

where we assume that Y_h , is positive, the state of affairs is just as above, except that the deformation takes place in the other co-ordinate direction. This transformation likewise preserves the sense of rotation (or reverses it, if Y_h < 0 instead of Y_h > 0).

By combining the two primitive transformations, we obtain the transformation

and from the theorem on Jacobians we see that

We now assert that an arbitrary (1.1) continuously differentiable transformation

of the region R in the xy-plane onto a region R' in the uv-plane can be resolved in the neighbourhood of any point interior to R into continuously differentiable primitive transformations, provided that, throughout the entire region R, the Jacobian

differs from zero.

It follows from the non-vanishing of the Jacobian that at no point can we have both f_x = 0 and f_y = 0. We consider a point with the co-ordinates (x₀, y₀) and assume that at that point f_x ¹ 0. Then, by the main theorem, we can mark off intervals x₁£x£x₂, y₁£y£y₂, u₁£u£u₂ about x₀ , y₀ and u₀ = u(x₀ , y₀), respectively, in such a way that within these bounds the equation u=f(x₀ , y₀) can be solved uniquely for x and defines x = g(u, y) as a continuously differentiable function of u and y. If we substitute this expression in v = f(x, y), we obtain v = y(g(u, y) =Y(u,y). Hence, in any neighbourhood of the point (x₀,, y₀), we may regard the given transformation as being composed of the two primitive transformations

Similarly, in a neighbourhood of a point (x₀, y₀) at which f_y ¹ 0, we can resolve the given transformation into two primitive transformations of the form

This pair of transformations is not exactly identical in form with the pairs considered above, each of which leaves one of the co-ordinate directions unaltered. However, it can easily be brought into that form by interchanging the letters u and v (this interchange is itself the resultant of three very simple primitive transformations (1.4.2 footnote ). However, for the purposes of the present chapter, it is more convenient not to carry out this resolution; we write instead the last set of equations in the form

These last equations represent two primitive transformations, each affecting one co-ordinate direction only, and also a rotation of the axes in the w-plane through an angle of 90°. The rotation is so easy to deal with that it need not be split up into primitive transformations.

It is not to be expected that we can resolve a transformation into primitive transformations in one and the same way throughout the whole region. However, since one of the two types of resolution can be carried out for every interior point of R, every closed region interior to R can be subdivided into a finite number of sub-regions (as follows from the covering theorem) in such a way that in each sub-region one of the resolutions is possible.

We can draw from the possibility of this resolution into primitive transformations an interesting conclusion. We have seen that, in the case of a primitive transformation, the sense of rotation is reversed or preserved according to whether the Jacobian is negative or positive. Hence, in the case of general transformations, the sense of rotation is reversed or preserved according to whether the Jacobian is negative or positive. In fact, if the sign of the Jacobian is positive, when the resolution into primitive transformations is carried out, the Jacobians of the primitive transformations will either be both positive or both negative. (The rotation of the u- and v-axes through 90°, required in some cases, has +1 for its Jacobian and leaves the sense of rotation unchanged, and accordingly does not affect the discussion at all.) In the first case, it is obvious that the sense of rotation is preserved; in the second case, it follows from the fact that two reversals of the sense bring us back to the original sense. However, if the Jacobian is negative, one, and only one, of the primitive transformations will have a negative Jacobian and will therefore reverse the sense, while the other will not affect it.

3.3.6 General Theorem on the Inversion of Transformations and Systems of Implicit Functions: The possibility of inverting a transformation depends on the general theorem:

If in the neighbourhood of point (x₀, y₀) the functions f(x, y) and y (x, y) are continuously differentiable (i.e. continuous and possess continuous derivatives) and u₀=f(x₀, y₀), v₀=y(x₀,y₀), and if, in addition, the Jacobian D = f_xy_y - f_yy_x, is not zero at (x₀, y₀), then, in a neighbourhood of the point (x₀, y₀), the system of equations u = f(x, y), v = y(x, y) has a unique inverse, i.e., there is a uniquely determined pair of functions x = g(u, v), y = h(u, v) such that x₀=g(u₀,v₀) and y₀=h(u₀, v₀) and also the equations

hold in some neighbourhood of the point (u₀, v₀).

