Euler's theorem for homogeneous functions of three variables:
Let $f(x, y, z) = \sin u = \frac{x + 2y + 3z}{x^8 + y^8 + z^8}$.
The degree of homogeneity of numerator is 1, and the denominator is 8.
Thus, the degree of $f$ is $n = 1 - 8 = -7$.
By Euler's theorem on homogeneous functions:
$x\frac{\partial f}{\partial x} + y\frac{\partial f}{\partial y} + z\frac{\partial f}{\partial z} = n f = -7 f$.
Since $f = \sin u$, we have $\frac{\partial f}{\partial x} = \cos u \frac{\partial u}{\partial x}$, and similarly for $y$ and $z$.
Therefore:
$\cos u \left(x\frac{\partial u}{\partial x} + y\frac{\partial u}{\partial y} + z\frac{\partial u}{\partial z}\right) = -7\sin u$
$\implies x\frac{\partial u}{\partial x} + y\frac{\partial u}{\partial y} + z\frac{\partial u}{\partial z} = -7\frac{\sin u}{\cos u} = -7\tan u$.
Option breakdown:
- $-7\tan u$ is the correct result.
- $7\tan u$ has the wrong sign (degree is $1 - 8 = -7$, not $+7$).
- $7\sin u$ fails to divide by $\cos u$.
- $-7\sin u$ fails to divide by $\cos u$.
समघात फलनों पर यूलर का प्रमेय:
माना $f(x, y, z) = \sin u = \frac{x + 2y + 3z}{x^8 + y^8 + z^8}$ है।
अंश की घात $1$ तथा हर की घात $8$ है। अतः फलन $f$ की समघातता की कोटि $n = 1 - 8 = -7$ है।
यूलर के प्रमेय के अनुसार:
$x\frac{\partial f}{\partial x} + y\frac{\partial f}{\partial y} + z\frac{\partial f}{\partial z} = n f = -7 f$.
चूँकि $f = \sin u$ है, अतः $\frac{\partial f}{\partial x} = \cos u \frac{\partial u}{\partial x}$ इत्यादि।
मान रखने पर:
$\cos u \left(x\frac{\partial u}{\partial x} + y\frac{\partial u}{\partial y} + z\frac{\partial u}{\partial z}\right) = -7\sin u$
$\implies x\frac{\partial u}{\partial x} + y\frac{\partial u}{\partial y} + z\frac{\partial u}{\partial z} = -7\tan u$.
विकल्पों का विश्लेषण:
- $-7\tan u$ पूर्णतः सही उत्तर है।
- चिह्न धनात्मक होने के कारण गलत है ($1 - 8 = -7$)।
- $\cos u$ से भाग न देने के कारण गलत है।
- $\tan u$ के स्थान पर $\sin u$ होने से गलत है।
