2.4.5. Application to the Calculus of Errors: The practical advantage of having the differential df = hf_x + kf_y as a convenient approximation to the increment of the function f(x, y), Du = f(x + h, y + k) - f(x, y), as we pass from (x, y) to (x+h, y + k), is exhibited particularly well in the so-called Calculus of Errors. For example, let it be required to find the possible error in the determination of the density of a solid body by the method of displacement. If m is the weight of the body in air and its weight in water, by Archimedes' principle, the loss of weight (m - ) is the weight of the water displaced. If we are using the c.g.s. system of units, the weight of the water displaced is numerically equal to its volume, and hence to the volume of the solid. The density a is thus given in terms of the independent variables m and by the formula s = m/(m-). The error in the measurement of the density s, caused by an error dm in the measurement of m and an error d in the measurement of is given approximately by the total differential

By the quotient rule, the partial derivatives are

hence the differential is

Thus, the error in s is largest if, say, dm is negative and d is positive, i.e., if we measure instead of m too small an amount m - dm and instead of too large an amount + d. For example, if a piece of brass weighs about 100 gm in air, with a possible error of 5 mg, and in water about 88 gm, with a possible error of 8 mg, the density is given by our formula to within an error of about

or about one per cent.

2.5. FUNCTIONS OF FUNCTIONS (COMPOUND FUNCTIONS) AND THE INTRODUCTION OF NEW INDEPENDENT VARIABLES

2.5.1 General Remarks. The Chain Rule: It often happens that the function u of the independent variables x, y is stated in the form of a compound function

where the arguments x , h, ··· of the function f are themselves functions of x and y:

We then say that

is given as a compound function of x and y.

For example, the function

may be written as a compound function by means of

Similarly, the function

can be expressed in the form

In order to make this concept more precise, we adopt, to start with, the assumption: The functions x = f (x,y), h = y (x,y), ··· are defined in a certain region R of the independent variables x, y. As the argument point (x, y) varies within this region, the point with the co-ordinates (x, h, ···) always lies in a certain region S of the x,h, ···-space, in which the function u=f(x, h, ···) is defined. The compound function

is then defined in the region R.

In many cases, detailed examination of the regions R and S will be quite unnecessary, e.g., in the first example given above, in which the argument point (x, y) can traverse the whole of the xy-plane and the function u = e^xsin h is defined throughout the xh-plane. On the other hand, the second example shows the need for considering the regions R and S in the definition of compound functions. In fact, the functions

are defined only in the region R of the points 0 < x² + y² £ 1, i.e., in the region consisting of the circle with unit radius and centre at the origin, the centre being removed. Within this region, |x| < 1, while h can have all negative values and the value 0. For the region S of points (x,h), defined by these relations, the function x arsin x is defined.

A continuous function of continuous functions is itself continuous. More precisely: If the function u=f(x, h, ···) is continuous in the region S, and the functions x = f (x,y), h = y (x,y), ··· are continuous in the region R, then the compound function u = F(x, y) is continuous in R.

The proof follows immediately from the definition of continuity. Let (x₀, y₀) be a point of R and x₀, h₀, ··· be the corresponding values of x, h, ···. Then, for any positive e, the difference

is numerically less than e, provided only that all the inequalities

are satisfied, where d is a sufficiently small positive number. However, by the continuity of f (x, y), h(x, y), ··· these last inequalities are all satisfied, if

where g is a sufficiently small positive quantity. This establishes the continuity of the compound function.

Moreover, we shall prove that a differentiable function of differentiable functions is itself differentiable. This statement is formulated more precisely in the following theorem which at the same time gives the rule for the differentiation of compound functions, or the so-called chain rule:

If x = f(x,y), h = y(x, y), ··· are differentiable functions of x and y in the region R and f(x, h, ···) is a differentiable function of x, h, ··· in the region S, then the compound function

is also a differentiable function of x and y, and its partial derivatives are given by

or, briefly, by

Thus, in order to form the partial derivative with respect to x, we must first differentiate the compound function with respect to all the functions x, h, ···, which depend on x, multiply each of these derivatives by the derivative of the corresponding function with respect to x, and then add all the products thus formed. This is the generalization of the chain rule for functions of one variable discussed in the Chain Rule.