In the neighbourhood of (u₀, v₀), the so-called inverse functions x = g(u, v), y = h(u, v) possess the continuous derivatives

The proof follows from the discussion in 3.3.5. In fact, in a sufficiently small neighbourhood of the point (x₀, y₀), we can resolve the transformation u = f(x, y), v = y(x, y) into continuously differentiable primitive transformations, possibly with an additional rotation by 90° of the u- and v-axes. Each of these transformations has a unique inverse, which is itself a continuously differentiable transformation. The combination of these inverse transformations yields at once the transformation, which is the inverse of the given one. As a combination of continuously differentiable transformations, this is itself continuously differentiable. It then follows from 3.3.4 that the differentiation formulae hold as stated.

This inversion theorem is a special case of a more general theorem which may be regarded as an extension of the theorem of implicit functions to systems of functions. The theorem of implicit functions applies to the solution of one equation for one of the variables. The general theorem is:

If f(x, y, u, v, ··· , w) and y(x, y, u, v, ··· , w) are continuously differentiable functions of x, y, u, v, ··· , w and the equations

are satisfied by a certain set of values x₀, y₀, u₀, v₀, ··· , w₀, and if in addition the Jacobian of f and y with respect to x and y differs from zero at that point (i.e., D =f_xy_y — f_yy_x ¹ 0), then, in the neighbourhood of that point, the equations f = 0 and y = 0 can be solved in one, and only one, way for x and y, and this solution gives x and y as continuously differentiable functions of u, v, ··· , w.

The proof of this theorem is similar to that of the inversion theorem above. From the assumption D ¹ 0, we can conclude without loss of generality that at the point in question f_x ¹ 0. Then, by the theorem of implicit functions, if we restrict x, y, u, v, ··· , w to sufficiently small intervals about x₀, y₀, u₀, v₀, ··· , w₀, respectively, the equation f(x, y, u, v, ··· , w) = 0 can be solved for x in exactly one way as a function of the other variables, and this solution x = g(y, u, v, ··· , w) is a continuously differentiable function of its arguments and has the partial derivative g_y= -f_y/f_x. If we substitute this function x=g(y,u,v,···,w) in x = y(y, u, v, ··· , w), we obtain a function y(y, u, v, ··· , w) = Y(y, u, v, ··· , w) and

Hence, by virtue of the assumption that D ¹ 0, we see that the derivative Y_y is not zero. Thus, if we restrict y, u, v, ··· , w to intervals about y₀, u₀, v₀, ··· , w₀ (which we take to be smaller than the intervals to which they were previously restricted), we can solve the equation Y = 0 in exactly one way for y as a function of u, v, ··· , w, and this solution is continuously differentiable. Substitution of this expression for y in the equation x = g(y, u, v, ··· , w) now yields x as a function of u,v,···,w, and this solution is continuously differentiable and unique, subject to the restriction of x, y, u, v, ··· , w to sufficiently small intervals about x₀, y₀ , u₀, ··· , w₀, respectively.

3.3.7 Non-independent Functions: It is worth mentionmg that, if the Jacobian D vanishes at a point (x₀, y₀), no general statement can be made about the possibility of solving the equations in the neighbourhood of that point. However, even if the

inverse functions do happen to exist, they cannot be differentiable, for then the product would vanish, while we know it must be equal to 1. For example, the equations

can be solved uniquely, the solution being

although the Jacobian vanishes at the origin; but the function is not differentiable at the origin.

On the other hand, the equations

cannot be solved uniquely in the neighbourhood of the origin, since both the points (x,y) and (-x,-y) of the xy-plane correspond to the same point of the uv-plane.

