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#### Elements of Partial Differential Equations

**By Ian N. Sneddon**

**Dover Publications, Inc.**

**Copyright © 2006 Dover Publications, Inc.**

All rights reserved.

ISBN: 978-0-486-16299-7

All rights reserved.

ISBN: 978-0-486-16299-7

CHAPTER 1

*ORDINARY DIFFERENTIAL EQUATIONS IN MORE THAN TWO VARIABLES*

In this chapter we shall discuss the properties of ordinary differential equations in more than two variables. Parts of the theory of these equations play important roles in the theory of partial differential equations, and it is essential that they should be understood thoroughly before the study of partial differential equations is begun. Collected in the first section are the basic concepts from solid geometry which are met with most frequently in the study of differential equations.

**1. Surfaces and Curves in Three Dimensions**

By considering special examples it is readily seen that if the rectangular Cartesian coordinates (*x,y,z*) of a point in three-dimensional space are connected by a single relation of the type

f(x, y, z) = 0 *(1)*

the point lies on a surface. For that reason we call the relation (1) the equation of a surface *S.*

To demonstrate this generally we suppose a point (*x,y,z*) satisfying equation (1). Then any increments ([partial derivative]*x*, [partial derivative]*y*, [partial derivative]*z*) in (*x,y,z*) are related by the equation

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

so that two of them can be chosen arbitrarily. In other words, in the neighborhood of P(*x,y,z*) there are points *P'* (*x* + [xi], *y* + η, *z* + ζ satisfying (1) and for which any two of [xi], η, [xi] are chosen arbitrarily and the third is given by

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

The projection of the initial direction *PP'* on the plane *xOy* may therefore be chosen arbitrarily. In other words, equation (1) is, in general, a relation satisfied by points which lie on a surface.

If we have a set of relations of the form

x = F1(u,v), y = F2(u,v), z = F3(u,v) *(2)*

then to each pair of values of *u, v* there corresponds a set of numbers (*x,y,z*) and hence a point in space. Not every point in space corresponds to a pair of values of *u* and *v,* however. If we solve the first pair of equations

x = F1(u,v), y = F2(u,v)

we may express *u* and *v* as functions of *x* and *y,* say

u = λ(x,y), v = μ(x,y)

so that *u* and *v* are determined once *x* and *y* are known. The corresponding value of z is obtained by substituting these values for *u* and *v* into the third of the equations (2). In other words, the value of *z* is determined once those of *x* and *y* are known. Symbolically

z = F3{λ(x,y), μ(x,y)}

so that there is a functional relation of the type (1) between the three coordinates *x, y,* and *z.* Now equation (1) expresses the fact that the point (*x,y,z*) lies on a surface. The equations (2) therefore express the fact that any point *(x,y,z)* determined from them always lies on a fixed surface. For that reason equations of this type are called *parametric equations* of the surface.

It should be observed that parametric equations of a surface are *not* unique; i.e., the same surface (1) can be reached from different forms of the functions *F*1, *F*2, *F*3 of the set (2). As an illustration of this fact we see that the set of parametric equations

x = a sin u cos v, y = a sin u sin v, z = a cos u

and the set

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

both yield the spherical surface

x2 + y2 + z2 = a2

A surface may be envisaged as being generated by a curve. A point whose coordinates satisfy equation (1) and which lies in the plane *z = k* has its coordinates satisfying the equations

z = k, f(x,y,k) = 0 *(3)*

which expresses the fact that the point (*x,y,z*) lies on a curve, Γ*k* say, in the plane *z = k* (cf. **Fig. 1**). For example, if S is the sphere with equation *x2 + y2 + z2 = a2*, then points of *S* with *z = k* have

z = k, x2 + y2 = a2 - k2

showing that, in this instance, Γ*k* is a circle of radius (*a*2 – *k*2)[??] which is real if *k*< *a.* As *k* varies from *-a* to +*a,* each point of the sphere is covered by one such circle. We may therefore think of the surface of the sphere as being "generated" by such circles. In the general case we can similarly think of the surface (1) as being generated by the curves (3).