Our statement can be written in a particularly simple and suggestive form if we use the notation of differentials, namely

This equation means that the linear part of the increment of the compound function u = f(x, h, ···) = F(x, y) can be found by first writing down this linear part as if x, h, ··· were the independent variables and subsequently replacing dx, dh, . . . by the linear parts of the increments of the functions x = f(x, y), h = y(x, y), ···. This fact demonstrates the convenience and flexibility of the differential notation.

In order to prove our statement, we must merely use the assumption that the functions concerned are differentiable, whence, if we denote the increments of the independent variables x and y by Dx and Dy, the quantities x, h, ··· change by

where the numbers e₁, e₂, ··· ,g₁, g₂, ··· tend to 0 with Dx and Dy, or as . Moreover, if the quantities x, h, ··· undergo changes Dx, Dh, ···, the function u = f(x, h, ···).is subject to an increment of the form

where the quantities d₁, d₂, ··· tend to 0 as Dx, Dh, ···,, or as (and may be taken as exactly equal to zero when the corresponding increments Dx, Dh vanish).

If we take in the last expression the increments Dx, Dh, ··· as those due to a change Dx in the value of x and a change of Dy in the value of y, as given above, we obtain

Here the quantities e and g have the values

On the right hand side, we have a sum of products, each of which contains at least one of the quantities e₁, e₂, ··· ,g₁, g₂, ···,d₁, d₂, ··· . Hence, we see that e and g also tend to 0 with Dx and Dy. Hhowever, by the results of the preceding section, this expresses the statement asserted in our theorem.

It is obvious that this result is quite independent of the number of independent variables x, y, ··· and remains valid, for example, if the quantities x, h, ··· depend on only one independent variable x, so that the quantity u is a compound function of the single independent variable x.

If we wish to calculate the higher partial derivatives, we need only differentiate the right-hand sides of our equations with respect to x and y, treating f_x , f_h , ··· as compound functions. Confining ourselves, for the sake of simplicity, to the case of three functions x, h and z , we obtain

2.5.2. Examples: Consider the function

Note that we the following differentiations can be carried out directly without use of the chain rule.

Set

and obtain

Hence

and

2. In the case of the function

set x = x² + y² and obtain

3. In the case of the function

the substitution

leads to

2.5.3 Change of the Independent Variables: A particularly important application of the methods developed above occurs in the process of changing independent variables. For example, let u = f(x , h) be a function of the two independent variables x, h, which we interpret as rectangular co-ordinates in the xh-plane. If we introduce new rectangular co-ordinates x, y in that plane (1.1.2) by the transformation

the function u = f(x , h) is transformed into a new function of x and y

and this new function is formed from f(x , h) by a process of compounding as was described in 2.5. We then say that new independent variables x and y have been introduced into f(x , h) between the independent variables x and h and the dependent variable u.

The rules of differentiation above yield at once

where the symbols u_x, u_y denote the partial derivatives of the function F(x, y) and the symbols u_x, u_h the partial derivatives of the function f(x, h).

Thus, the partial derivatives of any function are transformed according to the same law as the independent variables when the co-ordinate axes are rotated. This is also true for rotation of the axes in space.

Another important type of change of independent variables is the change from rectangular co-ordinates (x, y) to polar co-ordinates (r, q) which are connected with the rectangular co-ordinates by the equations

On introducing polar co-ordinates, we find that

and the quantity u appears as a compound function of the independent variables r and q. Hence, by the chain rule,

These formulae yield the equation

which is frequently of use. By the chain rule, the higher derivatives are given by

This leads to the following formula, which gives the expression appearing in the well-known Laplace or potential equation Du = 0 in terms of polar co-ordinates

The first of the formulae

expresss the rule for the differentiation of a function f(x, y) in terms of r and q in the direction of the radius vector r previously met in 2.4.2.

In general, whenever we are given a series of relations defining a compound function,

we may regard it as an introduction of new independent variables x, y in place of x , h, ···. Corresponding sets of values of the independent variables assign the same value to u, whether it is regarded as a function of x , h, ··· or of x, y.

In all cases involving the differentiation of compound functions

the following aspect must be carefully noted. We must distinguish clearly between the dependent variable u and the function f(x , h, ···) which links u to the independent variables x , h, ···. The symbols of differentiation u_x , u_h , ··· have no meaning until the functional connection between u and the independent variables is specified. Hence, when dealing with compound functions u=f(x,h, ···)=F{x, y), we really should not write u_x , u_y or u_x , u_h , but should instead write f_x , f_h , or F_x, F_y, respectively. Yet, for the sake of brevity, the simpler symbols u_x , u_h , u_x , u_y are often used, provided there is no risk that confusion will arise.