However, if the Jacobian vanishes identically, that is, not merely at the single point (x, y), but at every point in an entire neighbourhood of the point (x, y), then the transformation is of the type called degenerate. In this case, we say that the functions u = f(x, y) and v == y(x, y) are dependent.

We first consider the special, almost trivial, case in which the equations f_x = 0 and f_y = 0 hold everywhere, so that the function f(x, y) is a constant. We then see that, as the point (x,y) ranges over an entire region, its image (u, v) always remains on the line u = const, i.e., our region is mapped only onto a line, instead of onto a region, so that there is no possibility here of speaking of a (1.1) mapping of two two-dimensional regions onto one another. A similar situation arises in the general case in which at least one of the derivatives f_x or f_y does not vanish, but the Jacobian D is still zero. We suppose that at a point (x₀, y₀) of the region under consideration we have f_x ¹ 0. It is then possible to resolve our transformation into two primitive transformations x=f(x,y), h= y and u = x, v = y(x, h), because we only used earlier the assumption f_x ¹ 0. However, by virtue of the equation D = f_xy_h = 0, y_h must be identically zero in the region where f_x ¹ 0, i.e., the quantity y = v does not depend on h at all, and v is only a function of x = u. Our result is therefore:

If the Jacobian of a transformation vanishes identically, a region of the xy-plane is mapped by the transformation onto a curve in the uv-plane instead of onto a two-dimensional region, since in a certain interval of values of u only one value of v corresponds to each value of u. Thus, if the Jacobian vanishes identically, the functions are not independent, i.e. there exists a relation

which is satisfied for all systems of values (x, y) in the above-mentioned region. In fact, if F(u, v) = 0 is the equation of the curve in the uv-plane onto which the region of the xy-plane is mapped, then for all points of this region the equation

is satisfied, i.e., this equation is an identity in x and y.

The exceptional case discussed separately at the beginning is obviously included in this general statement. The curve in question is then just the curve u = const, which is a parallel to the v-axis.

An example of a degenerate transformation is

According to this transformation, all the points of the xy-plane are mapped onto the points of the parabola h = x² in the xh-plane. An inversion of the transformation is out of the question, for all the points of the line x + y = const are mapped onto a single point (x, h). As is easily verified, the value of the Jacobian is zero. The relation between the functions x and h, in accordance with the general theorem, is given by the equation F(x, h) = x² - h = 0.

3.3.8 Concluding Remarks: The generalization of the theory for three or more independent variables offers no particular difficulties. The main difference is that instead of the two-rowed determinant D we have determinants with three or more rows. In the case of transformations with three independent variables,

the Jacobian is given by

In the same way, the Jacobian for the ransformations

with n independent variables is

For more than two independent variables, it is still true that, when transformations are combined, the Jacobians are multiplied together. In symbols,

In particular, the Jacobian of the inverse transformation is the reciprocal of the Jacobian of the original transformation.

The theorems on the resolution and combination of transformations, on the inversion of a transformation and on the dependence of transformations remain valid for three and more independent variables. The proofs are similar to those for the case n = 2; in order to avoid unnecessary repetition, we shall omit them here.

In the preceding section, we have seen that the behaviour of a general transformation in many ways resembles that of an affine transformation and that the Jacobian has the same role as the determinant does in the case of affine transformations. The following remark makes this even clearer. Since the functions x = f(x, y) and h = y(x, y) are differentiable in the neighbourhood of (x₀, y₀), we can express them in the form

where e and d tend to zero with This shows that for sufficiently small values of |x - x₀| and |y - y₀| the transformation may be regarded, to a first approximation, as affine, since it can be approximated by the affine transformation

the determinant of which is the Jacobian of the original transformation.

Exercises 3.3

1. If f(x) is a continuously differentiable function, then the transformation

has a single inverse in every region of the xy-plane in which f '(x) ¹ 0. The inverse transformation has the form

2. A transformation is said to be conformal, if the angle between any two curves is preserved.

(a) Prove that the inversion

is a conformal transformation.
(b) Prove that the inverse of any circle is another circle or a straight line.
(c) Find the Jacobian of the inversion.