We can look at this in another way. The curve symbolized by the pair of equations (3) can be thought of as the intersection of the surface (1) with the plane *z = k.* This idea can readily be generalized. If a point whose coordinates are (*x,y,z*) lies on a surface *S*1, then there must be a relation of the form *f* (*x,y,z*) = 0 between these coordinates. If, in addition, the point (*x,y,z*) lies on a surface *S*2, its coordinates will satisfy a relation of the same type, say *g* (*x,y,z*) = 0. Points common to *S*1 and *S*2 will therefore satisfy a pair of equations

f(x,y,z) = 0, g(x,y,z) = 0 *(4)*

Now the two surfaces *S*1 and *S*2 will, in general, intersect in a curve *C,* so that, in general, the locus of a point whose coordinates satisfy a pair of relations of the type (4) is a curve in space (cf. **Fig. 2**).

A curve may be specified by parametric equations just as a surface may. Any three equations of the form

x = f1(t), y = f2(t), z = f3(t) *(5)*

in which *t* is a continuous variable, may be regarded as the parametric equations of a curve. For if *P* is any point whose coordinates are determined by the equations (5), we see that *P* lies on a curve whose equations are

Φ1(x,y) = 0, Φ2(x,z) = 0

where Φ1(*x,y*) = 0 is the equation obtained by eliminating *t* from the equations *x* =*f*1 (*t*), y = *f*2(*t*) and where Φ2(*x,z*) = 0 is the one obtained by eliminating *t* between the pair *x* = *f*1 (*t*), *z = f*3(*t*). A usual parameter *t* to take is the length of the curve measured from some fixed point. In this case we replace *t* by the symbol *s.*

If we assume that *P* is any point on the curve

x = x(s), y = y(s), z = z(s) *(6)*

which is characterized by the value s of the are length, then *s* is the distance *P*0*P* of *P* from some fixed point *P*0*measured along the curve* (cf. **Fig. 3**). Similarly if *Q* is a point at a distance *δs* along the curve from *P,* the distance *P*0*Q* will be *s* + δ*s*, and the coordinates of *Q* will be, as a consequence,

{x(s + δs), y(s + δs), z(s + δs)}

The distance δ*s* is the distance from *P* to *Q* measured along the curve and is therefore greater than δ*c*, the length of the chord *PQ.* However, in many cases, as *Q* approaches the point *P,* the difference δ*s* - δ*c* becomes relatively less. We shall therefore confine our attention to curves for which

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] *(7)*

On the other hand, the direction cosines of the chord *PQ* are

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

and by Maclaurin's theorem

x(s + δs) - x(s) = δs (dx/ds) + O(δs2)

so that the direction cosines reduce to

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

As δ*s* tends to zero, the point Q tends towards the point *P,* and the chord *PQ* takes up the direction to the tangent to the curve at *P.* If we let δ*s* -> 0 in the above expressions and make use of the limit (7), we see that the direction cosines of the tangent to the curve (6) at the point *P* are

(dx/ds, dy/ds, dz/ds) *(8)*

In the derivation of this result it has been assumed that the curve (6) is completely arbitrary. Now we shall assume that the curve *C* given by the equations (6) lies on the surface S whose equation is *F*(*x,y,z*) = 0 (cf. **Fig. 4**). The typical point {*x*(*s*),*y* (*s*),*z* (*s*)} of the curve lies on this surface if

F[x(s),y(s),z(s)] = 0 *(9)*

and if the curve lies entirely on the surface, equation (9) will be an identity for all values of *s.* Differentiating equation (9) with respect to *s,* we obtain the relation

[partial derivative]F/[partial derivative]x dx/ds + [partial derivative]F/[partial derivative]y dy/ds + [partial derivative]F/[partial derivative]z dz/ds = 0 *(10)*

Now by the formulas (8) and (10) we see that the tangent *T* to the curve *C* at the point *P* is perpendicular to the line whose direction ratios are

([partial derivative]F/[partial derivative]x, [partial derivative]F/[partial derivative]y, [partial derivative]F/[partial derivative]z) *(11)*

The curve *C* is arbitrary except that it passes through the point *P* and lies on the surface *S.* It follows that the line with direction ratios (11) is perpendicular to the tangent to every curve lying on S and passing through *P.* Hence the direction (11) is the direction of the *normal* to the surface *S* at the point *P.*

If the equation of the surface *S* is of the form

z = f(x,y)

and if we write

[partial derivative]z/[partial derivative]x = p, [partial derivative]z/[partial derivative]y = q *(12)*

then since *F = f(x,y) - z,* it follows that *F*x = *p, Fy* = *q, Fz* = - 1 and the direction cosines of the normal to the surface at the point (*x,y,z*) are

([p, q, -1]/√p2 + q2 + 1) *(13)*

The expressions (8) give the direction cosines of the tangent to a curve whose equations are of the form (6). Similar expressions may be derived for the case of a curve whose equations are given in the form (4).