The following example will serve to show that the result of differentiation of a quantity depends on the nature of the functional connection between it and the independent variables, that is, it depends on which of the independent variables are kept fixed during the differentiation. In the case of the identical transformation x = x, h = y, the function u = 2x + h becomes u = 2x + y, and we have u_x = 2, u_y= 1. However, if we introduce the new independent variables x = x (as before) and x + h = v, we find that u = x+ v, so that u_x = 1, u_y = 1, i.e., differentiation with respect to the same independent variable x gives different results in the two different cases.

Exercises 2.3

1. Prove that the tangent plane to the quadric

at the point (x₀, y₀, z₀) is

2. If u = u(x, y) is the equation of a cone, then

3. Prove that if a function f(x) is continuous and has a continuous derivative, then the derivative of the function

vanishes for a certain value between x₁ and x₂.

4. Let f(x, y, z) be a function depending only on i.e., let f(x, y, z) = g(r).

(a) Calculate f_xx + f_yy + f_zz.

(b) Prove that if f_xx + f_yy + f_zz = 0, then f = a/r + b (where a and b are constants).

5 If , calculate

(cf. Example 2 above).

6.* Find the expression f_xx + f_yy + f_zz in three-dimensional polar co-ordinates, i.e., transform to the variables r, q, r, defined by

Compare the result with Example 4(a).

7. Prove that the expression

does not change with rotation of the co-ordinate system.

8. Prove that with the linear transformation

f_xx(x,y), f_xy (x,y), f_yy(x,y), respectively are transformed by the same law as the coefficienta a, b, c of the polynomial

Hints and Answers

2.6 THE MEAN VALUE THEOREM AND TAYLOR'S THEOREM FOR FUNCTIONS OF SEVERAL VARIABLES

2.6.1 Statement of the Problem. Preliminary Remarks: We have seen in Volume 1 how a function of a single variable can be approximated within the neighbourhood of a given point with an accuracy of order higher than the n-th by means of a polynomial of degree n - the Taylor series, provided that the function possesses derivatives up to the (n + l)-th order. The approximation by means of the linear part of the function, as given by the differential, is only the first step towards this closer approximation. In the case of functions of several variables, for example, of two independent variables, we may also seek an approximate representation in the neighbourhood of a given point by means of a polynomial of degree n. In other words, we wish to approximate to f(x + h, y + k) by means of a Taylor expansion in terms of the differences h and k.

This problem can be reduced by a very simple device to what we already know from the theory of functions of one variable. Instead of considering the function f(x + h, y + k), we introduce yet another variable t and study the expression

as a function of t, keeping for the moment x, y, h and k fixed. As t varies between 0 and 1, the point with the co-ordinates (x+ht, y+kt) traverses the line-segment joining (x, y) and (x+h,y+k).

We begin by calculating the derivatives of F(t). If we assume hat all the derivatives of the function f(x, y), which we are about to write down, are continuous in a region entirely containing the line-segment, the chain rule (2.5.1) at once yields

*

And, in general, we find by mathematical induction that the n-th Derivative is given by

which, as in 2.4.5, can be written symbolically in the form

In this formula, the bracket on the right hand side is to be expanded by he binomial theorem and then the powers and products of the quantities are to be replaced by the corresponding n-th derivatives In all these derivatives, the arguments x+ht and y+kt are to be written in place of x and y.

2.6.2 The Mean Value Theorem: In forming the polynomial of approximation, we start from a mean value theorem analogous to that which we already know for functions of one variable. This theorem yields a relationship between the difference f(x + h, y + k) —f(x, y) and the partial derivatives f_x and f_y. We expressly assume that these derivatives are continuous. On applying the ordinary mean value theorem to the function F(t), we obtain

where q is a a number between 0 and 1, and it follows from this that

If we put t = 1, we obtain the required mean value theorem for functions of two variables in the form

Thus, the difference between the values of the function at the points (x + h, y + k) and (x, y) is equal to the differential at an intermediate point (x, h) on the line-segment joining the two points. Note that the same value of q occurs in both f_x and f_y.