3. Prove that in a curvilinear triangle, formed by three circles passing through one point O, the sum of the angles is p.

4. A transformation of the plane

is conformal if the functions f and y satisfy the identities

5. The equation

determines two values of t, depending on x and y:

(a) Prove that the curves t₁ = const and f₂ = const are ellipses and hyperbolas with the same foci (confocal conics).
(b) Prove that the curves t₁ = const and f₂ = const are orthogonal.
(c) t₁ and t₂ may be used as curvilinear co-ordinates (so-called focal co-ordinates). Express x and y in terms of these co-ordinates.
(d) Express the Jacobian in terms of x and y.
(e) Find the condition that two curves, which are represented parametrically in the system of focal co-ordinates by the equations

are mutually orthogonal.

6. (a) Prove that the equation in t

has three distinct real roots t₁, t₂, t₃, which lie, respectively, in the intervals

provided that the point (x, y, z) does not lie on a co-ordinate plane.
(b) Prove that the three surfaces t₁ = const, t₂ = const, t₃ = const passing through an arbitrary point are orthogonal to each other.
(c) Express x, y, z in terms of the focal co-ordinates t₁, t₂, t₃.

7. Prove that the transformation of the xy-plane given by the equations

(a) is conformal;
(b) Transforms straight lines through the origin and circles with the origin as centre in the xy-plane into confocal conics t=const, given by

8. Inversion in three dimensions is defined by the formulae

Prove that

(a) the angle between any two surfaces is unchanged;
(b) spheres are transformed either into spheres or into planes.

9. Prove that if all the normals of a surface z = u(x, y) meet the z-axis, then the surface is a surface of revolution.

Hints and Answers

3. 4 APPLICATIONS

3.4.1 Applications to the Theory of Surfaces: In. the study of surfaces, as in that of curves, parametric representation is frequently to be preferred to other types of representation. Here we need two parameters instead of one; we will denote them by u and v. A parametric representation may be expressed in the form

where f , y and c are given functions of the parameters u and v and the point (u, v) ranges over a given region R in the uv-plane. The corresponding point with the three rectangular co-ordinates (x,y,z) then ranges over a configuration in the xyz-space. In general, this configuration is a surface which can be represented, say, in the form z = f(x, y). In fact, we can seek to solve two of our three equations for u and v in terms of the two corresponding rectangular co-ordinates. If we substitute the expressions thus found for u and v in the third equation, we obtain an unsymmetrical representation of the surface, say, z = f(x, y).(This is actually a special case of the parametric form, as we see by setting x = u and y = v.) Hence, in order to ensure that the equations really do represent a surface, we have only to assume that not all the three Jacobians

vanish simultaneously, which can be expressed in the single formula

Then, in some neighbourhood of each point in space, represented by our three equations, it is certainly possible to express one of the three co-ordinates uniquely in terms of the other two.

A simple example of parametric representation is the representation of the spherical surface x²+y²+z²=r² of radius r by the equations

where v = q is the polar distance and u = f the geographical longitude of the point on the sphere (3.3.3).

This example shows one of the advantages of the parametric representation. The three co-ordinates are given explicitly as functions of u and v, and these functions are single-valued. If v runs from p/2 to p, we obtain the lower hemisphere, i.e., while values of v from 0 to p/2 give the upper hemisphere. Thus, with the parametric representation, it is unnecessary, as it is with the representation , to consider two single-valued branches of the function in order to obtain the entire sphere.

We obtain another parametric representation of the sphere by means of the stereographic projection. In order to project the sphere x² + y² + z² - r² = 0 stereographically from the north pole N (0, 0, r) onto the equatorial plane z = 0, we join each point of the surface to the north pole N by a straight line and call the intersection of this line with the equatorial plane the stereographic image of the corresponding point of the sphere (Fig. 14). We thus obtain a (1.1) correspondence between the points of the sphere and the points of the plane, except for the north pole N. Using elementary geometry, we readily find that this correspondence is expressed by

where (u, v) are the rectangular co-ordinates of the image-point in the plane. These equations may be regarded as a parametric representation of the sphere, the parameters u and v being rectangular co-ordinates in the uv-plane.