The equation of the tangent plane π1 at the point *P* (*x,y,z*) to the surface *S*1 (cf. **Fig. 5**) whose equation is *F*(*x,y,z*) = 0 is

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] *(14)*

where (*X, Y, Z*) are the coordinates of any other point of the tangent plane. Similarly the equation of the tangent plane π2 at *P* to the surface *S*2 whose equation is *G*(*x,y,z*) = 0 is

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] *(15)*

The intersection *L* of the planes π1 and π2 is the tangent at *P* to the curve C which is the intersection of the surfaces *S*1 and *S*2. It follows from equations (14) and (15) that the equations of the line *L* are

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] *(16)*

In other words, the direction ratios of the line *L* are

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (16)

**Example 1.***The direction cosines of the tangent at the point (x,y,z) to the conic ax2 + by2 + cz2 = 1, x + y + z = 1 are proportional to (by -cz, cz - ax, ax - by).*

In this instance

F = ax2 + by2 + cz2 - 1

and

G = x + y + z- 1

so that

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

etc., and the result follows from the expressions (16).

**PROBLEMS**

1. Show that the condition that the surfaces *F*(*x,y,z*) = 0, *G*(*x,y,z*) = 0 should touch is that the eliminant of *x, y,* and z from these equations and the equations *Fx: Gx = Fy: Gy = Fz: Gz* should hold.

Hence find the condition that the plane *lx + my + nz + p* = 0 should touch the central conicoid *ax*2 + *by*2 + *cz*2 = 1.

2. Show that the condition that the curve *u*(*x,y,z*) = 0, *v*(*x,y,z*) = 0 should touch the surface *w*(*x,y,z*) = 0 is that the eliminant of *x, y,* and z from these equations and the further relation

[partial derivative](u,v,w)/[partial derivative](x,y,z) = 0

should be valid.

Using this criterion, determine the condition for the line

[x – a]/l = [y – b]/m = [z – c]/n

to touch the quadric *ax*2 + β*y*2 + γ*z*2 = 1.

**2. Simultaneous Differential Equations of the First Order and the First Degree in Three Variables**

Systems of simultaneous differential equations of the first order and first degree of the type

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] *(1)*

arise frequently in mathematical physics. The problem is to find *n* functions *xi,* which depend on *t* and the initial conditions (i.e., the values of *x*1, *x*2, ..., *xn* when *t* = 0) and which satisfy the set of equations (1) identically in *t.*

For example, a differential equation of the *n*th order

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] *(2)*

may be written in the form

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

showing that it is a special case of the system (1).

Equations of the kind (1) arise, for instance, in the general theory of radioactive transformations due to Rutherford and Soddy.

A third example of the occurrence of systems of differential equations of the kind (1) arises in analytical mechanics. In Hamiltonian form the equations of motion of a dynamical system of *n* degrees of freedom assume the forms

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] *(3)*

where *H*(*q*1,*q*2, ..., *qn, p*1, *p*2, ..., *pn, t*) is the Hamiltonian function of the system. It is obvious that these Hamiltonian equations of motion form a set of the type (1) for the 2*n* unknown functions *q*1, *q*2, ..., *qn, p*1*p*2, ..., *pn,* the solution of which provides a description of the properties of the dynamical system at any time *t.*

In particular, if the dynamical system possesses only one degree of freedom, i.e., if its configuration at any time is uniquely specified by a single coordinate *q* (such as a particle constrained to move on a wire), then the equations of motion reduce to the simple form

dp/dt = -[partial derivative]H/[partial derivative]q, dq/dt = [partial derivative]H/[partial derivative]p *(4)*

where *H*(*p,q,t*) is the Hamiltonian of the system. If we write

-[partial derivative]H/[partial derivative]q = P(p,q,t)/R(p,q,t), [partial derivative]H/[partial derivative]p = Q(pq,t)/R(p,q,t)

then we may put the equations (4) in the form

dp/P(p,q,t) = dq/Q(p,q,t) = dt/R(p,q,t) *(5)*

For instance, for the simple harmonic oscillator of mass *m* and stiffness constant *k* the Hamiltonian is

H = p2/2m + kq2/2

so that the equations of motion are

dp/-kmq = dq/p = dt/m

*(Continues...)*

Excerpted fromElements of Partial Differential EquationsbyIan N. Sneddon. Copyright © 2006 Dover Publications, Inc.. Excerpted by permission of Dover Publications, Inc..

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