Question

Show that the variables u = x - y + z, v = x + y - z , w = x ^ 2 + xz - xy are functionally related. Find the relationship between them.​

Answer 1

Answer:

Step-by-step explanation:

UNIT 14.8 - PARTIAL DIFFERENTIATION 8

DEPENDENT AND INDEPENDENT FUNCTIONS

14.8.1 THE JACOBIAN

Suppose that

u ≡ u(x, y) and v ≡ v(x, y)

are two functions of two independent variables, x and y; then, in general, it is not possible

to express u solely in terms of v, nor v solely in terms of u.

However, on occasions, it may be possible, as the following illustrations demonstrate:

ILLUSTRATIONS

1. If

u ≡

x + y

x

and v ≡

x − y

y

,

then

u ≡ 1 +

x

y

and v ≡

x

y

− 1,

which gives

(u − 1)(v + 1) ≡

x

y

.

y

x

≡ 1.

Hence,

u ≡ 1 +

1

v + 1

and v ≡

1

u − 1

− 1.

2. If

u ≡ x + y and v ≡ x

2 + 2xy + y

2

,

then

v ≡ u

2

and u ≡ ±√

v.

If u and v are not connected by an identical relationship, they are said to be “independent

functions”.