As a further example, we give parametric representations of the surfaces

which are called the hyperboloid of one sheet and the hyperboloid of two sheets, respectively (Figs.15/16). The hyperboloid of one sheet is represented by

the hyperboloid of two sheets by

In general, we may regard the parametric representation of a surface as the mapping of the region R of uv-plane onto the corresponding surface; where, as always, the term mapping is understood to mean a point-to-point correspondence. There corresponds to each point of the region R of the uv-plane one point of the surface and, in general, the converse is also true.

Of course, this is not always the case. For example, in the representation of the sphere by polar co-ordinates , the poles of the sphere correspond to all of the line-segments v = 0 and v = p, respectively.

In the same way, a curve u = u(t), v = v(t) in the uv-plane corresponds,. by virtue of the equations x = f(u(t), v(t)) = x(t), ··· , corresponds to a curve on the surface (2.7.1). In particular, in the representation of the sphere by means of polar co-ordinates, the meridians are represented by the equation u == const, the parallels of latitude by « ~ const. Thus, this net of curves corresponds to the system of parallels to the axes in the uv -plane.

The representation of a curve on a given surface is one of the most important methods for a thorough investigation of the properties of the surface. Here, we shall give only the expression for the length of arc s of such a curve. As has been mentioned in 2.7.2, we have

so that, by virtue of the equations

we obtain

where, for the sake of compactness, we have introduced the Gaussian fundamental entities of the surface,

These entities are independent of the particular choice of the curve on the surface and depend only on the surface itself and on its parametric representation. The above expressions for the derivative of the length of arc with respect to the parameter are, as a rule, expressed symbolically by omitting the reference to the parameter t and saying that the line element ds on the surface is given by the quadratic differential form

We have already obtained in 3.2.3 for the direction cosines of the normal to a surface, given in the form F(x, y, z) = 0, the expressions

In order to obtain these direction cosines in the case of parametric representation, we assume that the surface, given by the equations x = f(u, v), y = y(u, v), z = c(u, v), is written in the form F(x, y, z) = 0. The equation

is then an identity in u and v, and we obtain by differentiation

Hence, it follows at once from 1.3.2 that

where r is a suitably chosen multiplier. From the definitions of E, F, G, we find by direct expansion that

and combining this with the preceding equation, we have

Thus, finally, we obtain the formulae for the direction cosines of the normal to the surface in the form

The equations u = g(t), v = h(t), as we have seen, represent a curve on the surface. The direction cosines of the tangent to this curve are given, according to the chain rule, by

For the sake of brevity, we have set here dg(t)/dt = u', dh(t)/dt = v'. If we now consider a second curve on the surface, given by the equations u = g₁(t), v = h₁(t), the tangent of which has the direction cosines cos a₁, cos b₁, cosg₁, and use the abbreviations

then the cosine of the angle between the two curves is given by the cosine of the angle between their tangents,i.e., by

where all the quantities on the right hand side are to be given the values which they have at the point of intersection of the two curves. In particular, we may consider those curves on the surface which are given by the equations u = const or v = const. If we substitute in our parametric representation a definite fixed value for u, we obtain a three-dimensional or twisted curve lying on the surface and having v as parameter; a corresponding statement holds good if we substitute a fixed value for v and allow u to vary. These curves u = const and v = const are the parametric curves on the surface. The net of parametric curves corresponds to the net of parallels to the axes in the uv-plane (Fig. 17)..

