14 Chapters
Medium 9789380298818

3

Parmananda Gupta Laxmi Publications PDF

18

DIFFERENTIAL GEOMETRY AND CALCULUS OF VARIATIONS

.

.

.

.

.

.

(Fx) x + (Fy) y + (Fz) z = 0

(Gx) x + (Gy) y + (Gz) z = 0

and

.

.

.

Solving these equations for x, y and z , we get

.

.

.

x y z

=

=

Fy G z − Fz G y Fz G x − Fx G z Fx G y − Fy G x

FyGz – FzGy , FzGx – FxGz , FxGy – FyGx are also d.r.’s of the tangent to the curve given by the equations F(x, y, z) = 0, G(x, y, z) = 0 at the point t.

Example 7. Show that the equation of the tangent to the curve of intersection of the ellipsoid

x2 a2

+

y2

+

b2

z2 c2

x( X − x )

2

x2

= 1 and the confocal

2

2

a (b − c )(a − λ )

y2 b2 − λ

y(Y − y)

=

2

a2 − λ

+

2

2

2

2

b (c − a ) (b − λ )

+

=

z2 c2 − λ

= 1 is

z(Z − z)

2

2

c (a − b 2 ) (c 2 − λ )

,

where (x, y, z) is an arbitrary point on the curve.

Sol. The given surfaces are x2 a2

x2

and

+

y2

y2

z2

+

b2

–1=0

c2

...(1)

z2

–1=0

...(2) a 2 − λ b2 − λ c 2 − λ

Let the equation of the curve of intersection of given surfaces be r = r(t), where t is an arbitrary parameter.

Differentiating the equations (1) and (2) w.r.t. t, we get

+

2x

2

+

.

x+

2y

2

.

y+

2z

2

.

z=0

a b c

2x .

2y .

2z . y+ 2 z =0 x+ 2 a2 − λ b −λ c −λ

See All Chapters
Medium 9789380298818

9

Parmananda Gupta Laxmi Publications PDF

3

Surfaces in Space

1. INTRODUCTION

A sphere, a portion of a cylinder are examples of surfaces in space. Till now we have been discussing the differential geometry of curves in space. In the present chapter , we shall learn few quantities regarding surfaces in space.

2. SURFACE IN SPACE

We know that a curve in space is the locus of a point whose coordinates x, y, z are functions of a single parameter. On the same lines, we shall define a surface in space as the locus of a point whose coordinates are functions of two independent parameters.

A surface in space is defined as the locus of a point whose position vector relative to a fixed origin may be expressed as a function of two independent parameters.

P(x, y, z)

Thus, a surface S in space may be represented by a

S vector function r(u, v) = x(u, v)i + y(u, v)j + z(u, v)k, r(u, v) where u and v are independent parameters such that

(u, v) ∈ R, where R is some region in the uv-plane.

O(origin)

Here r(u, v) is the position vector of the point P on the v surface S and (x(u, v), y(u, v), z(u, v)) are the cartesian coordinates of the point P. The above representation of a surface is called a parametric representation of a surface and is due to Gauss. Also, (u, v) are called the curvilinear coordinates of the point P. In particular cases,

See All Chapters
Medium 9789380298818

10

Parmananda Gupta Laxmi Publications PDF

102

DIFFERENTIAL GEOMETRY AND CALCULUS OF VARIATIONS

F(u) = u3 + lu2 + mu + n

Let

(F(u)) = 3u2 + 2lu + m

∂u

The equation of the envelope of the given family is obtained by eliminating u from the equations

(F(u)) = 0.

F(u) = 0 and

∂u

∴ u3 + lu2 + mu + n = 0

...(2)

2

3u + 2lu + m = 0

...(3)

Multiplying (2) by 3 and (3) by u, we get

3u3 + 3lu2 + 3mu + 3n = 0

...(4)

3u3 + 2lu2 + mu = 0

...(5)

2

(4) – (5) ⇒ lu + 2mu + 3n = 0

...(6)

Solving (3) and (6), we get

u2

6ln − 2m

2

=

u2 =

Eliminating u, we get

F ml − 9n I

GH 6m − 2l JK

2

2

=

u

1

= ml − 9n

6m − 2l 2

6ln − 2m 2

6m − 2l

2

=

3ln − m 2

3m − l

2

and

u=

ml − 9n

6m − 2l 2

3ln − m 2

3m − l 2

(ml – 9n)2 = 4(3ln – m2)(3m – l2).

This is the equation of the required equation of the envelope, where l, m and n are as given above.

6. IMPORTANT RESULTS

I. The envelope of the two-parameter family of surfaces F(x, y, z, a, b) = 0, where a, b are parameters is found by eliminating a and b from the equations:

∂f

∂f

= 0,

= 0.

∂a

∂b

The envelope of the two-parameter family of surfaces F(x, y, z, a, b) = 0, where a, b are parameters connected by the equation φ(a, b) = 0 is found by eliminating a and b from the equations:

See All Chapters
Medium 9789380298818

14

Parmananda Gupta Laxmi Publications PDF

160

DIFFERENTIAL GEOMETRY AND CALCULUS OF VARIATIONS

∴ The extremals of the functional J[y(x)] can be found by solving Euler’s equation of the transformed functional J1[v(u)]. This is called the principle of invariance of Euler’s equation under coordinates transformations.

In solving practical problems, the change of variables is made directly in the integral representing the functional. We then write and solve the Euler’s equation for the new integral involving new variables. In the extremals so obtained, the new variables are changed to the original variables to get the desired extremals.

Example. Find the extremals of the functional

Sol. Let

J[r(θ)] =

z

z

θ2

θ1

θ2

θ1

r 2 + r ′ 2 dθ , where r = r(θ).

r 2 + r ′ 2 dθ .

Let

x = r cos θ, y = r sin θ.

r=

−1 x 2 + y2 , θ = tan

y x

rx + ry y′ dr dr/ dx

=

= r′ =

= dθ dθ/ dx θ x + θ y y′

x

2

x +y

1 y2

1+ 2 x

2

+

FG − y IJ

H xK

2

y

y′ x + y2

1

1

+ y′

2 x y

1+ 2 x

2

FG IJ

H K

x + yy′ x2 + y2

= xy′ − y x2 + y2

=

Also,

dθ =

xy′ − y dθ dx

. dx = (θx + θy y′)dx = 2 dx x + y2

r2 + r′2 = x2 + y2 +

= (x2 + y2)

See All Chapters
Medium 9789380298818

11

Parmananda Gupta Laxmi Publications PDF

4

Variational Problems with

Fixed Boundaries

1. INTRODUCTION

By a function we mean a correspondence between the elements of sets A and B such that to each element of A there correspond exactly one element of the set B. We have also studied the methods of finding maximum and minimum values of real functions of one or more variables. In calculus of variations, we shall deal with functionals and their maximum and minimum values. A functional associates a unique real number to each function belonging to a certain class of functions.

2. FUNCTIONAL

Let A be a class of functions. A correspondence between the functions of class A and the set of real numbers such that to each function belonging to A, there correspond exactly one real number is called a functional. In other words, a functional is a correspondence which assigns a unique real number to each function belonging to some class. Thus a functional is a function, where the independent variable takes functions as values.

A functional is denoted by capital letter I (or J). If y(x) represents a function in the class of functions of a functional J, then we write J = J[y(x)].

See All Chapters

See All Chapters