The following fact, the proof of which we leave to the reader, is a simple consequence of the mean value theorem. A function f(x, y) the partial derivatives f_x and f_y of which exist and have the value 0 at every point of a region is a constant.

2.6.3 Taylor's Theorem for Several Independent Variables: If we apply Taylor's formula with Lagrange's form of the remainder (Estimation of the Remainder) to the function F(t) and then set t = 1, we obtain Taylor's theorem for functions of two independent variables

where R_n is the remainder term

The homogeneous polynomials of degree 1, 2, ··· , n, n + 1 into which the increment f(x + h, y + k) - f(x, y) is thus split, apart from the factors

are respectively the first, second, ··· , n-th differentials

of f(x, y) at the point (x, y) and the (n + l)-th differential dⁿ⁺¹f at an intermediate point on the line-segment joining (x, y) and (x + h, y + k). Hence Taylor's theorem can be written more compactly

where

In general, the remainder R_n vanishes to a higher order than the term dⁿf preceding it, that is, as h ® 0 and k ® 0, we have

In the case of Taylor's theorem for functions of one variable, the passage (n ® ¥) to infinite Taylor series had an important role, leading us to the expansions of many functions in power series. For functions of several variables, in general, such a process is too complicated. To an even greater degree than in the case of functions of one variable, we emphasize here rather the fact that by means of Taylor's theorem the increment f(x + h, y+k) —f(x, y) of a function is split into increments df, d²f, ··· of different orders.

Exercises 2.4

1. Find the polynomial of the second degree which best approximates the function sin x in the neighbourhood of the origin.

2. If f(x. y) is a continuous function with continuous first and second derivatives, then

3. Prove that the function can be expanded in a series of the form

which converges for all values of x and y and that

(a) H_n(x) is a polynomial of degree n (so-called Hermite polynomials). (b) H'_n(x) = 2nH_n-1(x).
(c) H_n+1 - 2xH_n + 2nH_n-1 = 0. (d) H"_n - 2xH'_n + 2nH_n = 0.

4. Find the Taylor series for the following functions and indicate their ranges of validity:

Hints and Answers

2.7 THE APPLICATION OF VECTOR METHODS

Many facts and relationships in the differential and integral calculus of several independent variables take a decidedly clearer and simpler form, if we apply the ideas and notation of vector analysis. We shall therefore conclude this chapter with a discussion of these matters.

2.7.1 Vector Fields and Families of Vectors: The step which links vector analysis with the subjects just discussed is as follows. Instead of considering a single vector or a finite number of vectors, as in 1.2, we will investigate a vector manifold, depending on one or more continuously varying parameters. For example, if we consider a solid body occupying a portion of space and in a state of motion, then at a given instant each point of the solid will have a definite velocity, represented by a vector u. We say that these vectors form a vector field in the region in question. The three components of the field vector then appear as three functions

of the three co-ordinates of position, which we here denote by (x₁, x₂, x₃) instead of by (x, y, z).

A velocity field is represented in Fig. 8, which shows the velocity field of a solid body rotating about an axis with constant angular velocity.

The forces acting on the points of a moving solid body are also a vector field. As an example of a force field, consider the attractive force per unit mass exerted by a heavy particle, according to Newton's law of gravitation. By this law, the vectors of the force field are directed towards the attracting particle and their lengths are inversely proportional to the square of the distance from the particle.

If we pass by rotation of axes to a new rectangular co-ordinate system, all the vectors of the field will have new components with respect to the new system of axes. If the two co-ordinate systems are connected by equations of the form (1.1.2)

respectively, then. the relations between the components u₁, u₂, u₃ with respect to the x-system and the components w₁(x₁,x₂,x₃), w₂(x₁,x₂,x₃), w₃(x₁,x₂,x₃) with respect to the new x-system are given by equations of transformation

respectively (1.1.2). Thus, the components w, w₂ , w₃ in the new system arise from the introduction of tlie new variables and the simultaneous transformation of the functions, representing the components in the old system.

When in physical applications each point of a portion of space has assigned to it a definite value of a function u = f(x₁,x₂,x₃), such as the density at the point, and we wish to emphasize that the property is not a component of a vector, but, on the contrary, a property which retains the same value although the co-ordinate system is changed, we say that the function is a scalar function or a scalar, or, if we wish to emphasize the association between the values of the function and the points of the portion of space, we speak of a scalar field. Thus, for every vector field u, the quantity |u|² = |u₁|² + |u₂|² + |u₃|² is a scalar; in fact, it represents the square of the length of the vector and therefore retains its value independently of the co-ordinate system to which the components of the vector are referred.