The mapping of one plane region onto another plane may be regarded as a special case of parametric representation. In fact, if the third of our functions c(u, v) vanishes for all values of u and v under consideration, then, as the point (u, v) ranges over its given region, the point (x, y, z) will range over a region in the xy-plane. Hence, our equations merely represent the mapping of a region of the uv-plane onto a region of the xy-plane; or, if we prefer to think in terms of transformations of co-ordinates, the equations define a system of curvilinear co-ordinates in the uv-region and the inverse functions (if they exist) define a curvilinear uv-system of co-ordinates in the plane xy-region. In terms of the curvilinear co-ordinates (u, v), the line element in the xy-plane is simply

where

As another example of the representation of a surface in parametric form, consider the anchor ring or torus. This is obtained by rotating a circle about a line which lies in the plane of the circle and does not intersect it (Fig.18). If we take this axis of rotation as the x-axis and choose the y-axis in such a way that it passes through the centre of the circle, the y-co-ordinate of which we denote by a, and if the radius of the circle is T < |a|, we obtain, in the first instance,

as a parametric representation of the circle in the yz-plane. Letting the circle rotate about the z-axis, we find that for each point of the circle x² + y² remains constant, i.e., x²+y²=(a+rcos q)². Thus, if the angle of rotation about the z-axis is denoted by j, we have

as a parametric representation of the anchor ring in terms of the parameters q and j. In this representation, it appears as the image of a square of side 2p in the qj-plane, where any pair of boundary points lying on the same line q=const or j=const corresponds to only one point on the surface and all the four corners of the square correspond to the same point.

For the line element on the anchor ring, we have

3.4.2 Conformal Representation in General:. A transformation

is called a conformal transformation if any two curves are transformed by it into two other curves which have the same angle between them as the original curves.

Tleorem: A necessary and sufficient condition for our (continuously differentiable) transformation to be conformal is that the Cauchy-Riemann equations

hold. In the first case, the direction of the angles is preserved, in the second case, their direction is reversed. (The last statement follows directly from an earlier statement concerning the sign of the Jacobian f_xf_y-f_yf_z.)

Proof: We assume that the transformation is conformal. Then the two orthogonal curves x = const, h = const in the xh-plane must correspond to orthogonal curves f(x, y) = const and y(x, y) = const in the xy-plane.

Hence, it follows immediately from the formula for the angle between two curves that

In the same way, the curves corresponding to x + h = const and x - h = const must be orthogonal. This yields

whence

The first of our equations can be written in the form

where l denotes a constant of proportionality. Introducing this in the second equation, we immediately get l² = 1, so that one or the other of our two systems of Cauchy-Riemann equations holds.

That the equations are a sufficient condition is confirmed by the observation:

If two curves in the xy-plane are given by equations F(x, y) = 0, G(x, y) = 0 and if, according to our transformation, F(x,y)=F(x,h), G(x, y)=G(x,h), then the Cauchy-Riemann equations readily yield

whence

Hence, the curves F = 0, G = 0 and their images F = 0, G = 0 form the same angle between them.

Exercises 3.4

1. Prove that
(a) the stereographic projection of the unit sphere onto the plane is conformal,
(b) then the circles on the sphere are transformed either into circles or into straight lines in the plane,
(c) in the stereographic projection reflection of the spherical surface in the equatorial plane corresponds to an inversion in the uv-plane.
(d) Find the expression for the line element on the sphere in terms of the parameters u, v.

2. Calculate the line element
(a) on the sphere

(b) on the hyperboloid

(c) on a surface of revolution given by

using the cylindrical co-ordinates z and q - artan y/x as co-ordinates on the surface;

(d)* on the quadric t₃ = const of the family of confocal quadrics given by

using t₁ and t₂ as co-ordinates on the quadric (cf. Example 6 in 3.3).

3. Prove that if a new system of curvilinear co-ordinates r, s is introduced on a surface with parameters u, v by means of the equations

then

where E', F', G' denote the fundamental quantities taken with respect to r, s and E, F, G those taken with respect to u, v.

4. Let t be a tangent to a surface S at the point P and consider the section of S made by all planes containing t. Prove that the oentres of curvature of the different sections lie on a circle.