In. the examples above, at the start, the vector field u is given and its components with respect to any system of rectangular co-ordinates are therefore determined. If, conversely, in a definite co-ordinate system, say an x-system, there are given three functions u₁(x₁,x₂,x₃), u₂(x₁,x₂,x₃), u₃(x₁,x₂,x₃), these three functions define a vector field with respect to that system, the components of the field being given by the three functions. In order to obtain the expressions for the components w ₁,w₂ , w₃ in any other system, we need only apply the equations of transformation derived above.

Besides vector fields, we also consider manifolds of vectors called vector families, which do not correspond to each point of a region in space, but are functions of a parameter t. We express this by writing u = u(t). If we think of u as a position vector measured from the origin of co-ordinates in u₁u₂u₃-space, then, as t varies, the final point of this vector describes a curve in space given by the three parametric equations

Vectors which depend in this way on a parameter t can be differentiated with respect to t. By the derivative of a vector u(t), we mean the vector u'(t) which is obtained by the passage to the limit

and which accordingly has the components

We see at once that the fundamental rules of differentiation apply to vectors. Firstly, it is obvious that if

Moreover, the product rule applied to the scalar product of two vectors

yields

In the same way, we obtain the rule

for the vector product.

2.7.2 Application to the Theory of Curves in Space. Resolution of a Motion into Tangential and Normal Components: We shall now make some simple applications of these ideas. If x(t) is a position vector in x₁x₂x₃-space which depends on a parameter t and therefore defines a curve in space, the vector x'(t) will be in the direction of the tangent to the curve at the point corresponding to t. In fact, the vector x(t+k)-x(t) is in the direction of the line-segment joining the points (t) and (t + k) (Fig. 9), whence so is the vector [x(t+h) - x(t)]/h which differs from it only by the factor 1/h. As h ® 0, the direction of this chord approaches the direction of the tangent. If we introduce as parameter instead of t the length of the arc of the curve measured from a definite starting-point and denote differentiation with . respect to s by means of a dot (·), we can prove that

this may also be written m the form

The proof follows exactly the same lines as the corresponding proof for plane curves (5.2.5), whence the vector has unit length. If we differentiate both sides of the equation with respect to s, we obtain

This equation states that the vector with components is perpendicular to the tangent. We call this vector the curvature vector or the principal normal vector, its absolute value, i.e., its length is

the curvature of the curve at the corresponding point. As before, the reciprocal r = l/k of the curvature is called the radius of curvature. The point obtained by measuring from the point on the curve a length r in the direction of the principal normal vector is called the centre of curvature.

We shall show that this definition of the curvature agrees with that given in 5.2.6 of Volume 1. In fact, is a vector of unit length. If we think of the vectors as measured from a fixed origin, then the difference will be represented, as in Fig. 9 above, by the vector joining the final points of the vectors . If a is the angle between the vectors , the length of the vector joining their final points is 2 sin a/2, since are both of unit length. Hence, if we divide the length of this vector by a and let h ® 0, the quotient tends to the limit 1. Consequently,

The limit on the right hand side is exactly But a/h is the ratio of the angle between the directions of the tangents at two points of the curve and the length of arc between those points, and the limit of that ratio is what we have previously defined as the curvature of the curve.

The curvature vector has an important role in Mechanics. Let a particle of unit mass move along a curve x(t), where t is the time. The velocity of the motion is then given both in magnitude and in direction by the vector x'(t), where the dash denotes differentiation with respect to t. Similarly, the acceleration is given by the vector x"(t). By the chain rule, we have

(where the dot denotes differentiation with respect to s), and also

In view of what we already know about the lengths of and , this equation expresses the following facts:

The acceleration vector of the motion is the sum of two vectors. One of these is directed along the tangent to the curve and its length is equal to d²s/dt², i.e., it is equal to the acceleration of the point on its path (tangential acceleration). The other is directed towards the centre of curvature and its length is equal to the square of the velocity multiplied by the curvature (normal acceleration).