5. If t is a tangent to the surface S at the point P, we call the curvature of the normal plane section through t (i.e., the section through t and the normal) at that point the curvature (k) of S in the direction t. Take for every tangent at P the vector with the direction of t, initial point P, and length Prove that the final points of these vectors lie on a conic.

6*. A curve is given as the intersetion of the two surfaces

Find the equations of
(a) the tangent,
(b) the osculating plane, at any point of the curve.

Hints and Answers

3.5 FAMILIES OF CURVES, FAMILIES OF SURFACES AND THEIR ENVELOPES

3.5.1 General Remarks: On various occasions, we have already considered curves or surfaces not as individual configurations, but as members of a family of curves or surfaces, such as f(x,y) = c, where there corresponds to each value of c a different curve of the family.

For example, the lines parallel to the y-axis m the xy-plane, that is, the lines x = c, form a family of curves. The same is true for the family of concentric circles x² + y² = c² about the origin; there corresponds to each value of c a circle of the family, namely the circle with radius c. Similarly, the rectangular hyperbolas xy = c form a family of curves. The particular value c=0 corresponds to the degenerate hyperbola consisting of the two co-ordinate axes. Another example of a family of curves is the set of all the normals to a given curve. If the curve is given in terms of the parameter t by the equations x=j (t), h=y(t), we obtain the equation of the family of normals in the form

where t is used instead of c to denote the parameter of the family.

The general concept of a family of curves can be expressed analytically as follows:

Let f(x, y, c) be a continuously differentiable function of the two independent variables x and y and of the parameter c, this parameter varying in a given interval. (Thus, the parameter is really a third independent variable, which is lettered differently simply because it has a different role.) Then, if the equation

for each value of the parameter c represents a curve, the aggregate of the curves obtained as c describes its interval is called a family of curves depending on the parameter c. The curves of such a family may also be represented in parametric form by means of a parameter t of the curve in the form

where c is again the parameter of the family. If we assign to c a fixed value, these equations represent a curve with the parameter t.

For example, the equations

represent the family of concentric circles mentioned above, the equations

the family of rectangular hyperbolas mentioned above, except for the degenerate hyperbola consisting of the co-ordinate axes.

Occasionally, we are led to consider families of curves which depend not on one but on several parameters. For example, the aggregate of all circles (x— a)² + (y - b)² = c² in the plane is a family of curves depending on the three parameters a, b, c. If nothing is said to the contrary, we shall always understand a family of curves to be a one-parameter family, depending on a single parameter. The other cases we shall distinguish by speaking of two-parameter, three-parameter, or multi-parameter families of curves.

Naturally, similar statements hold for families of surfaces in space. If we are given a continuously differentiable function f(a,y,z,c) and for each value of the parameter c in a certain definite interval the equation

represents a surface in the space with rectangular co-ordinates x, y, z, then the aggregate of the surfaces obtained by letting c describe its interval is called a family of surfaces, or, more precisely, a one-parameter family of surfaces with the parameter c. For example, the spheres x²+y²+z²=c² about the origin form such a family. As in the case of curves, we can also consider families of surfaces depending on several parameters.

Thus, the planes defined by the equation

form a two-parameter family, depending on the parameters a and b, if the parameters a and b range over the region a² b £1. This family of surfaces consists of the class of all planes which are at unit distance from the origin. (Sometimes a one-parameter family of surfaces is referred to as ¥ ² surfaces, a two-parameter family as ¥ ² surfaces, etc.

3.5.2 Envelopes of One-Parameter families of Curves: If a family of straight lines is identical with the aggregate of the tangents to a plane curve E - as, for example, the family of normals of a curve C is identical with the family of tangents to the evolute E of C - we shall say that the curve E is the envelope of the family of lines. In the same way, we shall say that the family of circles with radius 1 and centre on the x-axis, that is, the family of circles with the equation (x - c)² + y² -1= 0, has the pair of lines y = 1 and y = -1, which touch each of the circles, as its envelope (Fig, 19). In these cases, we can obtain

the point of contact of the envelope and the curve of the family by finding the intersection of two curves of the family with parameter values c and c + h and then letting h tend to zero. We may express this briefly by saying that the envelope is the locus of the intersections of neighbouring curves.

In cases of other families of curves, it may again happen that there exists a curve E which at each of its points touches a certain curve of the family, which depends, of course, on the point of E in question. We then call E the envelope of the family of curves. There arises now the question of finding the envelope E of a given family of curves f(x, y, c) = 0. We first make a few plausible observations, in which we assume that an envelope E does exist and that it can be obtained, as in the above cases, as the locus of the intersections of neighbouring curves. (Since this last assumption will be shown by examples to be too restrictive, we shall shortly replace these plausibilities by a more complete discussion). We then obtain the point of contact of the curve f(x,y, c) = 0 with the curve E in the following way. In addition to this curve, we consider a neighbouring curve f(x, y, c + h) = 0, find the intersection of these two curves and then let h tend to zero. The point of intersection must then approach the point of contact sought. At the point of intersection, the equation

is true as well as the equations f(x, y, c) = 0 and f(x, y, c) = 0. In the first equation, we perform the passage to the limit h®0. Since we have assumed the existence of the partial derivative f_c, this yield the two equations

for the point of contact of the curve f(x, y, c) = 0 with the envelope. If we can determine x and y as functions of c by means of these equations, we obtain the parametric representation of a curve with the parameter c, and this curve is the envelope. By elimination of the parameter c, it can also be represented in the form g(x, y) = 0. This equation is called the discriminant of the family, the curve given by this equation is called the discriminant curve.

We are thus led to the rule:

In order to obtain the envelope of a family of curves f(x, y, c) = 0, we consider the two equations f(x,y,c)=0 and f_c(x,y,c)=0 simultaneously and attempt to express x and y as functions of c by means of them or to eliminate the quantity between them.

We shall now replace the heuristic considerations above by a more complete and more general discussion, based on the definition of the envelope as the curve of contact. At the same time, we shall learn under what conditions our rule actually does yield the envelope and what other possibilities present themselves.

To begin with, we assume that E is an envelope which can be represented in terms of the parameter c by two continuously differentiable functions

which at the point with parameter c touches the curve of the family with the same value of the parameter c. In the first place, the equation f(x, y, c) = 0 is satisfied at the point of contact. If we substitute in this equation the expressions x(c) and y(c) for x and y, it remains valid for all values of c in the interval. On differentiating with respect to c, we at once obtain

The condition of tangency is now

in fact, the quantities dx/dc and dy/dc are proportional to the direction cosines of the tangent to E and the quantities f_x and f_y are proportional to the direction cosines of the normal to the curve f(x,y,c) = 0 of the family, and these directions must be at right angles to one another. It follows that the envelope satisfies the equation f_x = 0, whence we see that the role given above is a necessary condition for the envelope.

In order to find out in how far this condition is also sufficient, we assume that a curve E represented by two continuously differentiable functions x = x(c) and y = y(c) satisfies the two equations f(x, y, c) = 0 and f_c(x, y, c) = 0. We substitute again in the first equation x(c) and y(c) for x and y; this equation becomes then an identity in c. If we differentiate with respect to c and remember that f_c = 0, we obtain at once the relation

which therefore holds for all points of E. If both the expressions

differ from zero at a point of E, so that at that point both the curve E and the curve of the family have well-defined tangents, this equation states that the envelope and the curve of the family touch each another. With these additional assumptions, our rule is a sufficient as well as a necessary condition for the envelope. However, if both f_x and f_y vanish, the curve of the family may have a singular point and we can draw no conclusions about the contact of the curves.

Thus, after we have found the discriminant curve, we must in each case still make a further 'investigation, in order to discover whether it is really an envelope or to what extent it fails to be one.

In conclusion, we state the condition for the discriminant curve of a family of curves given in parametric form

with the curve parameter t. This is

We can readily obtain it, for example, by changing from the parametric representation of the family to the original expression by elimination of t.

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