2.7.3 The Gradient of a Scalar: We now return to the consideration of vector fields and present a brief discussion of certain concepts which frequently arise in this connection.

Let u = f(x₁, x₂, x₃) be any function defined in a region of x₁x₂x₃-space, i.e., in the terminology adopted previously, it is a scalar. We may now regard the three partial derivatives

in the x-system as forming the three components of a vector u. If we now pass on to a new system of rectangular co-ordinates, the x-system, by rotation of axes, the new components of the vector u are given, according to the formulae of 1.1.2 , by the equations

On the other hand, if we introduce the rectangular co-ordinates (x₁, x₂, x₃) as new independent variables into the function f(x₁, x₂, x₃), the chain rule yields

Hence,

and thus we see that in the new co-ordinate system also the components of the vector u are given by the partial derivatives of the function f with respect to the three co-ordinates. Thus, there corresponds to every function in three-dimensional space a definite vector, the components of which in any rectangular co-ordinate system are given by the three partial derivatives with respect to the co-ordinates. We call this vector the gradient of the function, and denote it by

For a function of three variables, the gradient is an analogue to the derivative for functions of one variable. In order to form a graphical idea of its meaning, we form the derivative of the function in the direction (a₁, a₂, a₃), where a₁, a₂, a₃ are the three angles which this direction makes with the axes, so that

We have already obtained for this derivative the expression

If we think of a vector e of unit length in the direction (a₁, a₂, a₃), this vector will have the components e₁=cosa₁, e₂=cosa₂,e₃=cos a ₃. Thus, we obtain for the derivative of the function in the direction (a₁, a₂, a₃)

the scalar product of the gradient and the unit vector in the direction (a₁, a₂, a₃), i.e., the projection of the gradient onto that vector (1.1.2).

It is this fact that accounts for the importance of the concept of gradient. For example, if we wish to find the direction in which the value of a function increases or decreases most rapidly, we must choose the direction in which the above expression has the largest or smallest value. This clearly occurs when the direction of e is the same as that of the gradient or is exactly opposite to it.

Thus, the direction of the gradient is the direction in which a function increases most rapidly, while the direction opposite to that of the gradient is that in which a function decreases most rapidly; the magnitude of the gradient gives the rate of increase or decrease.

We shall return to the geometrical interpretation of the gradient.(p. 124). However, we can immediately give an intuitive idea of the direction of the gradient. In the first instance, if we confine ourselves to vectors in two dimensions, we must consider the gradient of a function f(x, y). We shall assume that this function is represented by its contour lines (or level lines)

in the xy-plane. Then the derivative of the function f(x, y) in the direction of these level lines is obviously zero. In fact, if P and Q are two points on the same level line, the equation f(P) —f(Q) = 0 holds (the meaning of the symbols is obvious), and this equation will still hold if we divide both sides by h, the distance between P and Q, and then let h tend to 0. The projection of the gradient in the direction of the tangent to the level line is therefore zero, whence at every point the gradient is perpendicular to the level line through that point. An exactly analogous statement holds for the gradient in three dimensions. If we represent the function f(x₁, x₂, x₃) by its level surfaces

the gradient has the component zero in every direction tangent to a level surface and is therefore perpendicular to the level surface.

In applications, we frequently encounter vector fields which represent the gradient of a scalar function. The gravitational field of force may be taken as an example.

If we denote the co-ordinates of the attracting particle by (x₁, x₂, x₃), those of the attracted particle by (x₁, x₂, x₃) and their masses by m and M, the components of the force of attraction are given by

Here, C is a constant with the value gmM, where g is the gravitational constant. (The factors

are the cosines of the angles which the line through the two points makes with the axes.) By differentiation, we see at once that these components are the derivatives of the function

with respect to the co-ordinates x₁, x₂, x₃, respectively. The force vector, apart from a constant factor, is therefore the gradient of the function

If a field of force is obtained from a scalar function by forming the gradient, this scalar function is often called the potential function of the field. We shall consider this concept from a more general point of view in the study of work and energy (5.2.4, 6.1.2,, 6.6).

2.7.4. The Divergence and Curl of a Vector field:.By differentiation, we have assigned to every function or scalar a vector field, the gradient. Similarly, also by differentiation, we can assign to every vector field a certain scalar, known as the divergence of the vector field. Given a specific co-ordinate system, the x-system, we define the divergence of the vector u as the function

i.e., the sum of the partial derivatives of the three components with respect to the corresponding co-ordinates. Suppose now that we change the co-ordinate system to the x-system. If the divergence is really to be a scalar function, associated with the vector field and independent of the particular co-ordinate system, we must have

where w₁, w₂, w₃ are the components of u in the x-system. In fact, the truth of the equation

can be verified immediately by applying the chain rule and the transformation formulae of 2.7.1.

Here, we will be content with the formal definition of the divergence; its physico-geometrical interpretation will be discussed in Chapter V 388).

We shall adopt the same procedure for the so-called curl (sometimes called rotation with the abbreviation rot) of a vector field. The curl is itself a vector

with the components r₁, r₂, r₃ defined by the equations

In order to show that our definition actually gives a vector independent of the particular co-ordinate system, we could verify by direct differentiation that the quantities

which define the curl in terms of the new co-ordinates are connected with the quantities r₁, r₂, r₃ by the equations of transformation for vector components. However, we shall omit here these computations, since in 5.6.2 we shall give a physical interpretation of the curl which clearly demonstrates its vectorial character.

All the three concepts of gradient, divergence and curl can be related to each other, if we use a symbolic vector with the components This symbolic vector, also called nabla, is denoted by the symbol . The gradient of a scalar field f(x₁, x₂, x₃), grad f, is the product of the scalar quantity f and the symbolic vector , i.e., it is a vector with the components

The curl of a vector field u(x₁, x₂, x₃), curl u, is the vector product of the vector u and the symbolic vector ; finally, the divergence is the scalar product

In conclusion, we mention a few relations which constantly recur. The curl of a gradient is zero'; in symbols

As we readily see, this relation follows from the reversibility of the order of differentiation.

The divergence of a curl is zero; in symbols,

This also follows directly from the reversibility of the order of differentiation.

The divergence of a gradient is an extremely important expression frequently encountered in analysis, notably in the well-known Laplace or potential equation. It is the sum of the three principal second-order partial derivatives of a function; in symbols

where Df is written as an abbreviation for the expression on the right hand side and the notation ²f is also used.The symbol

is called the Laplace operator.

Finally, we may mention that the terminology of vector analysis is often used in connection with more than three independent variables; thus a system of n functions of n independent variables is sometimes called a vector field in n-dimensional space. The concepts of scalar multiplication and of the gradient then retain their meanings, but in other respects the state of affairs is more complicated than in the case of three dimensions.

Exercises 2.5

1. Find the equation of the so-called osculating plane of a curve x = f(t), y = g(t), z = h(f) at the point t₀, i.e., the limit of the planes passing through three points of the curve as these points approach the point with parameter t.

2. Show that the curvature vector and the tangent vector both lie in the osculating plane.

3.* Let x = x(s) be an arbitrary curve in space, such that the vector x(s) is three times continuously differentiable (s is the length of arc). Find the centre of the sphere of closest contact with the curve at the point s.

4. If C is a continuously differentiable closed curve and A a point not on C, there is a point B on C which has a shorter distance from A than any other point on C. Prove that the line AB is normal to the curve.

5. If x = x(s) is a curve on a sphere of unit radius, the equation

holds.

6. If x = x(t) is any parametric representation of a curve, then the vector d²x/dt² with initial point x lies in the osculating plane at x.

7. The limit of the ratio of the angle between the osculating planes at two neighbouring points of a curve and the length of arc between these two points, i.e., the derivative of the unit normal vector with respect to the arc (s), is called the torsion of the curve. Let x₁(s), x₂(s) denote the unit vectors along the tangent and the curvature vector of the curve x(s); we mean by x₃(s) the unit vector orthogonal to x₁ and x₂ (the so-called binormal vector), which is given by [x₁x₂]. Prove Frenet's formulae

where 1/r = k is the curvature and 1/t the torsion of x(s).

8. Using the vectors x₁, x₂ and x₃ of Exercise 7 as co-ordinate vectors, find expressions for (a) the vector , (b) the vector from the point x to the centre of the sphere of closest contact at x.

9. Show that a curve of zero torsion is a plane curve.

10*. Prove that, if z = u(x, y) represents the surface formed by the tangents to an arbitrary curve, then (a) every osculating plane of the curve is a tangent plane to the surface; (b) u(x, y) satisfies the equation

11. Prove that

Hints and Answer

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